2.1 Two defining consequences of ellipticity

We begin our description by providing two fundamental properties that lie at the heart of much of the theory for elliptic PDEs, namely energy considerations and comparison principles. We will see how these give rise to various concepts of solutions and how much of the existence and regularity theory stems from these two simple ideas.

To make matters precise, we set $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{d}$ with $d\geq 1$ to be a bounded domain with Lipschitz boundary. If further smoothness of the domain becomes necessary we will specify this at every stage. For simplicity, and because these ideas are better motivated in this case, let us consider the Laplacian which, for a function $u\in C^{2}(\unicode[STIX]{x1D6FA})$ , is defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}u=\mathop{\sum }_{i=1}^{d}\displaystyle \frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}x_{i}^{2}}.\end{eqnarray}$$

The first fundamental property that can be observed for this operator is a maximum principle.

Theorem 2.1 (maximum principle).

Let $u\in C^{2}(\unicode[STIX]{x1D6FA})\cap C(\bar{\unicode[STIX]{x1D6FA}})$ be such that $\unicode[STIX]{x1D6E5}u\geq 0$ . Then

$$\begin{eqnarray}\sup _{x\in \bar{\unicode[STIX]{x1D6FA}}}u(x)=\sup _{x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}u(x).\end{eqnarray}$$

While we will not provide a detailed proof here, we wish to provide some intuition into this fact. Namely, if we assume that a strict global maximum is attained at an interior point $z\in \unicode[STIX]{x1D6FA}$ , then elementary considerations from calculus will give us that

$$\begin{eqnarray}Du(z)=0,\quad D^{2}u(z)<0,\end{eqnarray}$$

where $Du$ denotes the gradient and $D^{2}u$ the Hessian of $u$ , respectively. Since it can be easily seen that $\unicode[STIX]{x1D6E5}u=\operatorname{tr}D^{2}u$ , a contradiction ensues.

An important consequence of this result is the following comparison principle.

This result easily follows by setting $w=u-v$ and observing that $w\leq 0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6E5}w\geq 0$ in $\unicode[STIX]{x1D6FA}$ . An application of the maximum principle then allows us to conclude the result.

The comparison principle is one of the fundamental properties of an elliptic operator, namely that an ordering on the boundary and a (reverse) ordering of the operators implies an order of the underlying functions. Throughout this survey the application of a similar principle will be a recurring feature.

Having understood comparison principles we now proceed to describe energy considerations. Consider the equation

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}u=f\end{eqnarray}$$

in $\unicode[STIX]{x1D6FA}$ and multiply it by a sufficiently smooth function $\unicode[STIX]{x1D711}$ that vanishes on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . An application of Green’s identity reveals that

(2.2) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}Du\cdot D\unicode[STIX]{x1D711}=-\int _{\unicode[STIX]{x1D6FA}}f\unicode[STIX]{x1D711},\end{eqnarray}$$

which we immediately recognize as the Euler–Lagrange equation for the minimization of the energy functional

$$\begin{eqnarray}J(v)=\int _{\unicode[STIX]{x1D6FA}}\biggl(\displaystyle \frac{1}{2}|Dv|^{2}+fv\biggr)\end{eqnarray}$$

subject to the condition $v=u$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ .

It is important to notice that, as opposed to (2.1), identity (2.2) only requires the existence of square-integrable first derivatives. Note also that the second variation of $J$ is non-negative:

(2.3) $$\begin{eqnarray}\unicode[STIX]{x2202}^{2}J(v)[w_{1},w_{2}]=\int _{\unicode[STIX]{x1D6FA}}Dw_{1}\cdot Dw_{2},\quad \unicode[STIX]{x2202}^{2}J(v)[w,w]=\int _{\unicode[STIX]{x1D6FA}}|Dw|^{2}\geq 0,\end{eqnarray}$$

which shows its positivity in a certain sense.

On the basis of this observation, we now introduce our first definition of ellipticity Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Chapter 3). In order to do so, in what follows we let $\mathbb{S}^{d}$ denote the space of symmetric $d\times d$ matrices. We endow $\mathbb{S}^{d}$ with the usual partial order

$$\begin{eqnarray}M,N\in \mathbb{S}^{d}:\quad M\leq N\quad \Longleftrightarrow \quad \unicode[STIX]{x1D743}\cdot M\unicode[STIX]{x1D743}\leq \unicode[STIX]{x1D743}\cdot N\unicode[STIX]{x1D743},\quad \text{for all}~\unicode[STIX]{x1D743}\in \mathbb{R}^{d}.\end{eqnarray}$$

We denote the identity matrix by $I\in \mathbb{S}^{d}$ .

Definition 2.3 (elliptic operator in divergence form).

Let $A:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{S}^{d}$ . We say that the operator

(2.4) $$\begin{eqnarray}Lu(x)=-D\cdot (A(x)Du(x))\end{eqnarray}$$

is elliptic at $x\in \unicode[STIX]{x1D6FA}$ if $0<\unicode[STIX]{x1D706}(x)I\leq A(x)\leq \unicode[STIX]{x1D6EC}(x)I$ , strictly elliptic if $\unicode[STIX]{x1D706}(x)\geq \unicode[STIX]{x1D706}_{0}>0$ for all $x\in \unicode[STIX]{x1D6FA}$ , and uniformly elliptic if $\unicode[STIX]{x1D6EC}(x)/\unicode[STIX]{x1D706}(x)$ is bounded in $\unicode[STIX]{x1D6FA}$ .

Example 2.4 (lower-order terms).

The concept of elliptic operators in divergence form can be extended to operators having lower-order terms. For instance, consider the operator

$$\begin{eqnarray}\tilde{L}u(x)=-D\cdot (A(x)Du(x)+\mathbf{b}(x)u(x))+\mathbf{c}(x)\cdot Du(x)+e(x)u(x),\end{eqnarray}$$

where $A$ is as in Definition 2.3, and the functions $\mathbf{b},\mathbf{c}:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}^{d}$ and $e:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ are assumed to be measurable.

Example 2.5 (quasi-linear operators).

Given a differentiable vector-valued function $\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\ni (x,z,\mathbf{p})\mapsto \mathbf{a}(x,z,\mathbf{p})\in \mathbb{R}^{d}$ and a scalar function $\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\ni (x,z,\mathbf{p})\mapsto b(x,z,\mathbf{p})\in \mathbb{R}$ , we consider the quasi-linear operator

$$\begin{eqnarray}Qu(x)=-D\cdot \mathbf{a}(x,u,Du)+b(x,u,Du)\end{eqnarray}$$

defined for $u\in C^{2}(\unicode[STIX]{x1D6FA})$ . We say that this operator is variational if it is the Euler–Lagrange operator of the energy functional

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}E(x,u,Du),\end{eqnarray}$$

that is, $\mathbf{a}(x,z,\mathbf{p})=D_{\mathbf{p}}E(x,z,\mathbf{p})$ and $b(x,z,\mathbf{p})=D_{z}E(x,z,\mathbf{p})$ . Following Definition 2.3 we realize that the ellipticity of $Q$ is equivalent to the strict convexity of $E$ with respect to the $\mathbf{p}$ variables. This immediately hints at the fact that tools from calculus of variations will be essential in the study of equations with this type of operator. Examples of quasi-linear operators can be given by choosing appropriate energies $E$ . For instance, setting

$$\begin{eqnarray}E=E(\mathbf{p})=(1+|\mathbf{p}|^{2})^{s/2}\end{eqnarray}$$

Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Chapter 10) for $s>1$ , we obtain a family of uniformly elliptic quasi-linear operators.

On the other hand, many problems cannot be cast in this form. The prototypical example is that given by the operator

(2.5) $$\begin{eqnarray}{\mathcal{L}}u(x)=A(x):D^{2}u(x),\end{eqnarray}$$

where, for $M,N\in \mathbb{S}^{d}$ , $M:N$ denotes the Fröbenius inner product:

$$\begin{eqnarray}M:N=\mathop{\sum }_{i,j=1}^{d}M_{i,j}N_{i,j}.\end{eqnarray}$$

The Fröbenius norm of a matrix $M$ will be denoted by $|M|:=\sqrt{M:M}$ .

If $A$ is sufficiently smooth, the operator (2.5) can be recast in divergence form and $-{\mathcal{L}}$ can be understood as an elliptic operator in the sense of Definition 2.3. However, this is not always possible, and consequently we must extend the notion of ellipticity to non-divergence-form operators Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Chapter 3).

The reader is encouraged to verify that, for an operator that is strictly elliptic in the sense of Definition 2.6, variants of Theorem 2.1 and Corollary 2.2 hold.

Example 2.7 (lower-order terms).

As in the divergence-form case, the notion of elliptic operators extend to those with lower-order terms, for example,

$$\begin{eqnarray}\tilde{{\mathcal{L}}}u(x)=A(x):D^{2}u(x)+\mathbf{b}(x)\cdot Du(x)+c(x)u(x),\end{eqnarray}$$

where $A$ is as in Definition 2.6, and the lower-order coefficients $\mathbf{b}$ and $c$ are (vector- and scalar-valued) functions defined on $\unicode[STIX]{x1D6FA}$ .

Example 2.8 (linear operators in divergence form).

Consider the operator $L$ of Definition 2.3 and assume that the coefficient matrix $A$ is differentiable. One can then rewrite $Lu(x)$ as

$$\begin{eqnarray}Lu(x)=-A(x):D^{2}u(x)-(D\cdot A(x))\cdot Du(x),\end{eqnarray}$$

where the divergence operator acts on $A$ column-wise. Consequently, the operator $-L$ is of the form $\tilde{{\mathcal{L}}}$ of Example 2.7.

Example 2.9 (quasi-linear operators in non-divergence form).

For a function $u\in C^{2}(\unicode[STIX]{x1D6FA})$ we define the quasi-linear operator

$$\begin{eqnarray}{\mathcal{Q}}u(x)=A(x,u,Du):D^{2}u(x)+b(x,u,Du)\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\ni (x,z,\mathbf{p})\mapsto A(x,z,\mathbf{p})\in \mathbb{S}^{d}$ and $\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\ni (x,z,\mathbf{p})\mapsto b(x,z,\mathbf{p})\in \mathbb{R}$ . We say that ${\mathcal{Q}}$ is elliptic at the function $u$ if $A(x,u,Du)$ satisfies the positivity conditions of Definition 2.6.

The previous definitions and examples entailed linear and quasi-linear operators. While, by linearization, one could extend Definitions 2.3 and 2.6 to more general nonlinear problems, we shall instead give a general definition of ellipticity, one that preserves the fundamental concept of comparison Crandall, Ishii and Lions (Reference Crandall, Ishii and Lions1992), Caffarelli and Cabré (Reference Caffarelli and Cabré1995).

Definition 2.10 (elliptic operator).

Let $F\in C(\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\times \mathbb{S}^{d})$ . We say that $F$ is elliptic in $\unicode[STIX]{x1D6FA}$ if $F$ satisfies the following monotonicity condition. If $r,s\in \mathbb{R}$ and $M,N\in \mathbb{S}^{d}$ with $r\geq s$ and $M\leq N$ then

$$\begin{eqnarray}F(x,r,\mathbf{p},M)\leq F(x,s,\mathbf{p},N).\end{eqnarray}$$

We will say, moreover, that $F$ is uniformly elliptic if there are constants $0<\unicode[STIX]{x1D706}\leq \unicode[STIX]{x1D6EC}$ such that for all $M\in \mathbb{S}^{d}$ and $r\geq s$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D706}|N|\leq F(x,r,\mathbf{p},M+N)-F(x,s,\mathbf{p},M)\leq \unicode[STIX]{x1D6EC}|N|,\quad \text{for all}~N\geq 0.\end{eqnarray}$$

Example 2.11 (linear and quasi-linear equations).

As a first example, let us show that the linear operator in non-divergence form ${\mathcal{L}}$ of (2.5) is elliptic in the sense of Definition 2.10. By doing so and following the considerations of the examples previously given, we see that all the other cases also fit into this framework. Define

$$\begin{eqnarray}F(x,r,\mathbf{p},M)=\operatorname{tr}(A(x)M).\end{eqnarray}$$

Since, for symmetric matrices, $A:B=\operatorname{tr}(AB)$ , we see that ${\mathcal{L}}u(x)=F(x,u(x),Du(x),D^{2}u(x))$ . Moreover, the positivity of $A$ implies the monotonicity of $F$ .

We now present several examples of fully nonlinear equations that fit into Definition 2.10.

Example 2.12 (Hamilton–Jacobi–Bellman operator).

Let ${\mathcal{A}}$ be any compact set and assume that for every $\unicode[STIX]{x1D6FC}\in {\mathcal{A}}$ we are given a uniformly elliptic linear operator

$$\begin{eqnarray}{\mathcal{L}}^{\unicode[STIX]{x1D6FC}}u(x)=A^{\unicode[STIX]{x1D6FC}}(x):D^{2}u(x)\end{eqnarray}$$

and a function $f^{\unicode[STIX]{x1D6FC}}\in C(\unicode[STIX]{x1D6FA})$ . Define

$$\begin{eqnarray}F(x,r,\mathbf{p},M)=\sup _{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}[\operatorname{tr}(A^{\unicode[STIX]{x1D6FC}}(x)M)-f^{\unicode[STIX]{x1D6FC}}(x)].\end{eqnarray}$$

Note immediately that

$$\begin{eqnarray}F(x,u(x),Du(x),D^{2}u(x))=\sup _{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}[{\mathcal{L}}^{\unicode[STIX]{x1D6FC}}u(x)-f^{\unicode[STIX]{x1D6FC}}(x)].\end{eqnarray}$$

Moreover, since for every $\unicode[STIX]{x1D6FC}\in {\mathcal{A}}$ , $x\in \unicode[STIX]{x1D6FA}$ and $M,N\in \mathbb{S}^{d}$ we have that $A^{\unicode[STIX]{x1D6FC}}(x)M\leq A^{\unicode[STIX]{x1D6FC}}(x)N$ whenever $M\leq N$ , we immediately conclude that the operator $F$ is monotone and thus elliptic in the sense of Definition 2.10. A similar argument shows that $F$ is uniformly elliptic whenever the family of linear operators $\{{\mathcal{L}}^{\unicode[STIX]{x1D6FC}}\}_{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}$ is uniformly elliptic. More importantly, we note that the function $F$ is convex with respect to $M$ .

Example 2.13 (Isaacs operator).

The previous example can be generalized as follows. Assume now that we have two index sets ${\mathcal{A}}$ and ${\mathcal{B}}$ and, for each $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\in {\mathcal{A}}\times {\mathcal{B}}$ , we have a uniformly elliptic linear operator

$$\begin{eqnarray}{\mathcal{L}}^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}u(x)=A^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}(x):D^{2}u(x).\end{eqnarray}$$

Define

$$\begin{eqnarray}F(x,r,\mathbf{p},M)=\inf _{\unicode[STIX]{x1D6FD}\in {\mathcal{B}}}\sup _{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}[\operatorname{tr}(A^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}(x)M)-f^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}(x)],\end{eqnarray}$$

and note that

$$\begin{eqnarray}F(x,u(x),Du(x),D^{2}u(x))=\inf _{\unicode[STIX]{x1D6FD}\in {\mathcal{B}}}\sup _{\unicode[STIX]{x1D6FC}\in {\mathcal{A}}}[{\mathcal{L}}^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}u(x)-f^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}(x)].\end{eqnarray}$$

One more time, the uniform ellipticity of the operators ${\mathcal{L}}^{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}}$ yields the uniform ellipticity of $F$ . Note that $F$ is neither convex nor concave with respect to $M$ .

Example 2.14 (Monge–Ampère operator).

As a final example, consider the operator

$$\begin{eqnarray}F(x,r,\mathbf{p},M)=\det M-f(x).\end{eqnarray}$$

Note that, in general, this operator does not satisfy the monotonicity condition of Definition 2.10. However, if we restrict it to positive definite matrices, then this operator is elliptic. Consequently, for a positive  $f$ and a strictly convex function $u\in C^{2}(\unicode[STIX]{x1D6FA})$ , we define the Monge–Ampère operator as

$$\begin{eqnarray}F(x,u(x),Du(x),D^{2}u(x))=\det D^{2}u(x)-f(x).\end{eqnarray}$$

With Definition 2.10 at hand, we turn our attention to boundary value problems for elliptic operators. In other words, for an elliptic operator $F$ we consider the following problem: find $u:\bar{\unicode[STIX]{x1D6FA}}\rightarrow \mathbb{R}$ such that

(2.6) $$\begin{eqnarray}F(x,u,Du,D^{2}u)=0\quad \text{in }\unicode[STIX]{x1D6FA},\quad u=g\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

The sense in which (2.6) is satisfied will give rise to the various existing concepts of solutions.

2.2 Classical solutions

The first, and obvious, notion of solution is when identity (2.6) is understood in a pointwise sense. This gives rise to so-called classical solutions.

An immediate consequence of ellipticity is that classical solutions are unique.

Proof. Let us prove this result under the assumption that the map $F$ is strictly decreasing in the $r$ variable. The remaining case can be found, for instance, in Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Corollary 17.2). Assume that $u$ and $v$ are classical solutions to (2.6) and set $w=u-v$ . Note that $w=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , and that if $w$ attains a (positive) maximum at $x_{0}\in \unicode[STIX]{x1D6FA}$ , then $Dw(x_{0})=0$ and $D^{2}w(x_{0})\leq 0$ . Therefore, if $u(x_{0})>v(x_{0})$ , ellipticity and the fact that the map is strictly decreasing imply

$$\begin{eqnarray}0=F(x_{0},u(x_{0}),Du(x_{0}),D^{2}u(x_{0}))<F(x_{0},v(x_{0}),Dv(x_{0}),D^{2}v(x_{0})),\end{eqnarray}$$

which is a contradiction. Similarly, the function $w$ cannot attain a negative minimum in $\unicode[STIX]{x1D6FA}$ , and consequently $w\equiv 0$ .

As an example, we mention that Theorem 2.16 holds for the operator $\tilde{{\mathcal{L}}}$ of Example 2.7 whenever the zero order coefficient $c\leq 0$ .

In the linear case of Example 2.11, the Dirichlet problem (2.6) reads as follows: find $u\in C^{2}(\unicode[STIX]{x1D6FA})\cap C(\bar{\unicode[STIX]{x1D6FA}})$ such that

(2.7) $$\begin{eqnarray}{\mathcal{L}}u=f\quad \text{in }\unicode[STIX]{x1D6FA},\quad u=g\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

In this case, the existence of classical solutions is guaranteed by what is known as Schauder estimates which, simply put, boil down to freezing the coefficients and a continuity argument; see Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Theorems 6.13–6.14).

Theorem 2.17 (existence).

Let $\unicode[STIX]{x1D6FA}$ satisfy an exterior sphere condition at every boundary point. Assume the operator (2.5) is strictly elliptic in the sense of Definition 2.6. If $g\in C(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA})$ and, for some $\unicode[STIX]{x1D6FC}\in (0,1)$ , $f$ and the coefficients of ${\mathcal{L}}$ are bounded and belong to $C^{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FA})$ , then the Dirichlet problem (2.7) has a unique classical solution $u\in C^{2,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FA})\cap C(\bar{\unicode[STIX]{x1D6FA}})$ . If, in addition, we assume that $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\in C^{2,\unicode[STIX]{x1D6FC}}$ , that $f$ and the coefficients of ${\mathcal{L}}$ belong to $C^{\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})$ ; and $g\in C^{2,\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})$ , then $u\in C^{2,\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})$ and

$$\begin{eqnarray}\Vert u\Vert _{C^{2,\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})}\leq C(\Vert f\Vert _{C^{\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})}+\Vert g\Vert _{C^{2,\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}})}),\end{eqnarray}$$

where the constant $C$ is independent of $u$ , $f$ and $g$ .

At this point, the reader may wonder if Hölder continuity is indeed necessary for these results. Example 2.23 below will show us that this is the case.

If more regularity is assumed on the domain and problem data, it can be shown that the (unique) classical solution is also more regular Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Theorem 6.19).

While Theorems 2.17 and 2.18 provide a satisfactory and conclusive answer for a linear operator with smooth coefficients, it does not cover rough coefficients or nonlinear problems, in which a classical solution may not exist. This is why we must depart from classical solutions and consider weakened or generalized concepts of solutions.

2.3 Weak (variational) solutions

We now turn our attention to the case of divergence-form operators as in Definition 2.3 and consider, for this particular operator, the Dirichlet problem (2.6). The natural solution concept in this case is called weak solution, and follows from an integration-by-parts argument, and an integral identity similar to (2.2).

Definition 2.19 (weak solutions of linear equations).

A function $u\in H^{1}(\unicode[STIX]{x1D6FA})$ is said to be a weak solution of the Dirichlet problem

(2.8) $$\begin{eqnarray}Lu=f\quad \text{in }\unicode[STIX]{x1D6FA},\quad u=g\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\end{eqnarray}$$

if $u-g\in H_{0}^{1}(\unicode[STIX]{x1D6FA})$ and, for every $\unicode[STIX]{x1D711}\in H_{0}^{1}(\unicode[STIX]{x1D6FA})$ , we have

(2.9) $$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}(ADu)\cdot D\unicode[STIX]{x1D711}=\int _{\unicode[STIX]{x1D6FA}}f\unicode[STIX]{x1D711}.\end{eqnarray}$$

Existence and uniqueness follow from the classical Lax–Milgram lemma or, more generally, from so-called inf-sup conditions.

Again, under additional smoothness assumptions on the domain and problem data, one can assert further differentiability of the solution. This is the content of the following result Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001), Grisvard (Reference Grisvard2011).

Theorem 2.21 (regularity).

Assume, in addition to the conditions of Theorem 2.20, that $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\in C^{2}$ or that $\unicode[STIX]{x1D6FA}$ is convex. If $A\in C^{0,1}(\bar{\unicode[STIX]{x1D6FA}},\mathbb{S}^{d})$ , $f\in L^{2}(\unicode[STIX]{x1D6FA})$ and $g\in H^{2}(\unicode[STIX]{x1D6FA})$ , then we have $u\in H^{2}(\unicode[STIX]{x1D6FA})\cap H^{1}(\unicode[STIX]{x1D6FA})$ and

$$\begin{eqnarray}\Vert u\Vert _{H^{2}(\unicode[STIX]{x1D6FA})}\leq C(\Vert u\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}+\Vert f\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}+\Vert g\Vert _{H^{2}(\unicode[STIX]{x1D6FA})}),\end{eqnarray}$$

where the constant $C$ is independent of $u$ , $f$ and $g$ .

We also wish to mention the remarkable result by E. De Giorgi concerning the Hölder regularity of weak solutions.

In light of the second part of the previous result, it is natural to ask whether $f\in L^{\infty }(\unicode[STIX]{x1D6FA})$ with appropriate assumptions on the boundary data $g$ and the coefficients of $L$ would yield that $Du\in C_{\text{loc}}^{1}(\unicode[STIX]{x1D6FA})$ . The following example shows that this is generally false Han and Lin (Reference Han and Lin2011, Section 3.4).

Example 2.23 (second derivatives are not continuous).

For $R<1$ let $\unicode[STIX]{x1D6FA}=\{x\in \mathbb{R}^{d}:|x|<R\}$ and consider

$$\begin{eqnarray}f(x)=\left\{\begin{array}{@{}ll@{}}0\quad & x=0,\\ \displaystyle \frac{x_{2}^{2}-x_{1}^{2}}{2|x|^{2}}\biggl(\displaystyle \frac{d+2}{\sqrt{-\ln |x|}}+\displaystyle \frac{1}{2(-\ln |x|)^{3/2}}\biggr)\quad & x\neq 0.\end{array}\right.\end{eqnarray}$$

Note that $f\in C(\bar{\unicode[STIX]{x1D6FA}})$ and that the function

$$\begin{eqnarray}u(x)=(x_{1}^{2}-x_{2}^{2})\sqrt{-\ln |x|}\in C(\bar{\unicode[STIX]{x1D6FA}})\cap C^{\infty }(\bar{\unicode[STIX]{x1D6FA}}\setminus \{0\})\end{eqnarray}$$

satisfies $\unicode[STIX]{x1D6E5}u=f$ with boundary conditions

$$\begin{eqnarray}g=\sqrt{-\ln R}(x_{1}^{2}-x_{2}^{2}).\end{eqnarray}$$

However, this function cannot be a classical solution since

$$\begin{eqnarray}\lim _{|x|\rightarrow 0}\displaystyle \frac{\unicode[STIX]{x2202}^{2}u(x)}{\unicode[STIX]{x2202}x_{1}^{2}}=\infty\end{eqnarray}$$

so that $u\not \in C^{2}(\unicode[STIX]{x1D6FA})$ . In fact, although the problem has a weak solution, it does not have a classical one. This example also shows that, in the classical solution theory given in Theorem 2.17, mere continuity of the data is not sufficient, thus justifying the need for Hölder continuity.

Let us now focus our attention on the quasi-linear operator $Q$ of Example 2.5 and consider the Dirichlet problem

(2.10) $$\begin{eqnarray}Qu=f\quad \text{in }\unicode[STIX]{x1D6FA},\quad u=g\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

The definition of weak solution is as follows.

Definition 2.24 (weak solutions of quasi-linear equations).

A function $u\in W^{1,p}(\unicode[STIX]{x1D6FA})$ ( $1<p<\infty$ ) is called a weak solution of (2.10) if $u-g\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ and

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}(\mathbf{a}(x,u,Du)\cdot Dv+b(x,u,Du)v)=\int _{\unicode[STIX]{x1D6FA}}fv\end{eqnarray}$$

for all $v\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ .

Note that the equation that defines weak solutions to (2.10) are the Euler–Lagrange equations of the functional

$$\begin{eqnarray}I(u)=\int _{\unicode[STIX]{x1D6FA}}(E(x,u,Du)-fu)\end{eqnarray}$$

over the set of functions $v\in W^{1,p}(\unicode[STIX]{x1D6FA})$ such that $u-g\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ . Consequently, the existence of weak solutions is tightly bound with the calculus of variations.

Theorem 2.25 (existence and uniqueness).

Assume that there is a $p\in (1,\infty )$ for which the function $E$ satisfies the coercivity condition: there are constants $C_{1}>0$ , $C_{2}\geq 0$ such that, for every $x\in \unicode[STIX]{x1D6FA}$ , $z\in \mathbb{R}$ and $\mathbf{p}\in \mathbb{R}^{d}$ , we have

$$\begin{eqnarray}E(x,z,\mathbf{p})\geq C_{1}|\mathbf{p}|^{p}-C_{2}.\end{eqnarray}$$

Assume, in addition, that $E$ is convex in the $\mathbf{p}$ variable. Then, for $f\in L^{p^{\prime }}(\unicode[STIX]{x1D6FA})$ , the functional $I$ has a minimizer $u\in W^{1,p}(\unicode[STIX]{x1D6FA})$ such that $u-g\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ . Finally, if $E$ does not depend on $z$ and is uniformly convex, then this minimizer is unique.

With this theorem at hand it can be readily shown that, in this setting, the (unique) minimizer $u$ of $I$ is a weak solution of (2.10) in the sense of Definition 2.24.

We can also establish, under additional assumptions on $E$ , further differentiability of minimizers. To shorten the exposition we confine ourselves to the case where $E$ is independent of $x$ and $z$ , it is coercive with $p=2$ , satisfies the growth condition

(2.11) $$\begin{eqnarray}|D_{\mathbf{p}}E(\mathbf{p})|\leq C(|\mathbf{p}|+1),\quad \text{for all}~\mathbf{p}\in \mathbb{R}^{d},\end{eqnarray}$$

and

(2.12) $$\begin{eqnarray}|D^{2}E(\mathbf{p})|\leq C,\quad \text{for all}~\mathbf{p}\in \mathbb{R}^{d}.\end{eqnarray}$$

With these additional assumptions we have the following regularity result Evans (Reference Evans2010, Theorem 8.3.1).

Theorem 2.26 (regularity).

In the setting of Theorem 2.25 assume, in addition, that $E$ depends only on $\mathbf{p}$ and satisfies (2.11) and (2.12). If $g=0$ , $f\in L^{2}(\unicode[STIX]{x1D6FA})$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\in C^{2}$ , we have that $u\in H^{2}(\unicode[STIX]{x1D6FA})$ with the estimate

$$\begin{eqnarray}\Vert u\Vert _{H^{2}(\unicode[STIX]{x1D6FA})}\leq C\Vert f\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

What is more interesting and remarkable is that the results of De Giorgi presented in Theorem 2.22 can be extended to this case as well.

Theorem 2.27 (De Giorgi II).

Let $u\in W^{1,p}(\unicode[STIX]{x1D6FA})$ be a minimizer of $I$ . If $E$ satisfies the growth and monotonicity conditions

$$\begin{eqnarray}|D_{\mathbf{p}}E(x,z,\mathbf{p})|\leq C_{1}(1+|\mathbf{p}|^{p-1}),\quad D_{\boldsymbol{ p}}E(x,z,\mathbf{p}).\mathbf{p}\geq C_{2}|\mathbf{p}|^{p}-C_{3},\end{eqnarray}$$

then there exists $\unicode[STIX]{x1D6FC}\in (0,1)$ for which $u\in C_{\text{loc}}^{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FA})$ .

Under suitable assumptions, local Hölder continuity of the gradients of the minimizers can also be established. For further regularity results for quasi-linear problems the reader is referred, for instance, to Mingione (Reference Mingione2006).

While, in this setting, we have a sufficiently rich theory, it only applies to divergence-form operators. Below, in Section 2.5, we will describe the right generalization of the notion of solutions for more general problems.

2.4 Strong solutions

We now describe a solution concept that, in a sense, lies in between classical and weak solutions, and that can also be applied to non-divergence-form operators such as (2.5) and that of Example 2.9. These solutions are called strong.

We immediately remark that every classical solution is a strong solution. Moreover, an integration-by-parts and density argument shows that a sufficiently regular weak solution (see Theorems 2.21 and 2.26) is also a strong solution. Therefore, strong solutions for the divergence-form equations (2.8) and (2.10) can be obtained from regularity considerations.

Let us now turn our attention to the non-divergence-form problem (2.7) and study the existence of strong solutions. In this case we have the following result.

Theorem 2.29 (existence).

Let $\unicode[STIX]{x1D6FA}$ be a $C^{1,1}$ domain and the coefficients of the operator $L$ belong to $C(\bar{\unicode[STIX]{x1D6FA}})$ . If $f\in L^{p}(\unicode[STIX]{x1D6FA})$ and $g\in W^{2,p}(\unicode[STIX]{x1D6FA})$  ( $1<p<\infty$ ), then the Dirichlet problem (2.7) has a unique strong solution $u\in W^{2,p}(\unicode[STIX]{x1D6FA})$ , and moreover

$$\begin{eqnarray}\Vert u\Vert _{W^{2,p}(\unicode[STIX]{x1D6FA})}\leq C(\Vert f\Vert _{L^{p}(\unicode[STIX]{x1D6FA})}+\Vert g\Vert _{W^{2,p}(\unicode[STIX]{x1D6FA})}),\end{eqnarray}$$

where $C$ is independent of $u$ , $f$ and $g$ but depends on $\Vert A\Vert _{C(\bar{\unicode[STIX]{x1D6FA}},\mathbb{S}^{d})}$ , the dimension $d$ and the exponent $p$ .

We must comment on the technique of proof for this result. First, for $A=I$ and $p=2$ , this follows from the regularity result of Theorem 2.21. An interpolation result, in conjunction with the celebrated Calderón–Zygmund decomposition technique Calderón and Zygmund (Reference Calderón and Zygmund1952), yields the result for any $p$ . Using the continuity of $A$ , the result can be extended to a general  ${\mathcal{L}}$ .

Remark 2.30 (Hölder regularity).

Let us briefly describe the results of Krylov and Safonov; see Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Section 9.8) and Theorem 2.65 below. To do so, we assume that $f\in L^{d}(\unicode[STIX]{x1D6FA})$ , $g\in C^{\unicode[STIX]{x1D6FD}}(\bar{\unicode[STIX]{x1D6FA}})$ for some $\unicode[STIX]{x1D6FD}\in (0,1)$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ satisfies a uniform exterior cone condition. Then, given $\unicode[STIX]{x1D714}\Subset \unicode[STIX]{x1D6FA}$ , there is a constant $\unicode[STIX]{x1D6FC}\in (0,1)$ such that

$$\begin{eqnarray}|u|_{C^{\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D714})}\leq C.\end{eqnarray}$$

The constants $\unicode[STIX]{x1D6FC}$ and $C$ depend, in particular, on the dimension $d$ and the ratio $\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D706}$ that defines the ellipticity of ${\mathcal{L}}$ . A natural question to ask is whether a similar estimate for the gradient $Du$ (possibly under stricter smoothness assumptions) is possible. A result by Nirenberg (see Theorem 2.57) showed that this is the case for $d=2$ . For higher dimensions, however, this turns out to be false. Safonov (Reference Safonov1987) showed that in $B_{1}\subset \mathbb{R}^{3}$ , the unit ball, for every $\unicode[STIX]{x1D6FC}\in (0,1]$ there are:

• ∙ a bounded function $v\in C^{\infty }(\mathbb{R}^{3}\setminus \{0\})$ ,

• ∙ a constant $\unicode[STIX]{x1D708}\in (0,1)$ ,

• ∙ a family $\{A_{\unicode[STIX]{x1D700}}\}_{\unicode[STIX]{x1D700}>0}\subset C^{\infty }(\bar{B}_{1},\mathbb{S}^{d})$ such that the associated non-divergence operators ${\mathcal{L}}_{\unicode[STIX]{x1D700}}$ are uniformly elliptic with $\unicode[STIX]{x1D6EC}/\unicode[STIX]{x1D706}=1/\unicode[STIX]{x1D708}^{2}$ .

With these objects at hand, he showed that the solution to the problem

$$\begin{eqnarray}{\mathcal{L}}_{\unicode[STIX]{x1D700}}u_{\unicode[STIX]{x1D700}}=0\quad \text{in }B_{1},\quad u_{\unicode[STIX]{x1D700}}=v\quad \text{on }\unicode[STIX]{x2202}B_{1},\end{eqnarray}$$

satisfies $u_{\unicode[STIX]{x1D700}}\in C^{\infty }(\bar{B}_{1})$ , $\Vert u_{\unicode[STIX]{x1D700}}\Vert _{L^{\infty }(B_{1})}=1$ but

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D700}\downarrow 0}|u_{\unicode[STIX]{x1D700}}|_{C^{\unicode[STIX]{x1D6FC}}(B_{1/2})}=\infty .\end{eqnarray}$$

From this it immediately follows that Hölder estimates on the derivatives are not possible.

Let us point out now that, in Theorem 2.29, the assumption that $A\in C(\bar{\unicode[STIX]{x1D6FA}},\mathbb{S}^{d})$ cannot generally be weakened. The following example is due to Pucci.

Example 2.31 (non-uniqueness).

Let us show, following Maugeri et al. (Reference Maugeri, Palagachev and Softova2000, Section 1.1), that for $d\geq 3$ there is a bounded measurable matrix $A$ such that problem (2.7) with $f=0$ and $g=0$ has more than one strong solution in $H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ . Let $\unicode[STIX]{x1D6FA}$ be the unit ball of $\mathbb{R}^{d}$ , and define

$$\begin{eqnarray}A(x)=I+b\displaystyle \frac{xx^{\intercal }}{|x|^{2}},\quad b=\displaystyle \frac{d-2+\unicode[STIX]{x1D706}}{1-\unicode[STIX]{x1D706}},\quad \max \{2-d/2,0\}<\unicode[STIX]{x1D706}<1.\end{eqnarray}$$

Obviously

$$\begin{eqnarray}|\unicode[STIX]{x1D743}|^{2}\leq \unicode[STIX]{x1D743}\cdot A\unicode[STIX]{x1D743}\leq (1+b)|\unicode[STIX]{x1D743}|^{2},\end{eqnarray}$$

so that $A$ is bounded and the associated operators ${\mathcal{L}}$ are uniformly elliptic. Define $u(x)=|x|^{\unicode[STIX]{x1D706}}-1$ and note that

$$\begin{eqnarray}D^{2}u(x)=\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-2)|x|^{\unicode[STIX]{x1D706}-4}xx^{\intercal }+\unicode[STIX]{x1D706}|x|^{\unicode[STIX]{x1D706}-2}I,\end{eqnarray}$$

which, since $\unicode[STIX]{x1D706}>2-d/2$ , shows that $u\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ . Moreover, due to the choice of $b$ , we have

$$\begin{eqnarray}{\mathcal{L}}u(x)=A:D^{2}u(x)=\unicode[STIX]{x1D706}|x|^{\unicode[STIX]{x1D706}-2}[(1+b)\unicode[STIX]{x1D706}+d-b-2]=0.\end{eqnarray}$$

Since this will be important in subsequent developments, we now focus on conditions weaker than continuity that allow for the existence and uniqueness of a strong solution for (2.7).

2.4.1 The Cordes condition

Since, as Example 2.31 shows, mere boundedness of the coefficients in the operator of (2.7) does not suffice to ensure uniqueness of strong solutions, here we study the so-called Cordes condition for linear operators in non-divergence form. The idea behind it and the theory that follows is to reformulate the operator in such a way that the result is ‘close’ to one in divergence form, in particular, the Poisson equation. This reformulation allows us to apply classical tools in functional analysis to study the existence, uniqueness and a priori estimates for problem (2.5).

To motivate and derive this condition, consider the following problem: given $x\in \unicode[STIX]{x1D6FA}$ , find $\unicode[STIX]{x1D6FE}(x)\in \mathbb{R}$ that minimizes the quadratic function

$$\begin{eqnarray}\unicode[STIX]{x1D70F}\mapsto |\unicode[STIX]{x1D70F}A(x)-I|^{2}.\end{eqnarray}$$

Simple arguments show that the minimum is attained at

(2.13) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(x)=\displaystyle \frac{\operatorname{tr}A(x)}{|A(x)|^{2}}\quad \text{and}\quad |\unicode[STIX]{x1D6FE}(x)A(x)-I|^{2}=d-\displaystyle \frac{(\operatorname{tr}A(x))^{2}}{|A(x)|^{2}}.\end{eqnarray}$$

In particular, this simple calculation shows that

(2.14) $$\begin{eqnarray}\displaystyle |\unicode[STIX]{x1D6FE}(x){\mathcal{L}}v(x)-\unicode[STIX]{x1D6E5}v(x)|^{2} & = & \displaystyle |(\unicode[STIX]{x1D6FE}(x)A(x)-I):D^{2}v(x)|^{2}\nonumber\\ \displaystyle & \leq & \displaystyle \biggl(d-\displaystyle \frac{\operatorname{tr}A(x)^{2}}{|A(x)|^{2}}\biggr)|D^{2}v(x)|^{2}.\end{eqnarray}$$

The Cordes condition ensures that the multiplicative constant on the right-hand side of (2.14) is less than one.

Definition 2.32 (Cordes condition).

A positive definite matrix $A\in L^{\infty }(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ satisfies the Cordes condition provided that there exists an $\unicode[STIX]{x1D716}\in (0,1]$ such that

(2.15) $$\begin{eqnarray}\displaystyle \frac{|A|^{2}}{(\operatorname{tr}A)^{2}}\leq \displaystyle \frac{1}{d-1+\unicode[STIX]{x1D716}}\quad \text{a.e.}~\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Note that the Cordes condition ensures that there exists $\unicode[STIX]{x1D6FE}>0$ such that, for all $v\in H^{2}(\unicode[STIX]{x1D6FA})$ , we have

(2.16) $$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FE}{\mathcal{L}}v-\unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}\leq \sqrt{1-\unicode[STIX]{x1D716}}\Vert D^{2}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

Remark 2.33 (the Cordes condition in spectral terms).

Since, by assumption, for $\text{a.e.}~x\in \unicode[STIX]{x1D6FA}$ we have that $A(x)\in \mathbb{S}^{d}$ and that it is positive definite, it is diagonalizable and all its eigenvalues $\{\unicode[STIX]{x1D706}_{i}\}_{i=1}^{d}=\{\unicode[STIX]{x1D706}_{i}(x)\}_{i=1}^{d}$ are positive. Using the well-known identities

$$\begin{eqnarray}|A|^{2}=\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}^{2},\quad \operatorname{tr}A=\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}\quad \text{and}\quad \biggl(\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}\biggr)^{2}\leq d\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}^{2},\end{eqnarray}$$

condition (2.15) can be recast in terms of the eigenvalues of $A$ as follows:

$$\begin{eqnarray}\displaystyle \frac{1}{d}\leq \displaystyle \frac{\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i}^{2}}{(\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D706}_{i})^{2}}\leq \displaystyle \frac{1}{d-1+\unicode[STIX]{x1D716}}\quad \text{a.e.}~\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

In other words, (2.15) is an anisotropy condition on $A$ that becomes more stringent in higher dimensions.

The considerations in Remark 2.33 show that the Cordes condition is always satisfied in two dimensions with $\unicode[STIX]{x1D716}=\inf _{x\in \unicode[STIX]{x1D6FA}}2\unicode[STIX]{x1D706}_{1}\unicode[STIX]{x1D706}_{2}/(\unicode[STIX]{x1D706}_{1}^{2}+\unicode[STIX]{x1D706}_{2}^{2})\in (0,1]$ . On the other hand, there exist symmetric positive definite matrices in three dimensions (and higher) that do not satisfy (2.15).

Example 2.34 (three dimensions).

Consider the matrix

$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}1 & 0 & c\\ 0 & 1 & b\\ c & b & 4\end{array}\right)\end{eqnarray}$$

with $b^{2}+c^{2}<4$ so that $\det (A)=4-(b^{2}+c^{2})>0$ . Sylvester’s criterion ensures that the matrix is positive definite, and a straightforward calculation shows that

$$\begin{eqnarray}\displaystyle \frac{|A|^{2}}{(\operatorname{tr}A)^{2}}=\displaystyle \frac{18+2(b^{2}+c^{2})}{6^{2}}\geq \displaystyle \frac{1}{2}.\end{eqnarray}$$

Thus $A$ does not satisfy the Cordes condition.

Example 2.35 (the example of Pucci).

As another example consider the matrix of Example 2.31 for $d=3$ . Note first of all that in this case we have

$$\begin{eqnarray}\displaystyle \frac{1}{2}<\unicode[STIX]{x1D706}<1,\quad b=\displaystyle \frac{3-2+\unicode[STIX]{x1D706}}{1-\unicode[STIX]{x1D706}}>3.\end{eqnarray}$$

Simple calculations then yield

$$\begin{eqnarray}\displaystyle |A|^{2} & = & \displaystyle d+2b+b^{2},\nonumber\\ \displaystyle (\operatorname{tr}A)^{2} & = & \displaystyle (d+b)^{2},\nonumber\end{eqnarray}$$

and therefore

$$\begin{eqnarray}|A|^{2}-\displaystyle \frac{1}{d-1}(\operatorname{tr}A)^{2}=d+2b+b^{2}-\displaystyle \frac{(d+b)^{2}}{d-1}>0\end{eqnarray}$$

Thus one concludes that $A$ does not satisfy the Cordes condition.

The Cordes condition is a key assumption to establish the well-posedness of the elliptic problem (2.5) with discontinuous coefficients. Another crucial ingredient is the Miranda–Talenti estimate, which is summarized in the next lemma.

Lemma 2.36 (Miranda–Talenti estimate).

Let $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{d}$ be a bounded convex domain. Then, for any $v\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ ,

(2.17) $$\begin{eqnarray}|v|_{H^{2}(\unicode[STIX]{x1D6FA})}\leq \Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

While this result can be understood as a regularity estimate in the spirit of Theorem 2.21, we remark that it can be obtained without appealing to this theory; we refer the reader to Maugeri et al. (Reference Maugeri, Palagachev and Softova2000, Lemma 1.2.2) for a proof. Moreover, while the aforementioned regularity results yield that, for functions in $H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ , the norm $v\mapsto \Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}$ is equivalent to the $H^{2}(\unicode[STIX]{x1D6FA})$ -norm, the important feature of estimate (2.17) is that the equivalence constant is exactly one on convex domains.

Remark 2.37 (polygonal domains).

In two dimensions the Miranda–Talenti estimate (2.17) holds for a polygonal domain $\unicode[STIX]{x1D6FA}$ without the convexity assumption. Indeed, assuming that $u\in C^{\infty }(\bar{\unicode[STIX]{x1D6FA}})$ , we have

$$\begin{eqnarray}|D^{2}u|^{2}=|\unicode[STIX]{x1D6E5}u|^{2}+2(|\unicode[STIX]{x2202}_{12}u|^{2}-\unicode[STIX]{x2202}_{11}u\unicode[STIX]{x2202}_{22}u),\end{eqnarray}$$

where we explicitly used that we are in two dimensions. In addition, integration by parts and some algebraic manipulations show that

$$\begin{eqnarray}\int _{\unicode[STIX]{x1D6FA}}(|\unicode[STIX]{x2202}_{12}u|^{2}-\unicode[STIX]{x2202}_{11}u\unicode[STIX]{x2202}_{22}u)=\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}(-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}u\unicode[STIX]{x2202}_{n}u+\unicode[STIX]{x2202}_{n\unicode[STIX]{x1D70F}}u\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}u)\end{eqnarray}$$

(see Ciarlet Reference Ciarlet2002, equation (1.2.9)), where $\unicode[STIX]{x1D749}$ is the unit tangential vector along the boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}$ is the derivative in its direction and $\unicode[STIX]{x2202}_{n}$ denotes the normal derivative. Now, if $u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ then we have that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}u=0$ , so that the second term on the right-hand side of this expression vanishes. If, in addition, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ is polygonal, this also implies that $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70F}}u=0$ , which allows us to obtain (2.17). By density, the same result holds for every $u\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ .

Identities (2.13) and (2.17) motivate the introduction of the bilinear form

$$\begin{eqnarray}a(\cdot ,\cdot ):(H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA}))^{2}\ni (v,w)\mapsto a(v,w)=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FE}{\mathcal{L}}v\unicode[STIX]{x1D6E5}w\in \mathbb{R}.\end{eqnarray}$$

The properties of $a$ are as follows.

Lemma 2.38 (properties of $a$ ).

Assume that the coefficient $A$ of the operator ${\mathcal{L}}$ satisfies the Cordes condition (2.15). If $\unicode[STIX]{x1D6FA}$ is convex, then the bilinear form $a$ is bounded and coercive on $H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ .

Proof. Since $\unicode[STIX]{x1D6FE}$ is bounded, the continuity immediately follows.

If the Cordes condition (2.15) is satisfied and $\unicode[STIX]{x1D6FA}$ is convex, then from the Miranda–Talenti estimate (2.17) and the Cauchy–Schwarz inequality we obtain

(2.18) $$\begin{eqnarray}\displaystyle a(v,v) & = & \displaystyle \Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}^{2}+\int _{\unicode[STIX]{x1D6FA}}(\unicode[STIX]{x1D6FE}{\mathcal{L}}v-\unicode[STIX]{x1D6E5}v)\unicode[STIX]{x1D6E5}v\nonumber\\ \displaystyle & \geq & \displaystyle \Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}^{2}-\sqrt{1-\unicode[STIX]{x1D716}}|v|_{H^{2}(\unicode[STIX]{x1D6FA})}\Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}\nonumber\\ \displaystyle & \geq & \displaystyle (1-\sqrt{1-\unicode[STIX]{x1D716}})\Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}^{2}.\end{eqnarray}$$

In conclusion, $a$ is coercive on $H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ .

The coercivity estimate of Lemma 2.38 allows us to show the existence and uniqueness of strong solutions under the Cordes condition.

Theorem 2.39 (existence and uniqueness).

Assume that the coefficient $A$ of the operator ${\mathcal{L}}$ satisfies the Cordes condition (2.15). If $\unicode[STIX]{x1D6FA}$ is convex, then the Dirichlet problem (2.7) with $f\in L^{2}(\unicode[STIX]{x1D6FA})$ and $g=0$ has a unique strong solution $u\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ . Moreover, we have

$$\begin{eqnarray}\Vert u\Vert _{H^{2}(\unicode[STIX]{x1D6FA})}\leq C\displaystyle \frac{\Vert \unicode[STIX]{x1D6FE}\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}}{1-\sqrt{1-\unicode[STIX]{x1D716}}}\Vert f\Vert _{L^{2}(\unicode[STIX]{x1D6FA})},\end{eqnarray}$$

where the constant $C$ is independent of $u$ and $f$ .

Proof. From Lemma 2.38 and the Lax–Milgram lemma, there exists a unique $u\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ satisfying

$$\begin{eqnarray}a(u,v)=\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FE}f\unicode[STIX]{x1D6E5}v,\quad \text{for all}~v\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA}).\end{eqnarray}$$

Since $u\in H^{2}(\unicode[STIX]{x1D6FA})$ and the Laplace operator $\unicode[STIX]{x1D6E5}:H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})\rightarrow L^{2}(\unicode[STIX]{x1D6FA})$ is surjective on convex domains, standard arguments show that $u$ satisfies ${\mathcal{L}}u=f$ almost everywhere, that is, it is a strong solution to the elliptic problem (2.7). The coercivity condition (2.18) also implies the a priori estimate

$$\begin{eqnarray}\Vert u\Vert _{H^{2}(\unicode[STIX]{x1D6FA})}\leq C\Vert \unicode[STIX]{x1D6E5}u\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}\leq C\displaystyle \frac{\Vert \unicode[STIX]{x1D6FE}\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}}{1-\sqrt{1-\unicode[STIX]{x1D716}}}\Vert f\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

Remark 2.40 (inf-sup conditions).

Since the Laplace operator is surjective from $H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})$ to $L^{2}(\unicode[STIX]{x1D6FA})$ on convex domains, the above arguments show that the inf-sup condition

$$\begin{eqnarray}\sup _{w\in L^{2}(\unicode[STIX]{x1D6FA})\setminus \{0\}}\displaystyle \frac{\int _{\unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FE}{\mathcal{L}}vw}{\Vert w\Vert _{L^{2}(\unicode[STIX]{x1D6FA})}}\geq (1-\sqrt{1-\unicode[STIX]{x1D716}})\Vert \unicode[STIX]{x1D6E5}v\Vert _{L^{2}(\unicode[STIX]{x1D6FA})},\quad \text{for all}~v\in H^{2}(\unicode[STIX]{x1D6FA})\cap H_{0}^{1}(\unicode[STIX]{x1D6FA})\end{eqnarray}$$

is satisfied. One can then appeal to the Babuška–Brezzi theorem to deduce the existence of strong solutions to (2.5). To our knowledge the use of this inf-sup condition for the numerical approximation has yet to be investigated.

Remark 2.41 (the case $p\neq 2$ ).

It is possible to show Maugeri et al. (Reference Maugeri, Palagachev and Softova2000, Theorem 1.2.3) that, if $\unicode[STIX]{x1D6FA}$ is convex and $A$ satisfies the Cordes condition, there are $1<p_{l}<2<p_{r}<\infty$ such that if $p\in (p_{l},p_{r})$ , $f\in L^{p}(\unicode[STIX]{x1D6FA})$ and $g=0$ , then problem (2.7) has a unique strong solution $u\in W^{2,p}(\unicode[STIX]{x1D6FA})\cap W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ . We also have an a priori estimate in which the constant now depends on $p$ .

Remark 2.42 (strong solutions under other conditions).

It is possible to obtain the existence and uniqueness of strong solutions for problem (2.7) under other assumptions. Let us discuss two of them.

• ∙ Assuming that $A\in W^{1,d}(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ one can write the operator $L$ in non-divergence form and extend the theory of weak solutions, described in Section 2.3, to coefficients in this class. What is remarkable is that, in Example 2.31, for every $\unicode[STIX]{x1D700}>0$ the parameter $\unicode[STIX]{x1D706}$ can be chosen so that $A\in W^{1,d-\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ , thus showing that $A\in W^{1,d-\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ is not sufficient for uniqueness. On the other hand, if $A\in W^{1,d+\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ , for some $\unicode[STIX]{x1D700}>0$ , then $A\in C^{0,\unicode[STIX]{x1D6FC}}(\bar{\unicode[STIX]{x1D6FA}},\mathbb{S}^{d})$ and, consequently, the classical Schauder theory applies (see Theorem 2.17).

• ∙ In essence, the case of uniformly continuous coefficients boils down to realizing that, locally, their oscillation in the $L^{\infty }(\unicode[STIX]{x1D6FA})$ -norm is small, and so they can be considered a constant. These ideas have been extended (see Maugeri et al. Reference Maugeri, Palagachev and Softova2000, Chapter 2) to the case of a coefficient $A\in \mathit{VMO}(\unicode[STIX]{x1D6FA},\mathbb{S}^{d})$ , thus showing that this is a sufficient condition to obtain strong solutions. Since $W^{1,d}(\unicode[STIX]{x1D6FA})$ is a proper subset of $\mathit{VMO}(\unicode[STIX]{x1D6FA})$ this result truly extends the case of Sobolev coefficients detailed above.

2.5 Viscosity solutions

At this point we wish to introduce one final notion of solution, the one that will be suited to the study of fully nonlinear equations. This is that of a viscosity solution. The reader may recall that the notion of weak solutions, introduced in Section 2.3, was based on an integration-by-parts argument (2.2) and the positivity (2.3) of the resulting operators. While this proved sufficient for linear and quasi-linear operators in divergence form, different arguments are necessary for fully nonlinear operators such as those of Examples 2.12–2.14. The fundamental property that will be used to define solutions in this case, as in Corollary 2.2, will be a comparison principle.

2.5.1 Definition and first properties

Let us begin by motivating the definition following Katzourakis (Reference Katzourakis2015). Let $F$ be an elliptic operator in the sense of Definition 2.10 and let $u\in C^{2}(\unicode[STIX]{x1D6FA})$ be a classical solution to

(2.19) $$\begin{eqnarray}F(x,u,Du,D^{2}u)=0\quad \text{in }\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Let $x_{0}\in \unicode[STIX]{x1D6FA}$ and assume that there is a smooth function $\unicode[STIX]{x1D711}\in C^{2}(\unicode[STIX]{x1D6FA})$ that can touch from above the graph of $u$ at $x_{0}$ . More precisely, we assume that

$$\begin{eqnarray}u(x)\leq \unicode[STIX]{x1D711}(x)\quad \text{for all}~x\in \unicode[STIX]{x1D6FA},\quad u(x_{0})=\unicode[STIX]{x1D711}(x_{0}).\end{eqnarray}$$

These conditions imply that the function $u-\unicode[STIX]{x1D711}$ has a local maximum at $x_{0}$ and, consequently,

$$\begin{eqnarray}D(u-\unicode[STIX]{x1D711})(x_{0})=0,\quad D^{2}(u-\unicode[STIX]{x1D711})(x_{0})\leq 0.\end{eqnarray}$$

Since the operator $F$ is assumed to be elliptic we obtain

$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle F(x_{0},u(x_{0}),Du(x_{0}),D^{2}u(x_{0}))=F(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}),D^{2}u(x_{0}))\nonumber\\ \displaystyle & \leq & \displaystyle F(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}),D^{2}\unicode[STIX]{x1D711}(x_{0})).\nonumber\end{eqnarray}$$

Similar considerations will give us that if $\unicode[STIX]{x1D713}\in C^{2}(\unicode[STIX]{x1D6FA})$ touches from below the graph of $u$ at $x_{0}$ , we would obtain

$$\begin{eqnarray}F(x_{0},\unicode[STIX]{x1D713}(x_{0}),D\unicode[STIX]{x1D713}(x_{0}),D^{2}\unicode[STIX]{x1D713}(x_{0}))\leq 0.\end{eqnarray}$$

Finally we note that it is possible to reach the same conclusions if we replace the equality in (2.19) with a corresponding inequality. These considerations motivate the following definition.

Definition 2.43 (viscosity solution).

Let $F$ be elliptic in the sense of Definition 2.10. We say that the function $u\in C(\unicode[STIX]{x1D6FA})$ is:

1. (a) a viscosity subsolution of (2.19) if, whenever $x_{0}\in \unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x1D711}\in C^{2}(\unicode[STIX]{x1D6FA})$ and $u-\unicode[STIX]{x1D711}$ has a local maximum at $x_{0}$ we have that

   $$\begin{eqnarray}F(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}),D^{2}\unicode[STIX]{x1D711}(x_{0}))\geq 0,\end{eqnarray}$$

2. (b) a viscosity supersolution of (2.19) if, whenever $x_{0}\in \unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x1D711}\in C^{2}(\unicode[STIX]{x1D6FA})$ and $u-\unicode[STIX]{x1D711}$ has a local minimum at $x_{0}$ we have that

   $$\begin{eqnarray}F(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}),D^{2}\unicode[STIX]{x1D711}(x_{0}))\leq 0,\end{eqnarray}$$

3. (c) a viscosity solution if it is a sub- and supersolution.

Remark 2.44 (viscosity solutions).

Several remarks must immediately be made about Definition 2.43.

• ∙ While the motivation provided assumed that the function $u$ is smooth, the definition only requires its continuity.

• ∙ By approximation and continuity of $F$ , it is sufficient to verify the condition for quadratic polynomials $\unicode[STIX]{x1D711}\in \mathbb{P}_{2}$ ; see Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Proposition 2.4).

• ∙ If $u\in C^{2}(\unicode[STIX]{x1D6FA})$ is a classical solution then it is a viscosity solution. This follows from the ellipticity of $F$ . Moreover, sufficiently smooth viscosity solutions are also classical; see Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Lemma 2.5) and Katzourakis (Reference Katzourakis2015, Theorem 2.11).

• ∙ This definition talks only about solutions to equation (2.19), not the boundary value problem (2.6). More details on this issue will be provided below; see Definition 2.48 and Section 2.5.3.

• ∙ The definition assumes that the candidate solution can be touched from above (below). At points where this is not possible there is nothing to verify, and the function automatically satisfies the equation at these points.

• ∙ For the divergence-form operators $L$ and $\tilde{L}$ of Definition 2.3 and Example 2.4, respectively, it is known Ishii (Reference Ishii1995) that the concepts of weak solution, in the sense of Definition 2.19, and viscosity solutions coincide.

• ∙ We will not provide a historical account of the origin and development of this definition. The interested reader can consult the classical reference by Crandall et al. (Reference Crandall, Ishii and Lions1992).

A remarkable property of viscosity solutions is their stability, which is detailed in the following two results. For a proof of the first one we refer to Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Proposition 2.8), Katzourakis (Reference Katzourakis2015, Theorem 3.2) or Crandall et al. (Reference Crandall, Ishii and Lions1992, Section 6). For the second one, we refer to Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Proposition 2.7) or Katzourakis (Reference Katzourakis2015, Theorem 3.12).

Theorem 2.45 (limits and viscosity solutions).

Let $\{F_{k}\}_{k\in \mathbb{N}}$ be a sequence of uniformly elliptic operators in the sense of Definition 2.10, and let $\{u_{k}\}_{k\in \mathbb{N}}\subset C(\unicode[STIX]{x1D6FA})$ , for each $k$ , be viscosity subsolutions to the equations

$$\begin{eqnarray}F_{k}(x,u_{k},Du_{k},D^{2}u_{k})=0.\end{eqnarray}$$

If, as $k\rightarrow \infty$ , $F_{k}\rightarrow F$ uniformly on compact subsets of $\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\times \mathbb{S}^{d}$ and $u_{k}\rightarrow u$ uniformly in compact subsets of $\unicode[STIX]{x1D6FA}$ , then $u$ is a viscosity subsolution of

$$\begin{eqnarray}F(x,u,Du,D^{2}u)=0.\end{eqnarray}$$

Theorem 2.46 (suprema of subsolutions).

Let ${\mathcal{U}}\subset C(\unicode[STIX]{x1D6FA})$ be a set of viscosity subsolutions of (2.19). For $x\in \unicode[STIX]{x1D6FA}$ define

$$\begin{eqnarray}\bar{u}(x)=\sup \{u(x):u\in {\mathcal{U}}\}.\end{eqnarray}$$

Suppose that $\bar{u}\in C(\unicode[STIX]{x1D6FA})$ and is bounded. Then $\bar{u}$ is a viscosity subsolution of (2.19).

2.5.2 Existence and uniqueness

Let us now turn our attention to the existence and uniqueness of viscosity solutions to (2.6). To do so we must specify in which sense the boundary conditions are being understood. We begin by introducing the notion of semi-continuity.

Definition 2.47 (semi-continuity).

We say the function $u\in \mathit{LSC}(\unicode[STIX]{x1D6FA})$ (is lower semi-continuous) if, for all $x\in \unicode[STIX]{x1D6FA}$ ,

$$\begin{eqnarray}u(x)\leq \liminf _{y\rightarrow x}u(y).\end{eqnarray}$$

On the other hand, we say $u\in \mathit{USC}(\unicode[STIX]{x1D6FA})$ (is upper semi-continuous) if $-u\in \mathit{LSC}(\unicode[STIX]{x1D6FA})$ .

Note that, in Definition 2.43 and the discussion that followed, nothing would have changed if we had only required subsolutions and supersolutions to be upper and lower semi-continuous, respectively. With this definition at hand, we may define viscosity solutions to the Dirichlet problem.

Definition 2.48 (solution to the Dirichlet problem).

Let $F$ be elliptic in the sense of Definition 2.10 and $g\in C(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA})$ .

1. (a) The function $u_{\star }\in \mathit{USC}(\bar{\unicode[STIX]{x1D6FA}})$ is a viscosity subsolution to (2.6) if it is a viscosity subsolution to the equation (2.19) and $u_{\star }(x)\leq g(x)$ for all $x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ .

2. (b) The function $u^{\star }\in \mathit{LSC}(\bar{\unicode[STIX]{x1D6FA}})$ is a viscosity supersolution to (2.6) if it is a viscosity supersolution to (2.19) and $u^{\star }(x)\geq g(x)$ for all $x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ .

3. (c) The function $u\in C(\bar{\unicode[STIX]{x1D6FA}})$ is a viscosity solution to (2.6) if it is a sub- and supersolution.

Note that this definition requires the boundary values to be attained in the classical sense. Different boundary conditions might require a different interpretation, and we will briefly comment on this below.

We now turn our attention to the existence of solutions and the so-called Perron’s method. Simply put, this method provides existence under the assumption that the problem cannot have more than one solution. While uniqueness of classical solutions is immediate, as shown in Theorem 2.16, in this more general setting we need one additional condition.

Definition 2.49 (comparison).

We say that the Dirichlet problem (2.6) satisfies a comparison principle if, whenever $u_{\star }\in \mathit{USC}(\unicode[STIX]{x1D6FA})$ and $u^{\star }\in \mathit{LSC}(\unicode[STIX]{x1D6FA})$ are sub- and supersolutions, respectively, we have

$$\begin{eqnarray}u_{\star }\leq u^{\star }\quad \text{in }\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Note that from Definition 2.49, it immediately follows that (2.6) cannot have more than one solution. Indeed, if $u$ and $v$ are two viscosity solutions, then, in particular, $u$ is a subsolution and $v$ a supersolution; consequently, $u\leq v$ . An analogous reasoning yields the reverse inequality.

With these two conditions at hand, we proceed to show existence of solutions. For a proof, we refer the reader, for instance, to Crandall et al. (Reference Crandall, Ishii and Lions1992, Theorem 4.1) and Katzourakis (Reference Katzourakis2015, Theorem 5.3).

Theorem 2.50 (Perron’s method).

Let $F$ be elliptic in the sense of Definition 2.10 and $g\in C(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA})$ . Assume that the Dirichlet problem (2.6) satisfies a comparison principle in the sense of Definition 2.49. If there exist a subsolution $u_{\star }$ and a supersolution $u^{\star }$ to (2.6) that satisfy the boundary condition, then

$$\begin{eqnarray}u(x)=\sup \{v(x):u_{\star }\leq v\leq u^{\star }~\text{and }v\text{ is a subsolution}\}\end{eqnarray}$$

defines a viscosity solution to (2.6).

Note that, while Theorem 2.50 provides a somewhat explicit construction of the unique solution to (2.6), one still needs to verify the existence of sub- and supersolutions that satisfy the boundary condition in a classical sense. This must be done on a case-by-case basis, and we refer the reader to Crandall et al. (Reference Crandall, Ishii and Lions1992, Example 4.6) and Katzourakis (Reference Katzourakis2015, Application 5.9) for two examples where these are constructed.

It remains to understand which operators satisfy the comparison principle of Definition 2.49. Loosely speaking, similar ideas to those presented in Theorem 2.16 should yield uniqueness of viscosity solutions. However, the arguments presented there cannot be applied directly since we are dealing with functions that are merely continuous, and additional structural conditions must be imposed. This is due to the subtle fact – which may have escaped the reader – that Definition 2.10 is too general. By this we mean that, for instance, first-order and parabolic equations fit into this definition. For this reason, many authors say that an operator is degenerate elliptic if it only satisfies the monotonicity condition with respect to the $M$ variable. This is in contrast with uniform ellipticity, which precludes these two degenerate cases.

For the sake of illustration, let us concentrate our efforts on finding a comparison principle for uniformly elliptic equations. We begin by showing, following Kawohl and Kutev (Reference Kawohl and Kutev2007, Example 1), that uniform ellipticity is not enough to ensure a comparison principle.

Example 2.51 (lack of comparison).

Consider the Dirichlet problem

$$\begin{eqnarray}u^{\prime \prime }+18x(u^{\prime })^{4}=0\quad \text{in }(-1,1),\quad u(-1)=-b,~u(1)=b,\end{eqnarray}$$

with $b>1$ . Clearly, the equation is uniformly elliptic. It is easy to check that the functions

$$\begin{eqnarray}u_{\star }(x)=\left\{\begin{array}{@{}ll@{}}\sqrt[3]{x}-1+b\quad & x\in [0,1],\\ \sqrt[3]{x}+1-b\quad & x\in [-1,0),\end{array}\right.\quad u^{\star }(x)=\left\{\begin{array}{@{}ll@{}}\sqrt[3]{x}-1+b\quad & x\in (0,1],\\ \sqrt[3]{x}+1-b\quad & x\in [-1,0],\end{array}\right.\end{eqnarray}$$

which differ only at the origin, are viscosity sub- and supersolutions, respectively, and that they satisfy the boundary values. Note, however, that

$$\begin{eqnarray}\max _{x\in (-1,1)}\{u_{\star }(x)-u^{\star }(x)\}=u_{\star }(0)-u^{\star }(0)=2b-2>0.\end{eqnarray}$$

While, to our knowledge, necessary and sufficient conditions for the existence of a comparison principle for a general elliptic operator are not known, there are several sufficient conditions. We collect these in the following result.

Theorem 2.52 (existence of comparison principle).

If the Dirichlet problem (2.6) satisfies any of the structural conditions given below, then it satisfies a comparison principle in the sense of Definition 2.49.

1. (a) Katzourakis (Reference Katzourakis2015, Theorem 6.1) The dependence with respect to $x$ is decoupled, that is, the equation reads

   $$\begin{eqnarray}F(u,Du,D^{2}u)=f,\end{eqnarray}$$
   with $F\in C(\mathbb{R},\mathbb{R}^{d},\mathbb{S}^{d})$ , $f\in C(\bar{\unicode[STIX]{x1D6FA}})$ . The operator $F$ is elliptic, and satisfies, for some $\unicode[STIX]{x1D6FE}>0$ ,
   $$\begin{eqnarray}F(r,\mathbf{p},M)\geq F(s,\mathbf{p},M)+\unicode[STIX]{x1D6FE}(s-r),\quad \text{for all}~r\leq s.\end{eqnarray}$$

2. (b) The operator $F$ is elliptic, independent of the $r$ and $\mathbf{p}$ variables, and satisfies, for some $\unicode[STIX]{x1D706}>0$ ,

   $$\begin{eqnarray}F(x,M+tI)\geq F(x,M)+\unicode[STIX]{x1D706}t,\quad \text{for all}~t\geq 0.\end{eqnarray}$$

3. (c) Trudinger (Reference Trudinger1988) The operator $F$ is uniformly elliptic, Lipschitz-continuous in $\mathbf{p}$ and the following continuity assumption holds:

   $$\begin{eqnarray}|F(x,r,\mathbf{p},M)-F(y,r,\mathbf{p},M)|\leq \unicode[STIX]{x1D707}_{2}|x-y|^{1/2}|M|+\unicode[STIX]{x1D714}(|x-y|),\end{eqnarray}$$
   for all $x,y\in \unicode[STIX]{x1D6FA}$ , $r$ and $\mathbf{p}$ in a suitable ball and $\unicode[STIX]{x1D714}(a)\rightarrow 0$ as $a\downarrow 0$ . Additionally, one must assume that sub- and supersolutions belong to $C^{0,1}(\unicode[STIX]{x1D6FA})$ .

4. (d) Koike (Reference Koike2004) The dependence with respect to the $r$ variable is decoupled, that is, the equation reads

   $$\begin{eqnarray}\unicode[STIX]{x1D708}u+F(x,Du,D^{2}u)=0,\end{eqnarray}$$
   with either $\unicode[STIX]{x1D708}<0$ and $F$ elliptic, or $\unicode[STIX]{x1D708}\leq 0$ , $F$ uniformly elliptic and Lipschitz in the $\mathbf{p}$ variable, for $\mathbf{p}\in \mathbb{R}^{d}$ .

5. (e) Silvestre (Reference Silvestre2015) The operator $F$ is independent of $x$ and $\mathbf{p}$ and is strictly decreasing in $r$ , that is, whenever $r>s$ ,

   $$\begin{eqnarray}F(r,M)<F(s,M),\quad \text{for all}~M\in \mathbb{S}^{d}.\end{eqnarray}$$

Other conditions can be found in the literature.

2.5.3 Other boundary conditions

So far, for all notions of solutions, we have only discussed the Dirichlet problem (see the second equation in (2.6)). Moreover, for viscosity solutions we have assumed that the boundary conditions are attained in a classical sense. Let us here consider other types of boundary conditions as well as generalized notions for them. Consider

(2.20) $$\begin{eqnarray}F(x,u,Du,D^{2}u)=0\quad \text{in }\unicode[STIX]{x1D6FA},\quad B(x,u,Du)=0\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where, as before, the map $F$ is elliptic, but its domain of definition on the $x$ variable is now $\bar{\unicode[STIX]{x1D6FA}}$ . The function $B:\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}$ is assumed to be non-increasing in its second argument, that is,

$$\begin{eqnarray}r\geq s\Rightarrow B(x,r,\mathbf{p})\leq B(x,s,\mathbf{p}),\quad \text{for all}~x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},~\mathbf{p}\in \mathbb{R}^{d}.\end{eqnarray}$$

The Dirichlet problem obviously falls into this description with $B(x,r,\mathbf{p})=g(x)-r$ , but others are also admissible. For instance, let $\mathbf{n}(x)$ denote the outer normal to $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ at $x$ and let $\unicode[STIX]{x1D742}:\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}^{d}$ be such that, for all $x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , we have $\unicode[STIX]{x1D742}(x)\cdot \mathbf{n}(x)>0$ . The boundary condition

$$\begin{eqnarray}B(x,\mathbf{p})=\unicode[STIX]{x1D742}(x)\cdot \mathbf{p}-g(x)\end{eqnarray}$$

gives rise to the so-called oblique derivative problem; if $\unicode[STIX]{x1D742}=\mathbf{n}$ , this is the Neumann problem. A nonlinear example is the capillarity condition

$$\begin{eqnarray}B(x,r,\mathbf{p})=\mathbf{n}\cdot \mathbf{p}-g(x,r)\sqrt{1+|\mathbf{p}|^{2}}.\end{eqnarray}$$

At the beginning of Section 2.5.1 the introduction of viscosity solution was motivated by the assumption that the function $u-\unicode[STIX]{x1D711}$ had a local maximum (minimum) at $x_{0}\in \unicode[STIX]{x1D6FA}$ . When dealing with boundary conditions, we must now allow for $x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . At these points the relations that led to the definition of viscosity solution no longer hold, and a modification is necessary. It turns out that the correct notion is as follows.

Definition 2.53 (viscosity subsolution).

With the functions $F$ and $B$ as above, we say that $u\in \mathit{USC}(\bar{\unicode[STIX]{x1D6FA}})$ is a viscosity subsolution to (2.20) if it is a viscosity subsolution to (2.19) and, whenever there is a $\unicode[STIX]{x1D711}\in C^{2}(\mathbb{R}^{d})$ that touches the graph of $u$ from above at $x_{0}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , then either

$$\begin{eqnarray}B(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}))\geq 0\quad \text{or}\quad F(x_{0},\unicode[STIX]{x1D711}(x_{0}),D\unicode[STIX]{x1D711}(x_{0}),D^{2}\unicode[STIX]{x1D711}(x_{0}))\geq 0.\end{eqnarray}$$

In an analogous manner we can consider supersolutions and, as before, a solution to (2.20) is a function $u\in C(\bar{\unicode[STIX]{x1D6FA}})$ that is both a sub- and supersolution.

It is important to realize that boundary conditions in the viscosity sense, in general, are not equivalent to those in the classical sense. The reason for this, once more, is that Definition 2.10 is rather general and, for instance, allows us to consider first-order equations for which Dirichlet conditions cannot be imposed on the whole boundary. It is therefore natural to ask when a boundary condition in the viscosity sense is attained classically. Let us briefly elaborate on this issue for the Dirichlet problem (2.6). We begin with the definition of a barrier.

Definition 2.54 (barrier).

We say that (2.6) has barriers at $x_{0}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ if there exist two continuous functions $\bar{u}$ , $\text{}\underline{u}$ that are super- and subsolutions to (2.6), respectively, and that satisfy $\bar{u}(x_{0})=\text{}\underline{u}(x_{0})=g(x_{0})$ .

Proposition 2.55 (viscosity versus classical).

Let $u$ be a viscosity solution to (2.6) in the sense of Definition 2.53. If barriers exist at $x_{0}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , then $u(x_{0})=g(x_{0})$ .

In other words, classical and viscosity conditions coincide at points where it is possible to construct a barrier. We conclude this discussion by providing a sufficient condition for the existence of barriers.

Proposition 2.56 (existence of barriers).

Let $\unicode[STIX]{x1D6FA}$ be such that it has a tangent ball from outside at every point of $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . If $F$ is uniformly elliptic, Lipschitz with respect to all its variables and, for every $x\in \bar{\unicode[STIX]{x1D6FA}}$ , we have $F(x,0,\mathbf{0},0)=0$ , then barriers exist at every point $x_{0}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ .

2.5.4 Regularity

To finalize the presentation on viscosity solutions, we elaborate on their regularity. This is important not only because these results will serve as a guide to establishing rates of convergence for numerical schemes, but also many of the ideas and techniques that we present here have a discrete analogue that will be detailed in subsequent sections.

We begin with a result by Nirenberg (Reference Nirenberg1953) that shows that in two dimensions, essentially, all solutions to elliptic equations are locally  $C^{2,\unicode[STIX]{x1D6FC}}$ .

Theorem 2.57 (regularity in two dimensions).

Let $d=2$ . Assume that $F$ is uniformly elliptic in the sense of Definition 2.10 and that it has bounded first derivatives with respect to all its arguments. If $u$ is a solution to (2.19), then for every $\unicode[STIX]{x1D714}\Subset \unicode[STIX]{x1D6FA}$ there are $C>0$ , $\unicode[STIX]{x1D6FC}\in (0,1)$ that depend only on the ellipticity constants of $F$ , the bounds on its first derivatives and the distance between $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , for which

$$\begin{eqnarray}\Vert u\Vert _{C^{2,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D714})}\leq C\Vert u\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}.\end{eqnarray}$$

It is remarkable that this result was obtained long before the development of the theory of viscosity solutions.

To obtain global regularity or results in more dimensions we begin by introducing several notions of a more or less geometric nature. Recall that a function $u:\unicode[STIX]{x1D6FA}\rightarrow \mathbb{R}$ is convex if

$$\begin{eqnarray}u(\unicode[STIX]{x1D6FC}x+(1-\unicode[STIX]{x1D6FC})y)\leq \unicode[STIX]{x1D6FC}u(x)+(1-\unicode[STIX]{x1D6FC})u(y),\quad \text{for all}~x,y\in \unicode[STIX]{x1D6FA},~\unicode[STIX]{x1D6FC}\in [0,1].\end{eqnarray}$$

For a convex function we define its subdifferential as follows.

Definition 2.58 (subdifferential).

Let $u\in C(\unicode[STIX]{x1D6FA})$ . The subdifferential of $u$ at the point $x\in \unicode[STIX]{x1D6FA}$ is

$$\begin{eqnarray}\unicode[STIX]{x2202}u(x)=\{\mathbf{p}\in \mathbb{R}^{d}:u(y)-u(x)\geq \mathbf{p}\cdot (y-x)~\forall y\in \unicode[STIX]{x1D6FA}\}.\end{eqnarray}$$

It is well known Ekeland and Témam (Reference Ekeland and Témam1999) that if $u$ is convex then $\unicode[STIX]{x2202}u(x)\neq \emptyset$ , and that if $u$ is differentiable at $x$ then $\unicode[STIX]{x2202}u(x)=\{Du(x)\}$ . Given a function $u$ , we can always construct the largest convex function lying below $u$ . This gives rise to the convex envelope. In what follows we will only need this concept for the negative part of a function, so we define the convex envelope in this restricted setting.

Definition 2.59 (convex envelope and contact set).

Let $B_{r}$ be a ball such that $\unicode[STIX]{x1D6FA}\subset B_{r}$ , and let $v\in C(\unicode[STIX]{x1D6FA})$ with $v\geq 0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . Extend $v^{-}$ by zero to $B_{r}\setminus \unicode[STIX]{x1D6FA}$ . The convex envelope of $v$ is defined, for $x\in B_{r}$ , by

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}(v)(x)=\sup \{L(x):L(z)\leq -v^{-}(z)~\forall z\in B_{r},~L\in \mathbb{P}_{1}\}.\end{eqnarray}$$

The points at which these two functions coincide are called contact points:

$$\begin{eqnarray}{\mathcal{C}}^{-}(v)=\{x\in B_{r}:v(x)=\unicode[STIX]{x1D6E4}(v)(x)\}.\end{eqnarray}$$

Figure 2.1.

Convex envelope (dotted red) and contact set (dashed blue).

An illustration of the convex envelope of a function and its contact set is given in Figure 2.1. From the figure it is intuitively clear that, for fixed values of $v$ on the boundary, how deep the graph of $v$ can go depends only on the values of $v$ at ${\mathcal{C}}^{-}(v)$ . The formalization of this observation is the so-called Alexandrov estimate.

Theorem 2.60 (Alexandrov estimate).

Let $v\in C(\bar{B}_{r})$ with $v\geq 0$ on $\unicode[STIX]{x2202}B_{r}$ . If $\unicode[STIX]{x1D6E4}(v)\in C^{1,1}(B_{r})$ , then

$$\begin{eqnarray}\sup _{B_{r}}v^{-}\leq Cr|\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))|^{1/d}.\end{eqnarray}$$

In other words, there is a set $A\subset B_{r}$ that satisfies $|B_{r}\setminus A|=0$ and for which we have

$$\begin{eqnarray}\sup _{B_{r}}v^{-}\leq Cr\biggl(\int _{A\cap {\mathcal{C}}^{-}(v)}\det D^{2}\unicode[STIX]{x1D6E4}(v)\biggr)^{1/d},\end{eqnarray}$$

where the constant $C$ depends only on $d$ .

Proof. For completeness, we sketch the proof for $v\in C^{2}(B_{r})$ , since in this case $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)$ is single-valued on ${\mathcal{C}}^{-}(v)$ .

Let $M=\sup _{B_{r}}v^{-}/2r$ and assume that $B_{M}\subset \unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))$ . If that is the case,

$$\begin{eqnarray}M^{d}\leq C|\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))|,\end{eqnarray}$$

for a constant that depends only on the dimension $d$ . This shows the first estimate. On the other hand, a simple change of variables yields

$$\begin{eqnarray}|\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))|=\int _{\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))}=\int _{{\mathcal{C}}^{-}(v)}\det D^{2}\unicode[STIX]{x1D6E4}(v),\end{eqnarray}$$

so that the second statement follows from the first one.

We now show the inclusion $B_{M}\subset \unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))$ . Let $z\in B_{r}$ be a point where $\sup _{B_{r}}v^{-}$ is attained. For $\mathbf{a}\in B_{M}$ , define the affine function

$$\begin{eqnarray}L(x)=-\sup _{B_{r}}v^{-}+\mathbf{a}\cdot (x-z)\end{eqnarray}$$

and note that $L(z)=-\sup _{B_{r}}v^{-}$ and, for all $x\in B_{r}$ ,

$$\begin{eqnarray}L(x)\leq -\sup _{B_{r}}v^{-}+|\mathbf{a}|\,|x-z|<-\sup _{B_{r}}v^{-}+2Mr=0.\end{eqnarray}$$

Since $Dv(z)=0$ there is a $x_{1}\in B_{r}$ such that $v(x_{1})<L(x_{1})<0$ . In addition, we have that $v(x)\geq 0>L(x)$ for $x\in \unicode[STIX]{x2202}B_{r}$ . This shows that, if $\bar{x}\in B_{r}$ is a point where $v-L$ attains its minimum, then $v(\bar{x})<L(\bar{x})\leq 0$ and $Dv(\bar{x})=DL(\bar{x})=\mathbf{a}$ .

Define $\tilde{L}(x)=L(x)+v(\bar{x})-L(\bar{x})$ and note that $v(\bar{x})=\tilde{L}(\bar{x})$ , $Dv(\bar{x})=D\tilde{L}(\bar{x})=\mathbf{a}$ and, for every $x\in B_{r}$ $v(x)\geq \tilde{L}(x)$ . In other words, $\tilde{L}$ is a supporting hyperplane for $v$ . This shows that $\bar{x}\in {\mathcal{C}}^{-}(v)$ and that $\mathbf{a}\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)(\bar{x})$ , that is, $B_{M}\subset \unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}(v)({\mathcal{C}}^{-}(v))$ .

Note that in Theorem 2.60 only the contact set is relevant. This is due to the fact that if for $x_{0}\in B_{r}$ we have $\unicode[STIX]{x1D6E4}(v)(x_{0})<v(x_{0})$ , then locally $\unicode[STIX]{x1D6E4}(v)$ is affine, and thus $D^{2}\unicode[STIX]{x1D6E4}(v)=0$ .

With this estimate at hand we proceed to obtain the fundamental a priori estimate for viscosity solutions, the so-called Alexandrov–Bakelman–Pucci estimate. We begin by providing some motivation for this result. To do so, assume that $u\in C^{2}(\bar{B}_{r})$ with $u\geq 0$ on $\unicode[STIX]{x2202}B_{r}$ satisfies ${\mathcal{L}}u\leq f$ in $B_{r}$ . In this setting we have that, for $x\in {\mathcal{C}}^{-}(u)$ , $D^{2}u(x)\geq 0$ , and consequently $A(x):D^{2}u(x)\geq 0$ as well. This, in particular, implies that $f(x)\geq 0$ . Let $D=\inf _{x\in \bar{B}_{r}}\det A(x)>0$ and observe that

$$\begin{eqnarray}\det A(x)D^{2}u(x)=\det A(x)\det D^{2}u(x)\geq D\det D^{2}u(x)\geq 0.\end{eqnarray}$$

Let $\unicode[STIX]{x1D70E}(A(x)D^{2}u(x))=\{\unicode[STIX]{x1D707}_{i}(x)\}_{i=1}^{d}$ . Then an application of the arithmetic–geometric inequality reveals that

$$\begin{eqnarray}\displaystyle & & \displaystyle \det A(x)D^{2}u(x)\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\prod }_{i=1}^{d}\unicode[STIX]{x1D707}_{i}(x)=\biggl(\mathop{\prod }_{i=1}^{d}\unicode[STIX]{x1D707}_{i}(x)^{1/d}\biggr)^{d}\leq \biggl(\displaystyle \frac{1}{d}\mathop{\sum }_{i=1}^{d}\unicode[STIX]{x1D707}_{i}(x)\biggr)^{d}\nonumber\\ \displaystyle & & \displaystyle \qquad =\biggl(\displaystyle \frac{1}{d}\operatorname{tr}A(x)D^{2}u(x)\biggr)^{d}=\biggl(\displaystyle \frac{1}{d}A(x):D^{2}u(x)\biggr)^{d}\leq \biggl(\displaystyle \frac{1}{d}f(x)\biggr)^{d},\nonumber\end{eqnarray}$$

where in the last step we used that ${\mathcal{L}}u\leq f$ . Theorem 2.60 then yields that

$$\begin{eqnarray}\sup _{B_{r}}u^{-}\leq Cr\biggl(\int _{{\mathcal{C}}^{-}(u)}(f^{+})^{d}\biggr)^{1/d},\end{eqnarray}$$

for a constant that depends only on $d$ and $D$ .

While the considerations presented assumed that we were working with a linear equation, we essentially used that the matrix was uniformly positive definite and bounded, that is, the operator ${\mathcal{L}}$ is elliptic. A similar conclusion can be drawn from the fact that an operator is elliptic in the sense of Definition 2.10. We begin by observing Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Lemma 2.2) that $F$ is uniformly elliptic if and only if

$$\begin{eqnarray}F(x,r,\mathbf{p},M+N)-F(x,r,\mathbf{p},M)\leq \unicode[STIX]{x1D6EC}|N^{+}|-\unicode[STIX]{x1D706}|N^{-}|,\end{eqnarray}$$

where $N=N^{+}-N^{-}$ with $N^{+},N^{-}\geq 0$ and $N^{+}N^{-}=0$ . Now, if $u$ is a sufficiently smooth subsolution of (2.19), from the observation above we have

$$\begin{eqnarray}\displaystyle f(x) & := & \displaystyle -F(x,0,\mathbf{0},0)\leq F(x,u(x),Du(x),D^{2}u(x))-F(x,0,\mathbf{0},0)\nonumber\\ \displaystyle & \leq & \displaystyle \unicode[STIX]{x1D6EC}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)>0}\unicode[STIX]{x1D706}_{i}(u)+\unicode[STIX]{x1D706}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)<0}\unicode[STIX]{x1D706}_{i}(u),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}(D^{2}u(x))=\{\unicode[STIX]{x1D706}_{i}(u)\}_{i=1}^{d}$ . Similarly, for a supersolution we have

$$\begin{eqnarray}f(x)\geq \unicode[STIX]{x1D6EC}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)<0}\unicode[STIX]{x1D706}_{i}(u)+\unicode[STIX]{x1D706}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)>0}\unicode[STIX]{x1D706}_{i}(u).\end{eqnarray}$$

This motivates the following definitions which, in a sense, describe the class of all possible viscosity solutions to uniformly elliptic equations.

Definition 2.61 (class ${\mathcal{S}}$ ).

Let the operator $F$ be uniformly elliptic in the sense of Definition 2.10 and let $f=-F(\cdot ,0,\mathbf{0},0)$ . We say that $u\in \text{}\underline{{\mathcal{S}}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ if $u\in C(\unicode[STIX]{x1D6FA})$ and the inequality

$$\begin{eqnarray}f(x)\leq \unicode[STIX]{x1D6EC}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)>0}\unicode[STIX]{x1D706}_{i}(u)+\unicode[STIX]{x1D706}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)<0}\unicode[STIX]{x1D706}_{i}(u)\end{eqnarray}$$

holds in the viscosity sense. Similarly, we say that $u\in \overline{{\mathcal{S}}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ if $u\in C(\unicode[STIX]{x1D6FA})$ and

$$\begin{eqnarray}f(x)\geq \unicode[STIX]{x1D6EC}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)<0}\unicode[STIX]{x1D706}_{i}(u)+\unicode[STIX]{x1D706}\mathop{\sum }_{\unicode[STIX]{x1D706}_{i}(u)>0}\unicode[STIX]{x1D706}_{i}(u)\end{eqnarray}$$

in the viscosity sense. Finally ${\mathcal{S}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)=\text{}\underline{{\mathcal{S}}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)\cap \overline{{\mathcal{S}}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ .

With this notation at hand we present the Alexandrov–Bakelman–Pucci (ABP) estimate.

Theorem 2.62 (ABP estimate).

Let $u\in \overline{{\mathcal{S}}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ in $\unicode[STIX]{x1D6FA}$ with $u\geq 0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and assume that $f$ is continuous and bounded in $\unicode[STIX]{x1D6FA}$ . Then

$$\begin{eqnarray}\sup _{\unicode[STIX]{x1D6FA}}u^{-}\leq Cr\biggl(\int _{{\mathcal{C}}^{-}(u)}(f^{+})^{d}\biggr)^{1/d},\end{eqnarray}$$

where the constant $C$ depends only on $d$ , $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D6EC}$ , and $r$ is such that $\unicode[STIX]{x1D6FA}\subset B_{r/2}$ , and we have extended $u$ by zero outside $\unicode[STIX]{x1D6FA}$ .

Note that, as in the Alexandrov estimate, only the contact set ${\mathcal{C}}^{-}(u)$ is relevant in this estimate. Note also that we obtain control of the $L^{\infty }$ -norm of $u$ in terms of the $L^{d}$ -norm of the data $f$ . While Theorem 2.62 is a sort of stability estimate, it is also useful in establishing regularity of solutions. To do so, we begin with the Harnack inequality of Krylov and Safonov; see Safonov (Reference Safonov1980). In what follows, we let $Q_{l}$ denote a cube with sides parallel to the coordinate axes and of length $l$ .

Theorem 2.63 (Harnack inequality).

Let $u\in {\mathcal{S}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ in $Q_{1}$ with $f\in C(Q_{1})\cap L^{\infty }(Q_{1})$ . If $u\geq 0$ in $Q_{1}$ , then

$$\begin{eqnarray}\sup _{Q_{1/2}}u\leq C\Bigl(\inf _{Q_{1}}u+\Vert f\Vert _{L^{d}(Q_{1})}\Bigr),\end{eqnarray}$$

where the constant $C$ depends only on $d$ , $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D6EC}$ .

Since this will be useful later on, let us now show how one can obtain interior Hölder continuity of functions in ${\mathcal{S}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ from a Harnack inequality. We begin with a technical result, commonly referred to as an iteration lemma; see Han and Lin (Reference Han and Lin2011, Lemma 3.4) and Gilbarg and Trudinger (Reference Gilbarg and Trudinger2001, Lemma 8.23).

Lemma 2.64 (iteration).

Let $\unicode[STIX]{x1D711}:(0,R]\rightarrow \mathbb{R}$ be non-decreasing. Assume that for some $A>0$ , $B\geq 0$ and $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FD}$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70C})\leq A\biggl[\biggl(\displaystyle \frac{\unicode[STIX]{x1D70C}}{r}\biggr)^{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D700}\biggr]\unicode[STIX]{x1D711}(r)+Br^{\unicode[STIX]{x1D6FD}}\end{eqnarray}$$

whenever $0<\unicode[STIX]{x1D70C}\leq r\leq R$ . Then, for every $\unicode[STIX]{x1D6FE}\in (\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC})$ there exists $\unicode[STIX]{x1D700}_{0}$ such that if $\unicode[STIX]{x1D700}<\unicode[STIX]{x1D700}_{0}$ then

$$\begin{eqnarray}\unicode[STIX]{x1D711}(r)\leq C\biggl[\displaystyle \frac{\unicode[STIX]{x1D711}(R_{0})}{R_{0}}r^{\unicode[STIX]{x1D6FE}}+Br^{\unicode[STIX]{x1D6FD}}\biggr],\quad \text{for all}~r\in [0,R_{0}),\end{eqnarray}$$

for some fixed constant $C$ .

With this result at hand we obtain local Hölder continuity.

Theorem 2.65 (local Hölder regularity).

Let $u\in {\mathcal{S}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)$ in $Q_{1}$ . Then there exists $\unicode[STIX]{x1D6FC}\in (0,1)$ for which $u\in C^{\unicode[STIX]{x1D6FC}}(\bar{Q}_{1/2})$ and

$$\begin{eqnarray}\Vert u\Vert _{C^{\unicode[STIX]{x1D6FC}}(\bar{Q}_{1/2})}\leq C(\Vert u\Vert _{L^{\infty }(Q_{1})}+\Vert f\Vert _{L^{d}(Q_{1})}),\end{eqnarray}$$

where the constant $C$ is independent of $u$ and $f$ .

Proof. The proof is rather standard, so we merely sketch it. Let $m_{r}=\inf _{Q_{r}}u$ , $M_{r}=\sup _{Q_{r}}u$ and $\unicode[STIX]{x1D71B}_{r}=M_{r}-m_{r}$ . Applying the Harnack inequality of Theorem 2.63 to the non-negative function $u-m_{1}$ yields

$$\begin{eqnarray}M_{1/2}-m_{1}\leq C(m_{1/2}-m_{1}+\Vert f\Vert _{L^{d}(Q_{1})}).\end{eqnarray}$$

Since the function $M_{1}-u\geq 0$ , we can once more apply the Harnack inequality to obtain

$$\begin{eqnarray}M_{1}-m_{1/2}\leq C(M_{1}-M_{1/2}+\Vert f\Vert _{L^{d}(Q_{1})}).\end{eqnarray}$$

Adding these two inequalities yields

$$\begin{eqnarray}\unicode[STIX]{x1D71B}_{1/2}\leq \unicode[STIX]{x1D707}\unicode[STIX]{x1D71B}_{1}+2\Vert f\Vert _{L^{d}(Q_{1})},\end{eqnarray}$$

where $\unicode[STIX]{x1D707}=(C-1)/(C+1)\in (0,1)$ .

A similar argument for the functions $u_{r}(y)=u(ry)/r^{2}$ and $f_{r}(y)=f(ry)$ with $y\in Q_{1}$ reveals that

$$\begin{eqnarray}\unicode[STIX]{x1D71B}_{r}\leq \unicode[STIX]{x1D707}\unicode[STIX]{x1D71B}_{r/2}+2r\Vert f\Vert _{L^{d}(Q_{1})}.\end{eqnarray}$$

An application of the iteration Lemma 2.64 with $\unicode[STIX]{x1D711}(r)=\unicode[STIX]{x1D71B}_{r}$ immediately yields Hölder continuity and the estimate.

In a similar fashion, we can establish smoothness up to the boundary; see Caffarelli and Cabré (Reference Caffarelli and Cabré1995, Proposition 4.14).

Theorem 2.66 (global regularity).

Let $\unicode[STIX]{x1D6FA}$ be sufficiently smooth and $u\in {\mathcal{S}}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},f)\cap C(\bar{\unicode[STIX]{x1D6FA}})$ with $f\in C(\unicode[STIX]{x1D6FA})$ . Let $g=u_{|\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}$ and $\unicode[STIX]{x1D71A}$ be a modulus of continuity of $g$ . Then there is a modulus of continuity $\unicode[STIX]{x1D71A}^{\ast }$ of $u$ in $\bar{\unicode[STIX]{x1D6FA}}$ which depends only on $\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC},\unicode[STIX]{x1D71A},\Vert f\Vert _{L^{d}(\unicode[STIX]{x1D6FA})}$ and $\Vert g\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}$ .

We now focus on Hölder estimates for first and second derivatives. To simplify the presentation, in problem (2.6), the PDE takes the form

(2.21) $$\begin{eqnarray}F(x,D^{2}u)=f\quad \text{in }\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

To quantify the smoothness of $F$ with respect to the $x$ variable, we introduce the function

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}(x)=\sup _{M\in \mathbb{S}^{d}}\displaystyle \frac{|F(M,x)-F(M,0)|}{|M|+1}.\end{eqnarray}$$

The local regularity is as follows.

Theorem 2.67 (local $C^{2,\unicode[STIX]{x1D6FC}}$ -regularity).

Assume that $F$ is uniformly elliptic in the sense of Definition 2.10, $\unicode[STIX]{x1D6FD},f\in C^{\unicode[STIX]{x1D6FC}}(B_{1})$ and that there is a constant $\bar{\unicode[STIX]{x1D6FC}}\in (0,1)$ such that for any $M\in \mathbb{S}^{d}$ with $F(0,M)=0$ and $w_{0}\in C(\unicode[STIX]{x2202}B_{1})$ there exists $w\in C^{2}(B_{1})\cap C(\bar{B}_{1})\cap C^{2,\bar{\unicode[STIX]{x1D6FC}}}(B_{1/2})$ which satisfies

(2.22) $$\begin{eqnarray}F(0,D^{2}w+M)=0\quad \text{in }B_{1},\quad w=w_{0}\quad \text{on }\unicode[STIX]{x2202}B_{1},\end{eqnarray}$$

with an a priori estimate. If $u$ is a viscosity solution of (2.21) then $u\in C^{2,\unicode[STIX]{x1D6FC}}(\bar{B}_{1/2})$ for some $\unicode[STIX]{x1D6FC}\in (0,1)$ with an a priori estimate.

To apply this theorem, one must verify that solutions to (2.22) have $C^{2,\bar{\unicode[STIX]{x1D6FC}}}$ -estimates. For an $F$ that is convex, the Evans–Krylov theorem Caffarelli and Silvestre (Reference Caffarelli and Silvestre2010a ) provides such an estimate.

Theorem 2.68 (Evans–Krylov).

Let $F$ be convex and depend only on $M$ . If $u$ is a viscosity solution of $F(D^{2}u)=0$ in $B_{1}$ , then

$$\begin{eqnarray}\Vert u\Vert _{C^{2,\bar{\unicode[STIX]{x1D6FC}}}(\bar{B}_{1/2})}\leq C(\Vert u\Vert _{L^{\infty }(B_{1})}+|F(0)|),\end{eqnarray}$$

for some constants $\bar{\unicode[STIX]{x1D6FC}}\in (0,1)$ and $C$ that depend only on the dimension $d$ and the ellipticity of $F$ .

We mention that Theorem 2.67 can be applied to the Hamilton–Jacobi–Bellman operators of Example 2.12. With the aid of the so-called method of continuity, this allows us to show that solutions to the Dirichlet problem (2.6) for this class of operators are classical.

On the other hand, it is natural to ask if the convexity of $F$ is essential for this result. To understand this, we begin by providing a $C^{1,\unicode[STIX]{x1D6FC}}$ -estimate without convexity assumptions.

Theorem 2.69 ( $C^{1,\unicode[STIX]{x1D6FC}}$ -regularity).

Let $u$ be a viscosity solution of

$$\begin{eqnarray}F(D^{2}u)=0\quad \text{in }B_{1}.\end{eqnarray}$$

Then

$$\begin{eqnarray}\Vert u\Vert _{C^{1,\bar{\unicode[STIX]{x1D6FC}}}(\bar{B}_{1/2})}\leq C(\Vert u\Vert _{L^{\infty }(B_{1})}+|F(0)|),\end{eqnarray}$$

where the constants $\bar{\unicode[STIX]{x1D6FC}}\in (0,1)$ and $C$ depend only on the dimension and ellipticity of $F$ .

A similar argument to Theorem 2.67 then allows us to conclude that, in this setting and under similar assumptions, solutions to (2.6) are locally $C^{1,\unicode[STIX]{x1D6FC}}$ with an a priori estimate. However, there exists a series of counterexamples Nadirashvili and Vlăduţ (Reference Nadirashvili and Vlăduţ2007), Nadirashvili and Vlăduţ (Reference Nadirashvili and Vlăduţ2008), Nadirashvili, Tkachev and Vlăduţ (Reference Nadirashvili, Tkachev and Vlăduţ2012) showing that, in general, the convexity assumption on $F$ cannot be removed.

Example 2.70 (non-classical viscosity solution).

Let $d=5$ . Define

$$\begin{eqnarray}P_{5}(x)=x_{1}^{3}+\displaystyle \frac{3}{2}x_{1}(x_{3}^{2}+x_{4}^{2}-2x_{5}^{2}-2x_{2}^{2})+\displaystyle \frac{3\sqrt{3}}{2}(x_{2}x_{3}^{2}-x_{2}x_{4}^{2}+2x_{3}x_{4}x_{5}),\end{eqnarray}$$

and, for $\unicode[STIX]{x1D6FF}\in [0,1)$ ,

$$\begin{eqnarray}w(x)=P_{5}(x)/|x|^{1+\unicode[STIX]{x1D6FF}}\in C^{1,1-\unicode[STIX]{x1D6FF}}(\bar{B}_{1})\setminus C^{2}(B_{1}).\end{eqnarray}$$

There exists an Isaacs operator $F$ that depends only on $M$ and is Lipschitz, such that $w$ is a viscosity solution of

$$\begin{eqnarray}F(D^{2}w)=0\quad \text{in }B_{1},\quad w=P_{5}\quad \text{on }\unicode[STIX]{x2202}B_{1}.\end{eqnarray}$$

The existence of non-classical solutions in dimensions $3\leq d<5$ is an open problem.

We conclude by providing global regularity results in the general case; see Caffarelli and Souganidis (Reference Caffarelli and Souganidis2008), Caffarelli and Silvestre (Reference Caffarelli and Silvestre2010b ) and Turanova (Reference Turanova2015).

Theorem 2.71 (global regularity).

Let $\unicode[STIX]{x1D6FA}$ be sufficiently regular and let $u$ be a viscosity solution to (2.6) with $F$ of the form (2.21) being Lipschitz and uniformly elliptic. If $f\in C^{0,1}(\unicode[STIX]{x1D6FA})$ and, for some $\unicode[STIX]{x1D6FE}\in (0,1]$ , $g\in C^{1,\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA})$ , then $u\in C^{1,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FA})\cap C^{0,1}(\bar{\unicode[STIX]{x1D6FA}})$ with

$$\begin{eqnarray}\Vert u\Vert _{C^{1,\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D6FA})}\leq C(\Vert f\Vert _{L^{\infty }(\unicode[STIX]{x1D6FA})}+\Vert g\Vert _{C^{1,\unicode[STIX]{x1D6FE}}(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA})}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}\in (0,1)$ and $C$ depend only on $d$ , $\unicode[STIX]{x1D706}$ , $\unicode[STIX]{x1D6EC}$ and the smoothness of $F$ .

3.1 Stability, consistency and monotonicity implies convergence

As before, let $\unicode[STIX]{x1D6FA}$ be an open subset of $\mathbb{R}^{d}$ with Lipschitz boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . We consider numerical approximations of elliptic problems of the most general form (2.20), which we recall for convenience:

(3.1) $$\begin{eqnarray}F(x,u,Du,D^{2}u)=0\quad \text{in }\unicode[STIX]{x1D6FA},\quad B(x,u,Du)=0\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}.\end{eqnarray}$$

Here $F\in C(\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d}\times \mathbb{S}^{d})$ is locally bounded and elliptic in the sense of Definition 2.10.

We further assume that $B\in C(\unicode[STIX]{x1D6FA}\times \mathbb{R}\times \mathbb{R}^{d})$ is non-increasing in its second argument. For the moment, we consider viscosity solutions that satisfy the boundary conditions only in a viscosity sense; see Definition 2.53. In this case, and to simplify the presentation, we define the operator

$$\begin{eqnarray}\mathsf{F}(x,r,\mathbf{p},M)=\left\{\begin{array}{@{}ll@{}}F(x,r,\mathbf{p},M)\quad & \text{if }x\in \unicode[STIX]{x1D6FA},\\ B(x,r,\boldsymbol{ p})\quad & \text{if }x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

so that (3.1) becomes

(3.2) $$\begin{eqnarray}\mathsf{F}[u]:=\mathsf{F}(x,u,Du,D^{2}u)=0\quad \text{in }\bar{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

Note that, since $B$ is non-increasing in its second argument and $F$ is elliptic, the operator $\mathsf{F}$ is elliptic in the sense of Definition 2.10. We further see that, according to Definition 2.53, $u\in C(\bar{\unicode[STIX]{x1D6FA}})$ is a viscosity solution to (3.1) (equivalently (3.2)) if it is a viscosity solution to

$$\begin{eqnarray}F(x,u,Du,D^{2}u)=0\quad \text{in }\unicode[STIX]{x1D6FA},\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle \max \{F(x,u,Du,D^{2}u),B(x,u,Du)\} & \geq & \displaystyle 0\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\nonumber\\ \displaystyle \min \{F(x,u,Du,D^{2}u),B(x,u,Du)\} & \leq & \displaystyle 0\quad \text{on }\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}\nonumber\end{eqnarray}$$

in the viscosity sense.

We now consider approximation schemes. Find $u_{h}\in X_{h}$ satisfying

(3.3) $$\begin{eqnarray}\mathsf{F}_{h}[u_{h}](z)=0\quad \text{in }\bar{\unicode[STIX]{x1D6FA}}_{h},\end{eqnarray}$$

where $\mathsf{F}_{h}$ is a locally bounded operator which we may think of as an approximation to $\mathsf{F}$ , and $X_{h}$ is some finite-dimensional space. The operator is parametrized by $h>0$ , which we may view as a discretization parameter, or in some cases, a regularization parameter. The discrete domain $\overline{\unicode[STIX]{x1D6FA}}_{h}$ is an approximation to $\overline{\unicode[STIX]{x1D6FA}}$ with the property $\lim _{h\rightarrow 0^{+}}\bar{\unicode[STIX]{x1D6FA}}_{h}=\bar{\unicode[STIX]{x1D6FA}}$ ; namely, for all $z_{0}\in \bar{\unicode[STIX]{x1D6FA}}$ , there exists a sequence $\{z_{h}\}_{h>0}\subset \bar{\unicode[STIX]{x1D6FA}}_{h}$ such that $\lim _{h\rightarrow 0^{+}}z_{h}=z_{0}$ .

We now address the well-posedness of (3.3), and the sufficient structure conditions on $\mathsf{F}_{h}$ to ensure that the discrete solutions to (3.3) (if they exist) converge. As a first step we state the fundamental notions of consistency and stability.

Definition 3.1 (consistency).

The discrete problem (3.3) is said to be consistent with (3.2) if there exists an operator $I_{h}:C(\bar{\unicode[STIX]{x1D6FA}})\rightarrow X_{h}$ such that $I_{h}$ converges uniformly to the identity operator as $h\rightarrow 0^{+}$ , and for all sequences $\{z_{h}\}_{h>0}$ with $z_{h}\in \bar{\unicode[STIX]{x1D6FA}}_{h}$ and $z_{h}\rightarrow z_{0}\in \bar{\unicode[STIX]{x1D6FA}}$ and $\unicode[STIX]{x1D719}\in C^{2}(\bar{\unicode[STIX]{x1D6FA}})$ ,

$$\begin{eqnarray}\lim _{h\rightarrow 0^{+}}\mathsf{F}_{h}[I_{h}\unicode[STIX]{x1D719}](z_{h})=\mathsf{F}[\unicode[STIX]{x1D719}](z_{0}).\end{eqnarray}$$

Remark 3.2 (envelopes).

If the operators are not continuous, then the notion of consistency is changed to

$$\begin{eqnarray}\displaystyle \limsup _{h\rightarrow 0^{+}}\mathsf{F}_{h}[I_{h}\unicode[STIX]{x1D719}](z_{h}) & \leq & \displaystyle \mathsf{F}^{\ast }[\unicode[STIX]{x1D719}](z_{0}),\nonumber\\ \displaystyle \liminf _{h\rightarrow 0^{+}}\mathsf{F}_{h}[I_{h}\unicode[STIX]{x1D719}](z_{h}) & \geq & \displaystyle \mathsf{F}_{\ast }[\unicode[STIX]{x1D719}](z_{0}),\nonumber\end{eqnarray}$$

where $\mathsf{F}^{\ast }$ (resp.  $\mathsf{F}_{\ast }$ ) denote the upper (resp. lower) semi-continuous envelope of $\mathsf{F}$ ; see Definition 2.47.

The following theorem states the well-known result that, for linear problems with classical solutions, consistent and stable schemes converge.

Proof. By the given assumptions, there exists a linear operator $\mathsf{G}_{h}$ and a function $\mathsf{l}_{h}$ such that

$$\begin{eqnarray}\mathsf{F}_{h}[v_{h}](z)=\mathsf{G}_{h}[v_{h}](z)-\mathsf{l}_{h}(z)\quad z\in \bar{\unicode[STIX]{x1D6FA}}_{h}.\end{eqnarray}$$

The stability of the scheme shows that $\mathsf{G}_{h}[\cdot ]$ is an isomorphism whose inverse is bounded independent of $h$ . Since

$$\begin{eqnarray}\mathsf{G}_{h}[u_{h}-I_{h}u]=\mathsf{l}_{h}-\mathsf{G}_{h}[I_{h}u]=-\mathsf{F}_{h}[I_{h}u],\end{eqnarray}$$

the consistency of the scheme implies that $\lim _{h\rightarrow 0^{+}}\Vert u_{h}-I_{h}u\Vert _{L^{\infty }(\bar{\unicode[STIX]{x1D6FA}}_{h})}=0$ , and thus, since $I_{h}$ converges to the identity operator, $u_{h}$ converges locally uniformly to $u$ .

While Theorem 3.4 is a useful result for a large class of problems, it is not applicable to fully nonlinear problems nor to weaker notions of solutions, in particular, viscosity solutions. The issue is that, if $u$ is not a classical solution, then the approximation $I_{h}u$ may not be well-defined, and the consistency $\mathsf{F}_{h}[I_{h}u]\rightarrow 0$ used in the proof of Theorem 3.4 is no longer valid. Rather, to prove convergence to viscosity solutions, an additional structure condition is required. This requirement is summarized in the following definition.

Definition 3.5 (monotone operator).

The discrete operator $\mathsf{F}_{h}$ is said to be monotone if, whenever $u_{h}-v_{h}$ has a global non-negative maximum at $z\in \bar{\unicode[STIX]{x1D6FA}}_{h}$ , we have

(3.4) $$\begin{eqnarray}\mathsf{F}_{h}[u_{h}](z)\leq \mathsf{F}_{h}[v_{h}](z).\end{eqnarray}$$

Remark 3.6 (monotonicity).

The notion of monotonicity is essentially a discrete version of ellipticity. Indeed, following the proof of Theorem 2.16, if $F$ is an elliptic operator in the sense of Definition 2.10, and if $u-v$ , with $u,v\in C^{2}(\bar{\unicode[STIX]{x1D6FA}})$ , has a global non-negative maximum at $x\in \bar{\unicode[STIX]{x1D6FA}}$ , then $u(x)\geq v(x)$ , $Du(x)=Dv(x)$ and $D^{2}u(x)\leq D^{2}v(x)$ . Since $\mathsf{F}$ is non-increasing in its second argument, and non-decreasing in its fourth, we have

$$\begin{eqnarray}\mathsf{F}[u](x)=\mathsf{F}(x,u(x),Du(x),D^{2}u(x))\leq \mathsf{F}(x,v(x),Dv(x),D^{2}v(x))=\mathsf{F}[v](x).\end{eqnarray}$$

Conversely, it can be shown that the operator $\mathsf{F}$ is elliptic if, whenever $u-v$ has a global non-negative maximum at $x\in \bar{\unicode[STIX]{x1D6FA}}$ , then $\mathsf{F}[u](x)\leq \mathsf{F}[v](x)$ .

Consistency, stability and monotonicity are the three sufficient ingredients to guarantee convergence to viscosity solutions. The following result closely follows Barles and Souganidis (Reference Barles and Souganidis1991, Theorem 2.1).

Proof. Define $\bar{u}\in \mathit{USC}(\unicode[STIX]{x1D6FA})$ and $\text{}\underline{u}\in \mathit{LSC}(\unicode[STIX]{x1D6FA})$ by

(3.5) $$\begin{eqnarray}\bar{u}(x):=\limsup _{\substack{ y\rightarrow x \\ h\rightarrow 0^{+}}}u_{h}(y),\quad \text{}\underline{u}(x):=\liminf _{\substack{ y\rightarrow x \\ h\rightarrow 0^{+}}}u_{h}(y),\quad x\in \bar{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

Note that the stability of the scheme implies that both $\bar{u}$ and $\text{}\underline{u}$ are well-defined. The proof proceeds by showing that $\bar{u}$ and $\text{}\underline{u}$ are (viscosity) subsolutions and supersolutions to (3.2), respectively, and then appealing to the comparison principle.

To this end, suppose that $z_{0}\in \unicode[STIX]{x1D6FA}$ is a strict local maximum of $\text{}\underline{u}-\unicode[STIX]{x1D719}$ for some $\unicode[STIX]{x1D719}\in C^{2}(\unicode[STIX]{x1D6FA})$ . Then standard arguments show that there exist sequences $\{h_{n}\}_{n=1}^{\infty }$ and $\{z_{h_{n}}\}_{n=1}^{\infty }$ such that

$$\begin{eqnarray}h_{n}\rightarrow 0,\quad z_{h_{n}}\rightarrow x_{0},\quad \text{as }n\rightarrow \infty ,\end{eqnarray}$$

and $u_{h_{n}}-I_{h_{n}}\unicode[STIX]{x1D719}$ obtains a strict local maximum at $z_{h_{n}}$ . Since $\mathsf{F}_{h_{n}}[u_{h_{n}}](z_{h_{n}})=0$ , the monotonicity of the discrete operator implies that

$$\begin{eqnarray}\mathsf{F}_{h_{n}}[I_{h_{n}}\unicode[STIX]{x1D719}](z_{h_{n}})\geq 0.\end{eqnarray}$$

Passing to the limit, together with consistency of the scheme, yields

$$\begin{eqnarray}0\leq \lim _{n\rightarrow \infty }\mathsf{F}_{h_{n}}[I_{h_{n}}\unicode[STIX]{x1D719}](z_{h_{n}})=\mathsf{F}[\unicode[STIX]{x1D719}](z_{0}).\end{eqnarray}$$

Similar arguments show that if $\bar{u}-\unicode[STIX]{x1D719}$ obtains a strict minimum at $z_{0}\in \unicode[STIX]{x1D6FA}$ , then $0\geq \mathsf{F}[\unicode[STIX]{x1D719}](z_{0})$ . Thus, $\bar{u}$ and $\text{}\underline{u}$ are subsolutions and supersolutions to (3.2), respectively. Since $\text{}\underline{u}\leq \bar{u}$ , the comparison principle of $\mathsf{F}$ implies that $\text{}\underline{u}=\bar{u}=u$ , and $u$ is the viscosity solution to (3.2).

We once again mention that the problems and discretizations considered so far take into account the boundary conditions in a viscosity sense. While this set-up simplifies the proof of convergence, it may have practical limitations since the framework requires a consistent and monotone discretization of both the boundary conditions and the differential operator for all $x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ and smooth functions $\unicode[STIX]{x1D719}$ . Also recall that, in general, viscosity boundary conditions are not equivalent to those imposed pointwise unless other conditions are assumed; see Proposition 2.55. Here we turn our attention to the elliptic boundary value problem (2.6), where the Dirichlet boundary condition is understood in the classical sense; see Definition 2.48.

To this end, we consider approximations of the form

(3.6) $$\begin{eqnarray}F_{h}[u_{h}]=0\quad \text{in }\unicode[STIX]{x1D6FA}_{h}^{I},\quad u_{h}=g_{h}\quad \text{on }\unicode[STIX]{x1D6FA}_{h}^{B},\end{eqnarray}$$

where $g_{h}=I_{h}g\in X_{h}$ is a discrete approximation to the Dirichlet data $g\in C(\bar{\unicode[STIX]{x1D6FA}})$ , and $\unicode[STIX]{x1D6FA}_{h}^{I}$ and $\unicode[STIX]{x1D6FA}_{h}^{B}$ are disjoint sets with $\unicode[STIX]{x1D6FA}_{h}^{I}\rightarrow \unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D6FA}_{h}^{B}\rightarrow \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ as $h\rightarrow 0^{+}$ . Note that with minor notational changes, the notions of consistency, stability, and monotonicity are applicable to the operator $F_{h}$ . A natural question is whether the results of Theorem 3.7 carry over to the discrete problem (3.6). This issue is addressed in the next theorem.

3.2 Monotonicity in finite difference schemes

Here we discuss basic monotonicity results of finite difference schemes. The main message given in this section is that for any uniformly elliptic operator, one can construct a consistent and monotone finite difference scheme. The drawback, however, is that monotonicity requires a wide stencil, which may severely impact its practical use. Much of the material in this section is found in Kuo and Trudinger (Reference Kuo and Trudinger1990, Reference Kuo and Trudinger1992), Kocan (Reference Kocan1995), Motzkin and Wasow (Reference Motzkin and Wasow1953) and Oberman (Reference Oberman2006).

For simplicity we assume that the domain $\unicode[STIX]{x1D6FA}$ is discretized on an equally spaced Cartesian grid and that each coordinate direction is discretized uniformly; in particular, by a possible change of coordinates, we assume that the grid is given by

$$\begin{eqnarray}\bar{\unicode[STIX]{x1D6FA}}_{h}=\mathbb{Z}_{h}^{d}\cap \overline{\unicode[STIX]{x1D6FA}},\quad \text{with }\mathbb{Z}_{h}^{d}:=\{he:e\in \mathbb{Z}^{d}\},\end{eqnarray}$$

where $h>0$ is the grid scale and $\mathbb{Z}^{d}$ is the set of $d$ -tuples of integers. A finite subset $S\subset \mathbb{Z}^{d}\backslash \{0\}$ is called a stencil, and the space of nodal functions, denoted by $X_{h}^{fd}$ , consists of real-valued functions with domain $\bar{\unicode[STIX]{x1D6FA}}_{h}$ . The canonical interpolant $I_{h}^{fd}:C^{0}(\bar{\unicode[STIX]{x1D6FA}})\rightarrow X_{h}^{fd}$ is the operator satisfying $I_{h}^{fd}v(z)=v(z)$ for all $z\in \bar{\unicode[STIX]{x1D6FA}}_{h}$ . We assume the existence of a positive integer $m$ such that $S$ is of the form

(3.7) $$\begin{eqnarray}S=\{y:y\in \mathbb{Z}^{d}\backslash \{0\}:|y|_{\ell ^{\infty }}\leq m\}.\end{eqnarray}$$

The value $m$ satisfying (3.7) is called the stencil size of $S$ . The cardinality of $S$ is $|S|:=(2m+1)^{d}-1$ .

We consider finite difference schemes with stencil $S$ acting on grid functions. These discrete operators are thus of the (implicit) form

(3.8) $$\begin{eqnarray}F_{h}[v_{h}](z)=F_{h}(z,v_{h}(z),Tv_{h}(z)),\end{eqnarray}$$

where $Tv_{h}(x)=\{v_{h}(x+hy):y\in S\}$ is the set of translates of $v_{h}(x)$ with respect to the stencil. The method (3.8) is called a one-step scheme if $m=1$ , that is, the value $F_{h}[v_{h}](z)$ depends only on $z$ , $v_{h}(z)$ , and the values of $v_{h}$ at neighbouring points of $z$ . Otherwise, we call the scheme a wide-stencil scheme if $m\geq 2$ .

To construct monotone schemes, we first reformulate this property so that it is easier to work with.

Definition 3.9 (non-negative operator).

The operator $F_{h}$ is of non-negative type (or simply non-negative) if

(3.9) $$\begin{eqnarray}F_{h}(z,r,q+\unicode[STIX]{x1D70F})\geq F_{h}(z,r,q)\geq F_{h}(z,r+t,q+\unicode[STIX]{x1D70F}),\end{eqnarray}$$

for all $z\in \mathbb{R}^{d}$ , $r,t\in \mathbb{R}$ , and $q,\unicode[STIX]{x1D70F}\in \mathbb{R}^{|S|}$ satisfying

(3.10) $$\begin{eqnarray}0\leq \unicode[STIX]{x1D70F}_{i}\leq t\quad i=1,2,\ldots ,|S|.\end{eqnarray}$$

We see that if $F_{h}$ is of non-negative type, then $F_{h}$ is non-increasing in its second argument and non-decreasing in its third argument. If $F_{h}$ is differentiable, then it is of non-negative type provided that

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}F_{h}}{\unicode[STIX]{x2202}q_{i}}\geq 0~(i=1,2,\ldots ,|S|),\quad \displaystyle \frac{\unicode[STIX]{x2202}F_{h}}{\unicode[STIX]{x2202}r}+\mathop{\sum }_{i=1}^{|S|}\displaystyle \frac{\unicode[STIX]{x2202}F_{h}}{\unicode[STIX]{x2202}q_{i}}\leq 0.\end{eqnarray}$$

Remark 3.10 (reformulation).

Alternatively, as in Oberman (Reference Oberman2006), one can consider finite difference schemes of the form

$$\begin{eqnarray}F_{h}[u](z)=G_{h}(z,u(z),u(z)-Tu(z)).\end{eqnarray}$$

Let $\mathbf{1}\in \mathbb{R}^{|S|}$ denote the vector of ones. Then using the correspondence $F_{h}(z,r,q)=G_{h}(z,r,r\mathbf{1}-q)$ , one sees that $F_{h}$ is of non-negative type if and only if $G_{h}$ is non-increasing in its second and third arguments.

Let us now show that non-negativity is simply a reformulation of monotonicity.

Proof. Suppose that $F_{h}$ is of non-negative type. Let $u_{h}$ and $v_{h}$ be two grid functions such that $u_{h}-v_{h}$ has a global non-negative maximum at some grid point $z$ . Set $r=v_{h}(z)$ , $t=u_{h}(z)-v_{h}(z)\geq 0$ , $q_{i}=v_{h}(z+hy_{i})$ , and $\unicode[STIX]{x1D70F}_{i}=\max \{0,u_{h}(z+hy_{i})-v_{h}(z+hy_{i})\}$ , so that $t\geq \unicode[STIX]{x1D70F}_{i}$ . Noting that $q_{i}+\unicode[STIX]{x1D70F}_{i}=v_{h}(z+hy_{i})+\max \{0,u_{h}(z+hy_{i})-v_{h}(z+hy_{i})\}\geq u_{h}(z+hy_{i})$ , and $F_{h}$ is non-decreasing in its third argument, we find that $F_{h}[u_{h}](z)\leq F_{h}(z,r+t,q+\unicode[STIX]{x1D70F}).$ Therefore, by the second inequality in (3.9) we have

$$\begin{eqnarray}F_{h}[u_{h}](z)\leq F_{h}(z,r+t,q+\unicode[STIX]{x1D70F})\leq F_{h}(z,r,q)=F_{h}[v_{h}](z).\end{eqnarray}$$

Thus, $F_{h}$ is monotone.

Now suppose that $F_{h}$ is monotone. Let $z\in \mathbb{R}^{d}$ be fixed, and let $r,t\in \mathbb{R}$ and $q,\unicode[STIX]{x1D70F}\in \mathbb{R}^{|S|}$ satisfy (3.9). Then define the grid functions $u_{h},v_{h}$ (locally) as

$$\begin{eqnarray}v_{h}(z)=r,\quad v_{h}(z+hy_{i})=q_{i},\quad u_{h}(z)=r+t,\quad u_{h}(z+y_{i})=q_{i}+\unicode[STIX]{x1D70F}_{i}.\end{eqnarray}$$

Then

$$\begin{eqnarray}u_{h}(z)-v_{h}(z)=t\geq \unicode[STIX]{x1D70F}_{i}=u_{h}(z+hy_{i})-v_{h}(z+hy_{i}),\end{eqnarray}$$

that is, $u_{h}-v_{h}$ has a non-negative maximum at $z$ . The monotonicity of $F_{h}$ yields $F_{h}[u_{h}](z)\leq F_{h}[v_{h}](z)$ ; thus

(3.11) $$\begin{eqnarray}F_{h}(z,r+t,q+\unicode[STIX]{x1D70F})\leq F_{h}(z,r,q).\end{eqnarray}$$

On the other hand, with $v_{h}$ as before, we consider the grid function $w_{h}$ with

$$\begin{eqnarray}w_{h}(z)=r,\quad w_{h}(z+hy_{i})=q_{i}+\unicode[STIX]{x1D70F}_{i}.\end{eqnarray}$$

Then $v_{h}-w_{h}$ has a global maximum at $z$ and thus

(3.12) $$\begin{eqnarray}F_{h}(z,r,q)=F_{h}[v_{h}](z)\leq F_{h}[w_{h}](z)=F_{h}(z,r,q+\unicode[STIX]{x1D70F}).\end{eqnarray}$$

We conclude from (3.11)–(3.12) that $F_{h}$ is of non-negative type.

Following the framework given in Kuo and Trudinger (Reference Kuo and Trudinger1990, Reference Kuo and Trudinger1992) and Kocan (Reference Kocan1995), we consider discrete operators constructed from the first- and second-order difference operators

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{y,h}^{+}u(z) & := & \displaystyle \displaystyle \frac{1}{h}(u(z+hy)-u(z)),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FF}_{y,h}^{-}u(z) & := & \displaystyle \displaystyle \frac{1}{h}(u(z)-u(z-hy)),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FF}_{y,h}u(z) & := & \displaystyle \displaystyle \frac{1}{2}(\unicode[STIX]{x1D6FF}_{y}^{+}+\unicode[STIX]{x1D6FF}_{y}^{-})u(z)=\displaystyle \frac{1}{2h}(u(z+hy)-u(z-hy)),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FF}_{y,h}^{2}u(z) & := & \displaystyle \displaystyle \frac{1}{h^{2}}(u(z+hy)-2u(x)+u(z-hy)),\nonumber\end{eqnarray}$$

with $y\in S$ . Taylor’s theorem shows that the differences $\unicode[STIX]{x1D6FF}_{y,h}^{\pm }u(z)$ are first-order approximations to

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u(z)}{\unicode[STIX]{x2202}y}:=Du(z)\cdot y,\end{eqnarray}$$

whereas $\unicode[STIX]{x1D6FF}_{y,h}u(z)$ and $\unicode[STIX]{x1D6FF}_{y,h}^{2}u(z)$ , respectively, are second-order approximations to

$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u(z)}{\unicode[STIX]{x2202}y}\quad \text{and}\quad \displaystyle \frac{\unicode[STIX]{x2202}^{2}u(z)}{\unicode[STIX]{x2202}y^{2}}:=y\cdot D^{2}u(z)y;\end{eqnarray}$$

by this we mean that

$$\begin{eqnarray}\displaystyle \biggl|\displaystyle \frac{\unicode[STIX]{x2202}^{\pm }u(z)}{\unicode[STIX]{x2202}y}-\unicode[STIX]{x1D6FF}_{y,h}^{\pm }u(z)\biggr| & = & \displaystyle O(h|y|^{2}),\nonumber\\ \displaystyle \biggl|\displaystyle \frac{\unicode[STIX]{x2202}u(z)}{\unicode[STIX]{x2202}y}-\unicode[STIX]{x1D6FF}_{y,h}u(z)\biggr| & = & \displaystyle O(h^{2}|y|^{3}),\nonumber\\ \displaystyle \biggl|\displaystyle \frac{\unicode[STIX]{x2202}^{2}u(z)}{\unicode[STIX]{x2202}y^{2}}-\unicode[STIX]{x1D6FF}_{y,h}^{2}u(z)\biggr| & = & \displaystyle O(h^{2}|y|^{4}),\nonumber\end{eqnarray}$$

for sufficiently smooth $u$ .

Let

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{h}u_{h}(z)=\{\unicode[STIX]{x1D6FF}_{y,h}u_{h}(z):y\in S\}\quad \text{and}\quad \unicode[STIX]{x1D6FF}_{h}^{2}u_{h}(z)=\{\unicode[STIX]{x1D6FF}_{y,h}^{2}u_{h}(z):y\in S\}.\end{eqnarray}$$

Then a consistent and monotone finite difference scheme can be constructed in the form

(3.13) $$\begin{eqnarray}F_{h}[u_{h}](z)={\mathcal{F}}_{h}(z,u_{h}(z),\unicode[STIX]{x1D6FF}_{h}u_{h}(z),\unicode[STIX]{x1D6FF}_{h}^{2}u_{h}(z))\end{eqnarray}$$

Kuo and Trudinger (Reference Kuo and Trudinger1990, Reference Kuo and Trudinger1992), Motzkin and Wasow (Reference Motzkin and Wasow1953), where ${\mathcal{F}}_{h}:\unicode[STIX]{x1D6FA}_{h}\times \mathbb{R}\times \mathbb{R}^{|S|}\times \mathbb{R}^{|S|}\rightarrow \mathbb{R}$ . Let $(z,r,q,s)$ denote points in the domain of ${\mathcal{F}}_{h}$ , and assume that ${\mathcal{F}}_{h}$ is symmetric with respect to $\pm q_{\pm i}$ and $s_{\pm i}$ . Then from Definition 3.9 and Lemma 3.11 we see that $F_{h}$ of the form (3.13) is monotone provided that

(3.14) $$\begin{eqnarray}\displaystyle \frac{h}{2}\biggl|\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}_{h}}{\unicode[STIX]{x2202}q_{i}}\biggr|\leq \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}_{h}}{\unicode[STIX]{x2202}s_{i}}\quad i=1,2,\ldots ,|S|,\quad \text{and}\quad \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}_{h}}{\unicode[STIX]{x2202}r}\leq 0.\end{eqnarray}$$

Below, we will require slightly stronger conditions on the operator  ${\mathcal{F}}_{h}$ .

Definition 3.12 (positive operator).

An operator of the form (3.13) is of positive type (or simply positive) if (3.14) is satisfied and there exists $\unicode[STIX]{x1D706}_{0,h}>0$ and an orthogonal set of vectors $\{y_{i}\}_{i=1}^{d}\subset S$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{0,h}+\displaystyle \frac{h}{2}\biggl|\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}_{h}}{\unicode[STIX]{x2202}q_{i}}\biggr|\leq \displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{F}}_{h}}{\unicode[STIX]{x2202}s_{i}}.\end{eqnarray}$$

3.3 Finite difference stability estimates: Alexandrov estimates and Alexandrov–Bakelman–Pucci maximum principle

We now turn our attention to maximum principles of discrete operators, and correspondingly, stability estimates. As a starting point, we discuss monotone finite difference schemes for the linear non-divergence-form PDEs of Example 2.11

(3.15) $$\begin{eqnarray}F[u]={\mathcal{L}}u-f=A:D^{2}u-f=0.\end{eqnarray}$$

While this setting may seem overly simplistic, as we shall see, the construction and theoretical results for the linear problem form all of the necessary tools to approximate viscosity solutions of nonlinear elliptic equations.

We assume that $f\in C(\bar{\unicode[STIX]{x1D6FA}})$ and that the coefficient matrix $A$ is bounded and uniformly symmetric positive definite. It is then reasonable to assume that $F_{h}$ is linear and thus has the form

(3.16) $$\begin{eqnarray}F_{h}[u_{h}](z)={\mathcal{L}}_{h}u_{h}(z)-f(z):=\mathop{\sum }_{y\in S}a_{y}(z)\unicode[STIX]{x1D6FF}_{y,h}^{2}u_{h}(z)-f(z)\end{eqnarray}$$

for nodal functions (or coefficients) $a_{y}$ . Applying Definition 3.9 to (3.16), we see that $F_{h}$ is non-negative (and hence monotone) provided

(3.17) $$\begin{eqnarray}a_{y}(z)\geq 0,\end{eqnarray}$$

and of positive type if

$$\begin{eqnarray}a_{y_{i}}(z)\geq \unicode[STIX]{x1D706}_{0,h}>0\quad i=1,2,\ldots ,d.\end{eqnarray}$$

for some orthogonal basis $\{y_{i}\}_{i=1}^{d}\subset S$ . These inequalities suggest that the negation of the ensuing system is an $M$ -matrix, and hence solutions to the discrete problem satisfy certain maximum principles, analogous to the continuous setting. This issue is discussed in the next section.