1 Introduction
Let
$(\overline {M}, g)$
be an n-dimensional compact connected Riemannian manifold with smooth boundary
$\partial M$
. Denote
$\overline {M}:=M\cup \partial M$
, where
$n\geq 2$
and M is the interior of
$\overline {M}$
. Let
$\chi $
be a smooth symmetric
$(0,2)$
-tensor on
$\overline {M}$
, and let
$\psi \in C^\infty (\overline {M})$
and
$\varphi \in C^{\infty }(\partial M)$
. Moreover, denote
$\vec {\mathbf {1}}=(1,1,\ldots ,1)\in \mathbb {R}^n.$
In this paper, we propose a continuity method for the Dirichlet problem
$$ \begin{align} \begin{aligned} F(\nabla^2u+\chi) = \psi \quad\text{in } \overline{M}, \end{aligned} \end{align} $$
$$ \begin{align} \begin{aligned} u=\varphi \quad\text{on } \partial M, \end{aligned} \end{align} $$
where
$\nabla ^2u$
is the real Hessian of u with respect to the Levi–Civita connection
$\nabla $
of g. The nonlinear operator
$F(\nabla ^2u+\chi )$
is generated by a pair
$(f,\Gamma )$
and has the form
$$ \begin{align*} \begin{aligned} F(\nabla^2u+\chi) = f(\lambda[\nabla^2u+\chi]), \end{aligned} \end{align*} $$
where
$\lambda [\nabla ^2u+\chi ]$
is an n-tuple of eigenvalues of
$\nabla ^2u+\chi $
with respect to g, f is a smooth symmetric function of n real variables, defined in an open symmetric convex cone
$\Gamma $
with vertex at the origin, and
$$ \begin{align*}\Gamma_n:= \{(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n: \text{ each } \lambda_i>0\}\subseteq\Gamma,\end{align*} $$
as well as
$\partial \Gamma \neq \emptyset $
. The function f is supposed to satisfy
$$ \begin{align} \begin{aligned} f \text{ is a concave function in } \Gamma, \end{aligned} \end{align} $$
$$ \begin{align} \begin{aligned} f_{i}(\lambda) = f_{\lambda_i}(\lambda) :=\frac{\partial f}{\partial \lambda_{i}}(\lambda)> 0 \quad\text{in } \Gamma \text{ for all } 1\leq i\leq n. \end{aligned} \end{align} $$
Definition 1.1. For (1-1), following [Reference Caffarelli, Nirenberg and Spruck2], we say that
$u\in C^{2}(\overline {M})$
is an admissible function if
$\lambda [\nabla ^2u+\chi ]\in \Gamma $
.
Equation (1-1) includes the Monge–Ampère equation and the more general k-Hessian equation corresponding to
$$ \begin{align*} f=\sigma_k^{1/k}, \quad \Gamma=\Gamma_k, \end{align*} $$
where
$\sigma _k$
is the k th elementary symmetric function and
$\Gamma _k$
is the k th Gårding cone. Recently, an example introduced by [Reference Guan and Zhang12] has attracted considerable interest:
$$ \begin{align} \begin{aligned} f= \frac{\sigma_{k}}{\sigma_{k-1}} -\sum_{j=0}^{k-2} \frac{\beta_j \sigma_j}{\sigma_{k-1}} \quad \text{for all } \beta_j\geq0,\, \sum_{j=0}^{k-2} \beta_j>0, \, \Gamma=\Gamma_{k-1}. \end{aligned} \end{align} $$
Moreover, (1-1) also includes the following as a special case:
$$ \begin{align} \begin{aligned} {f}(\lambda[ \Delta u \cdot g- \nabla^2u +\chi]) =\psi. \end{aligned} \end{align} $$
When
$ {f}(\lambda )=\sum _{i=1}^n \log \lambda _i$
, it is a Monge–Ampère equation for
$(n-1)$
-PSH functions in the sense of [Reference Harvey and Lawson14]. Such a type of equation is of special interest in complex geometry, since its close connection to the form-type Calabi–Yau equation [Reference Fu, Wang and Wu5] and Gauduchon’s conjecture [Reference Popovici22, Reference Tosatti and Weinkove24].
When
$\chi =0$
and
$M=\Omega \subset \mathbb {R}^n$
, the Dirichlet problem was initially treated by Caffarelli et al. [Reference Caffarelli, Nirenberg and Spruck2] under some additional assumptions on f, including
$$ \begin{align} \begin{aligned} \lim_{t\rightarrow +\infty}f(t\lambda)=+\infty \quad \text{for any } \lambda \in \Gamma, \end{aligned} \end{align} $$
and the unbounded condition
$$ \begin{align} \begin{aligned} \lim_{t\rightarrow+\infty} f(\lambda_1,\ldots,\lambda_{n-1},\lambda_n+t)=+\infty \quad \text{for all }\lambda=(\lambda_1,\ldots,\lambda_n)\in\Gamma. \end{aligned} \end{align} $$
Similar results were also obtained in the literature [Reference Ivochkina16, Reference Krylov17]. Caffarelli et al.’s result has been further extended by Trudinger [Reference Trudinger25] to the bounded case. In addition, their works require a certain assumption on the principal curvatures of
$\partial \Omega $
.
Without geometric restriction to the boundary, as developed by [Reference Guan7, Reference Guan and Spruck11] and others, in general, the existence result for the Dirichlet problem requires an appropriate subsolution assumption.
Definition 1.2 (Subsolution)
We say that
$\underline {u}\in C^2(\overline {M})$
is an admissible subsolution to the Dirichlet problem (1-1)–(1-2) if it is an admissible function and satisfies
$$ \begin{align*} \begin{aligned} f(\lambda[ \nabla^2\underline{u} +\chi]) \geq \psi \quad\text{in } \overline{M}, \underline{u}=\varphi \quad\text{on } \partial M. \end{aligned} \end{align*} $$
Condition (1-4) ensures that (1-1) be elliptic at any admissible solutions, while (1-3) implies that the operator
$F(A)=f(\lambda (A))$
is concave with respect to A when
$\lambda (A)\in \Gamma $
. Therefore, under the assumption
$$ \begin{align} \begin{aligned} \inf_M \psi> \sup_{\partial\Gamma} f \quad\text{where } \underset{\partial \Gamma}\sup f = \underset{\lambda_{0}\in \partial \Gamma}\sup \underset{\lambda\rightarrow \lambda_{0}}\limsup f(\lambda), \end{aligned} \end{align} $$
once the following a priori estimate of the Hessian is in place,
$$ \begin{align} \begin{aligned} \sup_{\overline{M}} |\nabla^2 u|\leq C, \end{aligned} \end{align} $$
the higher regularities follow from the Evans–Krylov theorem and Schauder theory. Building on this, one can use the standard continuity method to derive the existence.
This paper is devoted to developing a continuity method for the Dirichlet problem. First, we review some notation. For a given pair
$(f,\Gamma )$
, as in [Reference Yuan34], denote
$$ \begin{align*} \begin{aligned} \Gamma_{\mathcal{G}}^{f} =\Big\{\lambda\in\Gamma: \lim_{t\to +\infty}f(t\lambda)>-\infty \Big\}. \end{aligned} \end{align*} $$
In addition,
$\mathring {\Gamma }_{\mathcal {G}}^{f}$
denotes the interior of
$\Gamma _{\mathcal {G}}^{f}$
. For
${\Gamma }_{\mathcal {G}}^{f}$
, as in [Reference Yuan33, Reference Yuan36], define
$$ \begin{align} \begin{aligned} (1,\ldots,1, 1-\varrho_{{\Gamma}_{\mathcal{G}}^f}) \in \partial\Gamma_{\mathcal{G}}^{f}. \end{aligned} \end{align} $$
Following the convention in [Reference Caffarelli, Nirenberg and Spruck2], as in [Reference Yuan34, Reference Yuan36], we say that
$(f,\Gamma )$
is a type-2 pair if
$$ \begin{align} \begin{aligned} (0,\ldots,0,1)\in\mathring{\Gamma}_{\mathcal{G}}^{f}. \end{aligned} \end{align} $$
Otherwise, it is a type-1 pair. Accordingly, we call (1-1) a type-2 equation if the corresponding pair
$(f,\Gamma )$
satisfies (1-12). Otherwise, it is a type-1 equation.
Remark 1.3. When f satisfies (1-7), we know
${\Gamma }_{\mathcal {G}}^{f}{\kern-1pt}={\kern-1pt}\Gamma $
. Notably,
${\mathring {\Gamma }_{\mathcal {G}}^{f}{\kern-1pt}\neq{\kern-1pt} \Gamma _n {\kern-1pt}\iff{\kern-1pt} \, \varrho _{{\Gamma }_{\mathcal {G}}^f}{\kern-1pt}>{\kern-1pt}1}$
. For (1-5), we have
$\mathring {\Gamma }_{\mathcal {G}}^{f}=\Gamma _k$
and
$\varrho _{{\Gamma }_{\mathcal {G}}^f}=({n}/{k})$
.
The type-2 pair is much simpler, as detailed in Lemma 2.2. This is a special case of the partial uniform ellipticity developed by the author in a series of works [Reference Yuan31, Reference Yuan32, Reference Yuan36]. In this type-2 case, together with a Morse theory technique that was first introduced in [Reference Yuan32] as a tool to construct admissible functions, we obtain the existence result for the Dirichlet problem without extra assumptions on the boundary or the subsolution.
Theorem 1.4. In addition to (1-3), (1-4), and (1-9), we assume that
$(f,\Gamma )$
is of type 2. Then, the Dirichlet problem (1-1)–(1-2) has a unique smooth admissible solution.
This is a fully nonlinear analogue of the existence theorem for Poisson’s equation. Significantly, in the type-2 case, we can solve the Dirichlet problem without any extra assumption regarding an admissible subsolution or admissible function. This stands in contrast with the vast existing literature.
However, using the constant
$\varrho _{{\Gamma }_{\mathcal {G}}^f}$
, we can give a characterization of
$(f,\Gamma )$
in Lemmas 2.4 and 2.2. Thus, we can recast (1-1) in the following form:
$$ \begin{align} \begin{aligned} \tilde{f}(\lambda[ \Delta u \cdot g-\varrho \nabla^2u+\tilde{\chi}]) =\psi, \end{aligned} \end{align} $$
corresponding to
$(\tilde {f},\tilde {\Gamma })$
, and vice versa, where
$$ \begin{align*} \varrho\leq \varrho_{\tilde{\Gamma}_{\mathcal{G}}^{\tilde{f}}},\quad \varrho\neq0 \quad \text{and }\quad \tilde{\chi}=(\mathrm{tr}_{g}\chi)\cdot g-\varrho \chi.\end{align*} $$
In the critical case
$\varrho =1$
, it reads as (1-6). Significantly, when
$\varrho <\varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}$
and
$ \varrho \neq 0$
, (1-13) can be reformulated as an equation (1-1) of type 2, and vice versa. See Proposition 2.6. Motivated by this, in Theorem 2.5, we propose an approximation of a pair
$(f,\Gamma )$
via type-2 pairs
$(f^t,\Gamma ^t)$
. An analogous approximation was also considered in [Reference Gursky and Viaclovsky13, Reference Li and Li18]. Our strategy is different and new. It is based on the applications of Morse theory to fully nonlinear equations, which was first observed by the author [Reference Yuan32] (see also [Reference Yuan27, Reference Yuan33, Reference Yuan35–Reference Yuan37]).
By integrating Theorems 1.4 and 2.5, we propose a continuity method for the Dirichlet problem and apply it to establish a proof of the following result. This is a real analogue of [Reference Yuan34, Theorem 1.6].
Theorem 1.5. Let f satisfy (1-3), (1-4), (1-9), and (1-8). Suppose that there exists an admissible subsolution
$\underline {u}\in C^2(\overline {M})$
to the Dirichlet problem (1-1)–(1-2). Then, the Dirichlet problem admits a unique smooth admissible solution.
In contrast to the usual approaches, our method leverages the a priori estimates directly, eliminating the necessity of smooth approximations to the
$C^2$
-regularity admissible subsolution. In the unbounded-condition case, a similar approach also works on closed manifolds.
Definitely not too surprisingly, things become more subtle in the degenerate case
$$ \begin{align} \begin{aligned} \inf_M \psi=\sup_{\partial\Gamma}f. \end{aligned} \end{align} $$
The optimal regularity one can expect is
$C^{1,1}$
. For a type-2 degenerate equation, we derive a
$C^{1,1}$
-weak solution to the Dirichlet problem. To this end, we further assume
$$ \begin{align} \begin{aligned} f\in C^\infty(\Gamma)\cap C(\overline{\Gamma}) \quad \text{where } \overline{\Gamma} =\Gamma\cup\partial\Gamma. \end{aligned} \end{align} $$
Theorem 1.6. Assume (1-3), (1-4), (1-12), (1-14), and (1-15) hold. Then, the Dirichlet problem (1-1)–(1-2) possesses a weak solution
$u\in C^{1,1}(\overline {M})$
with
$\Delta u \in L^{\infty }(\overline {M})$
and
$\lambda (\nabla ^2 u+\chi )\in \overline {\Gamma }$
in
$\overline {M}$
.
Remark 1.7. Theorems 1.4 and 1.6 allow (1-6) when
$\mathring { {\Gamma }}_{\mathcal {G}}^{ {f}}\neq \Gamma _n$
, in which case
$\varrho _{ {\Gamma }_{\mathcal {G}}^{ {f}}}>1$
.
One of the key ingredients in deriving (1-10) is to derive the gradient estimate
$$ \begin{align*} \begin{aligned} \sup_M |\nabla u|\leq C. \end{aligned} \end{align*} $$
For (1-1) with
$\chi =g$
, the gradient estimate has been studied by Li [Reference Li20] when the sectional curvature of g is nonnegative, and by Urbas [Reference Urbas26] when replacing such a curvature condition by certain extra assumptions on f including
$$ \begin{align*} \begin{aligned} f_i(\lambda)\geq \delta \sum f_j(\lambda) \quad\text{if } \lambda_i\leq 0. \end{aligned} \end{align*} $$
See also [Reference Guan9]. In 2020, the author [Reference Yuan31] confirmed such a key inequality when f satisfies (1-7), thereby extending (1-10) to a large number of real Hessian equations. A straightforward proof of the gradient estimate for (1-1) was carried out by Guan [Reference Guan10] under a rather weak assumption. The above-mentioned gradient estimate is based on the Bernstein method. Another effective method is the blow-up argument. Such an argument was developed by Chen [Reference Chen3] for the Dirichlet problem of a complex Monge–Ampère equation, and by Dinew and Kołodziej [Reference Dinew and Kołodziej4], building on Hou et al.’s [Reference Hou, Ma and Wu15] second-order estimate of the form
$$ \begin{align} \begin{aligned} \sup_{M}|\Delta u|\leq C\Big(1+\sup_M|\nabla u|^2\Big), \end{aligned} \end{align} $$
in the case of complex k-Hessian equations on closed (compact without boundary) Kähler manifolds. Recently, under the extra assumption (1-7), the blow-up argument was extended extensively by Székelyhidi [Reference Székelyhidi23] to Hessian equations on closed Hermitian manifolds. The technical assumption (1-7) was subsequently removed by the author [Reference Yuan34], thereby extending Székelyhidi’s
$C^{2,\alpha }$
-estimate to the optimal setting. Moreover, in the unbounded case (1-8), the author [Reference Yuan34] also established a prior estimate (1-16) and then solved the Dirichlet problem of Hessian equations on complex manifolds. In this paper, we employ the strategy of [Reference Yuan34] to derive a gradient estimate in the type-1 case.
The rest of this paper is organized as follows. In Section 2, we discuss the structure of a pair and then propose an approximation of the pair
$(f,\Gamma )$
by type-2 pairs. In Section 3, we prove Theorems 1.4 and 1.6 based on Morse theory and the characterization of type-2 pairs. In Section 4, we establish (1-16) and (1-10). In Section 5, we propose a continuity method for the Dirichlet problem and employ it to give a proof of Theorem 1.5. In Appendix A, we summarize some useful lemmas.
2 Approximate Hessian equations using type-2 equations
2.1 Structure of a type-2 pair
We review a property of
$ \Gamma _{\mathcal {G}}^{f}$
proposed in [Reference Yuan34].
Lemma 2.1 [Reference Yuan34]
Suppose that f satisfies (1-3) in
$\Gamma $
. Then,
$$ \begin{align} \begin{aligned} \sum_{i=1}^n f_i(\lambda)\mu_i \geq 0 \quad\text{and }\quad f(\lambda+\mu)\geq f(\lambda) \quad \text{for all } \lambda\in\Gamma, \quad \text{for all } \mu\in \Gamma_{\mathcal{G}}^{f}. \end{aligned} \end{align} $$
Proof. We start with an observation
$$ \begin{align} \begin{aligned} \sum_{i=1}^n f_i(\lambda)\mu_i \geq \limsup_{t\rightarrow+\infty} f(t\mu)/t \quad \text{for all } \lambda, \, \mu\in\Gamma. \end{aligned}\end{align} $$
This inequality follows from the concavity of f. Moreover, by (1-3), we get
$$ \begin{align} \begin{aligned} \sum_{i=1}^n f_i(\lambda)(\mu_i-\lambda_i)\geq f(\mu)-f(\lambda) \quad \text{for all } \lambda, \, \mu \in \Gamma. \end{aligned} \end{align} $$
By (2-2),
$\sum f_i(\lambda )\mu _i \geq 0\, \text { for all }\lambda \in \Gamma $
,
$\text { for all }\mu \in \Gamma _{\mathcal {G}}^{f}$
. Combining (2-3), we get
${f(\lambda +\mu )\geq f(\lambda )}$
.
As in the introduction,
$\vec {\mathbf {1}}=(1,\ldots ,1).$
According to [Reference Yuan36, Theorem 5.6], we obtain the following lemma.
Lemma 2.2 [Reference Yuan36]
In addition to (1-3), we assume
$(-\delta _0,0,\ldots ,0,1)\in \mathring {\Gamma }_{\mathcal {G}}^f$
for some
${\delta _0>0}$
. Then,
$$ \begin{align} \begin{aligned} f_i(\lambda)\geq \frac{\delta_0}{n}\sum_{j=1}^n f_j(\lambda) \quad\text{in } \Gamma \quad \text{for all } 1\leq i\leq n. \end{aligned} \end{align} $$
Moreover,
$\underset {t\rightarrow +\infty } \lim f(\lambda _1,\ldots ,\lambda _{n-1},\lambda _n+t)=\sup _\Gamma f \text { for all } \lambda =(\lambda _1,\ldots , \lambda _n)\in \Gamma .$
Remark 2.3. When f further satisfies (1-7), the result was obtained in [Reference Yuan31].
Proof. Suppose
$\lambda _1 \leq \cdots \leq \lambda _n$
. The concavity and symmetry of f implies
$f_n\leq \cdots \leq f_1$
and
$f_1\geq ({1}/{n})\sum f_j$
. By Lemma 2.1,
$f_n\geq \delta _0 f_1\geq ({\delta _0}/{n})\sum f_j$
. Finally, for
$R_0>0$
from
$(0,\ldots ,0,1)\in \mathring {\Gamma }_{\mathcal {G}}^f$
, one can find
$t\gg 1$
so that
$t(0,\ldots ,0,1)+\lambda -R_0\vec {\mathbf {1}}\in \mathring {\Gamma }_{\mathcal {G}}^f$
. Thus,
$ f(\lambda _1,\ldots ,\lambda _{n-1},\lambda _n+t) \geq f(R_0\vec {\mathbf {1}})$
according to (2-1) from Lemma 2.1.
2.2 Structure of a pair and an application to approximation
The results of this subsection are based on an observation. The observation is noted in [Reference Yuan36] and closely aligns with the results presented in [Reference Yuan35, Section 4]. Fix a pair
$({f},{\Gamma })$
. Let
$\varrho _{{\Gamma }_{\mathcal {G}}^f}$
be as in (1-11), and pick a positive constant
$\varrho $
with
$n-\varrho _{{\Gamma }_{\mathcal {G}}^f} < \varrho <n$
. Define
$\mu ^{\lambda } := \sum _{j=1}^n \lambda _j \vec {\mathbf {1}}-\varrho \lambda \, \text { for all } \lambda \in \mathbb {R}^n.$
Let P be a linear map from
$\mathbb {R}^n$
to
$\mathbb {R}^n$
with
$P(\lambda )= \mu ^\lambda .$
Since
$ 0<\varrho <n$
, the linear map is well defined and invertible, and
$P^{-1}(\mu )=({1}/{\varrho (n-\varrho )})(\sum _{j=1}^n \mu _j \vec {\mathbf {1}}-(n-\varrho )\mu )$
. Take
$\tilde {\Gamma }:=P({\Gamma })$
and define
$ \tilde {f}: \tilde {\Gamma } \to \mathbb {R}$
by
$$ \begin{align*} \begin{aligned} \tilde{f}(\mu^\lambda)= {f}(\lambda). \end{aligned}\end{align*} $$
So,
$\tilde {\Gamma }$
is an open, convex, symmetric cone with vertex at origin. We prove the following lemma, which is noted in [Reference Yuan36].
Lemma 2.4. Given a pair
$({f},{\Gamma })$
obeying (1-3) and (1-4), let
$\varrho $
,
$\tilde {\Gamma }$
, and
$\tilde {f}$
be as above. Then,
$\tilde {\Gamma }$
is an open symmetric convex cone with vertex at the origin and
$$ \begin{align*} \Gamma_n\subseteq \mathring{\tilde{\Gamma}}_{\mathcal{G}}^{\tilde{f}}\subseteq \tilde{\Gamma}\subseteq \Gamma_1, \quad \varrho\leq \varrho_{\tilde{\Gamma}_{\mathcal{G}}^{\tilde{f}}},\end{align*} $$
and
$\tilde {f}$
satisfies (1-3) and (1-4) in
$\tilde {\Gamma }$
. Moreover,
$(\tilde {f},\tilde {\Gamma })$
is of type 2 and we have the following.
-
(1) If
$(0,\ldots ,0,1)\in \mathring {{\Gamma }}_{\mathcal {G}}^{{f}}$
, then
$\varrho <\varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}.$
-
(2) If
$(0,\ldots ,0,1)\in \partial {{\Gamma }}_{\mathcal {G}}^{{f}}$
, then
$\varrho =\varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}.$
Proof. From
$\sum _{j=1}^n \mu _j^{\lambda }=(n-\varrho ) \sum _{j=1}^n\lambda _j$
, we see
$\tilde {\Gamma }\subseteq \Gamma _1.$
By
$ n-\varrho _{{\Gamma }_{\mathcal {G}}^f}< \varrho $
, we know
$(1,\ldots ,1,1-n+\varrho )\in \mathring {\Gamma }_{\mathcal {G}}^f,$
thereby
$(0,\ldots ,0,1)\in \mathring {\tilde {\Gamma }}_{\mathcal {G}}^{\tilde {f}}$
. By
${\tilde {\Gamma }}_{\mathcal {G}}^{\tilde {f}}=P({\Gamma }_{\mathcal {G}}^f)$
and
${(0,\ldots ,0,1)\in \overline {{\Gamma }}_{\mathcal {G}}^{{f}}}$
, we get
$(1,\ldots ,1,1-\varrho )\in \overline {\tilde {\Gamma }}_{\mathcal {G}}^{\tilde {f}}$
. Thus,
$\varrho \leq \varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}.$
However, we can check the concavity of
$\tilde {f} $
in
$\tilde {\Gamma }$
, thanks to
$$ \begin{align*} & \tilde{f}(t\mu^{\lambda}+(1-t)\mu^{\lambda'}) = f(t\lambda+(1-t)\lambda') \geq tf(\lambda)+(1-t)f(\lambda')\\ & \quad=t\tilde{f}(\mu^{\lambda}) + (1-t)\tilde{f}(\mu^{\lambda'}) \quad \text{for all } 0<t<1, \quad \text{for all } \mu^{\lambda},\,\mu^{\lambda'}\in \tilde{\Gamma}. \end{align*} $$
Finally, we see
$(\tilde {f},\tilde {\Gamma })$
satisfies (1-4) thanks to
$$ \begin{align*} \frac{\partial \tilde{f}}{\partial \mu_i^\lambda} =\frac{1}{\varrho(n-\varrho)} \sum_{j=1}^n \frac{\partial f}{\partial\lambda_j}-\frac{1}{\varrho} \frac{\partial f}{\partial\lambda_i}> \frac{1}{\varrho \varrho_{\Gamma_{\mathcal{G}}^f}} \bigg(\sum_{j=1}^n \frac{\partial f}{\partial\lambda_j}- \varrho_{\Gamma_{\mathcal{G}}^f} \frac{\partial f}{\partial\lambda_i}\bigg)\geq0. \end{align*} $$
The above two inequalities follow from
$0< n- \varrho <\varrho _{{\Gamma }_{\mathcal {G}}^f}$
and Lemma 2.1.
Theorem 2.5. Given a type-1 pair
$(f,\Gamma )$
with
$(0,\ldots ,0,1)\in \partial {\Gamma }_{\mathcal {G}}^f$
, let
$(\tilde {f}, \tilde {\Gamma })$
be the pair as asserted in Lemma 2.4. For
$t\geq 1$
, denote
$$ \begin{align} \begin{aligned} {f}^t(\lambda)=\tilde{f} \bigg(t\sum_{i=1}^n \lambda_i\vec{\mathbf{1}}-\varrho_{\tilde{\Gamma}_{\mathcal{G}}^{\tilde{f}}} \lambda\bigg), \quad {\Gamma}^t= \bigg\{\lambda\in\mathbb{R}^n: t\sum_{i=1}^n \lambda_i\vec{\mathbf{1}}-\varrho_{\tilde{\Gamma}_{\mathcal{G}}^{\tilde{f}}} \lambda \in\tilde{\Gamma} \bigg\}. \end{aligned} \end{align} $$
Then, we have the following.
-
(1)
$\Gamma \subseteq \Gamma ^t$
and
$\Gamma _{\mathcal {G}}^f\subseteq {\Gamma ^t}_{\mathcal {G}}^{f^t}$
for
$t>1$
. When
$t=1$
,
$({f}^t,{\Gamma }^t)$
coincides with
$({f},\Gamma )$
. -
(2) For each
$t>1$
,
$(f^t,\Gamma ^t)$
is a type-2 pair, that is,
$(0,\ldots ,0,1)\in \mathring {\Gamma ^t}_{\mathcal {G}}^{f^t}$
. -
(3)
${f}^t(\lambda )\geq {f}(\lambda ) \, \text { for all } \lambda \in {\Gamma }$
,
$\text { for all } t\geq 1$
.
From this, each equation of the form (1-1) can be reformulated as in the following form:
$$ \begin{align*} \begin{aligned} \tilde{f}(\lambda[ \Delta u \cdot g-\varrho\nabla^2u +\tilde{\chi}])=\psi, \quad \varrho\neq0, \end{aligned} \end{align*} $$
and vice versa.
Proposition 2.6. Fix a pair
$(f,\Gamma )$
and pick a positive constant
$\varrho $
with
${n-\varrho _{{\Gamma }_{\mathcal {G}}^f} {\kern-1pt}\leq{\kern-1pt} \varrho {\kern-1pt}<{\kern-1pt}n}$
. Let
$(\tilde {f},\tilde {\Gamma })$
be as above such that
$\varrho \leq \varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}.$
Given a symmetric
$(0,2)$
-tensor
$\chi $
, denote
$\tilde {\chi }:=(\mathrm {tr}_g\chi )\cdot g-\varrho \, \chi .$
Then, for any
$C^2$
-function u with
$\lambda [\nabla ^2u+\chi ]\in \Gamma $
,
$$ \begin{align*} \begin{aligned} {f}(\lambda[ \nabla^2u+\chi ])=\tilde{f}(\lambda[ \Delta u \cdot g-\varrho\nabla^2u +\tilde{\chi}]). \end{aligned} \end{align*} $$
Moreover,
$(0,\ldots ,0,1)\in \mathring {{\Gamma }}_{\mathcal {G}}^{{f}} \iff \varrho <\varrho _{\tilde {\Gamma }_{\mathcal {G}}^{\tilde {f}}}.$
3 On the type 2 equations
3.1
$C^0$
-estimate and construction of admissible functions
We use Morse theory to construct admissible functions. The strategy follows some ideas of [Reference Yuan32] (see also [Reference Yuan33, Reference Yuan36]). For our purpose, we need the following lemma.
Lemma 3.1. Let
$\overline {M}$
be a compact connected manifold of real dimension
$n\geq 2$
with smooth boundary. Then, there is a smooth function v without any critical points.
Proof. We include a proof, following [Reference Yuan32]. Let X be the double of
$\overline {M}$
. It is obtained by gluing two copies of
$\overline {M}$
together along their common boundary. Let w be a smooth Morse function on X with the critical set
$\{p_i\}_{i=1}^{m+k}$
, among which
$p_1,\ldots , p_m$
are all the critical points being in
$\overline {M}$
. Pick
$q_1,\ldots , q_m\in X\setminus \overline {M}$
, but not the critical point of w. By the homogeneity lemma (see [Reference Milnor21]), one can find a diffeomorphism
$h: X\to X$
, which is smoothly isotopic to the identity, such that
$h(p_i)=q_i$
for
$1\leq i\leq m$
and
$h(p_i)=p_i$
for
$m+1\leq i\leq m+k$
. Then,
$v=w\circ h^{-1}|_{\overline {M}}$
is the desired function.
By Lemma 3.1, we can take a smooth function v with
$v\geq 1$
and
${d} v\neq 0$
on
$\overline {M}$
. Let
$$ \begin{align} \begin{aligned} \underline{v}=e^{tv}. \end{aligned} \end{align} $$
The straightforward computation yields that
$$ \begin{align*} \chi+\nabla^2 \underline{v} =\chi +te^{tv} (\nabla^2 v+t\,{d}v\otimes {d}v). \end{align*} $$
Let
$R_0$
be a positive constant such that
$f(R_0\vec {\mathbf {1}})>\psi \text { in }\overline {M}.$
By
$(0,\ldots ,0,1)\in \mathring {{\Gamma }}_{\mathcal {G}}^{{f}}$
, we may pick
$t\gg 1$
such that
$$ \begin{align*} \begin{aligned} \lambda\bigg[\chi +te^{tv} \bigg( \nabla^2 v+\frac{t}{2}{\,{d}} v\otimes {\,{d}} v\bigg)\bigg]\in\mathring{{\Gamma}}_{\mathcal{G}}^{{f}}, \,\, \lambda\bigg[\frac{t^2 e^{tv}}{2} {\,{d}} v\otimes {\,{d}} v -R_0 g\bigg] \in\mathring{{\Gamma}}_{\mathcal{G}}^{{f}}. \end{aligned} \end{align*} $$
In particular,
$$ \begin{align} \begin{aligned} \lambda[ \nabla^2 \underline{v}+\chi]\in \mathring{\Gamma}_{\mathcal{G}}^f \quad\text{in }\overline{M}. \end{aligned} \end{align} $$
Using Lemma 2.1 twice, we conclude that
$$ \begin{align} \begin{aligned} f( \lambda[ \nabla^2 \underline{v}+\chi]) \geq f\bigg(\lambda\bigg[\frac{t^2 e^{tv}}{2} {\,{d}} v\otimes {\,{d}} v\bigg]\bigg) \geq f(R_0\vec{\mathbf{1}})>\psi. \end{aligned} \end{align} $$
The comparison principle then implies that
$$ \begin{align} \inf_{\overline{M}} (u-\underline{v})= \inf_{\partial M} (\varphi-\underline{v}). \end{align} $$
The upper bound of u is standard. Let h be the solution to
$$ \begin{align*} \begin{aligned} \Delta h +\mathrm{tr}_g \chi=0 \quad \text{ in } \overline{M}, \quad h= \varphi \quad\text{on } \partial M. \end{aligned} \end{align*} $$
The existence and regularity of h can be found in standard textbooks; see [Reference Gilbarg and Trudinger6, Theorem 6.14] and [Reference Aubin1, Theorem 4.8]. Then,
$$ \begin{align} u \leq h \quad \text{ in } \overline{M}, \quad u= h= \varphi \quad \text{ on } \partial M. \end{align} $$
3.2 Local and global estimates for type-2 equations
Proposition 3.2. In addition to (1-3), (1-4), and (1-9), we assume
$(-\delta _0,0,\ldots ,0,1)\in \mathring {\Gamma }_{\mathcal {G}}^f$
for some
$\delta _0>0$
. Let
$B_r\subset M$
be a geodesic ball with radius
$r>0$
. Let
$u\in C^4(M)\cap C^2(\overline {M})$
be an admissible solution to (1-1). Then,
$$ \begin{align} \begin{aligned} \sup_{B_{r/2}} (|\nabla^2u|+|\nabla u|^2)\leq C_1, \end{aligned} \end{align} $$
where
$C_1$
depends on
$\delta _0^{-1}$
,
$r^{-1}$
,
$|u|_{C^0(B_r)}$
,
$|\psi |_{C^2(B_r)}$
, and geometric quantities on
$B_r$
. Moreover,
$$ \begin{align} \begin{aligned} \sup_{M}(|\nabla^2u|+|\nabla u|^2)\leq C_2\Big(1+\sup_{\partial M}|\nabla^2u|+\sup_{\partial M}|\nabla u|^2\Big), \end{aligned} \end{align} $$
where
$C_2$
depends on
$\delta _0^{-1}$
,
$|u|_{C^0(M)}$
,
$|\psi |_{C^2(\overline {M})}$
, and other known data. Furthermore, both
$C_1$
and
$C_2$
are independent of
$\delta _{\psi ,f}^{-1}$
, where and hereafter
$$ \begin{align} \begin{aligned} \delta_{\psi,f}:=\inf_M\psi-\sup_{\partial\Gamma} f. \end{aligned} \end{align} $$
Proof. By Lemma 2.2,
${f}$
is of fully uniform ellipticity, that is, f satisfies (2-4). As in [Reference Guan8] (see also [Reference Yuan31, Reference Yuan32]), we can derive local estimates (3-6) for second- and first-order derivatives. When taking the cutoff function
$\eta \equiv 1$
in the proof, we get (3-7). Since the proof is somewhat standard, we omit the details here.
3.3 Boundary estimate for first derivatives
Let
$\sigma $
be the distance function to
$\partial M$
, let
$M_{\delta }:=\{z\in M: \sigma (z)<\delta \}$
. In the following, we use
$\sigma $
to construct a local
$\mathring {{\Gamma }}_{\mathcal {G}}^{{f}}$
-admissible function
$\underline {w}$
subject to
$$ \begin{align} \begin{aligned} f(\lambda[\nabla^2 \underline{w}+\chi])>\psi \quad\text{ in } M_{\delta_0}, \end{aligned} \end{align} $$
$$ \begin{align} \begin{aligned} u\geq \underline{w} \quad \text{ in } M_{\delta_0}, \,\, \underline{w}=\varphi \quad\text{on } \partial M \end{aligned} \end{align} $$
for some
$\delta _0>0$
. As in [Reference Guan8], take
$ w(z)= 2\log ({\delta ^2}/{\delta ^2 +\sigma (z)})$
. The computation gives
$$ \begin{align*} \begin{aligned} \chi+\nabla^2 (w+\varphi) = \chi +\nabla^2\varphi+\frac{2{\,{d}} \sigma\otimes {\,{d}} \sigma}{(\delta^2+ \sigma)^2}-\frac{2\nabla^2\sigma}{\delta^2+\sigma}. \end{aligned} \end{align*} $$
Notice
$|\nabla \sigma | =1$
on
$\partial M$
and
$(0,\ldots ,0,1)\in \mathring {{\Gamma }}_{\mathcal {G}}^{{f}}$
. Take
$0<\delta _1\ll 1$
so that
$$ \begin{align*} \begin{aligned} \lambda\bigg[\chi+\nabla^2\varphi+ \frac{{\,{d}}\sigma \otimes {\,{d}}\sigma}{(\delta^2+ \sigma)^2}-\frac{2\nabla^2\sigma }{\delta^2+ \sigma}\bigg] \in \mathring{{\Gamma}}_{\mathcal{G}}^{{f}} \quad\text{in } M_{\delta} \end{aligned} \end{align*} $$
for
$0<\delta \leq \delta _1$
. By Lemma 2.1, there is
$0<\delta _2\leq \delta _1$
such that for
$0<\delta \leq \delta _2$
,
$$ \begin{align*} \begin{aligned} f(\lambda[\chi+\nabla^2 (w+\varphi)]) \geq & f\bigg(\lambda\bigg[\frac{{\,{d}}\sigma\otimes{\,{d}}\sigma}{(\delta^2+ \sigma)^2}\bigg]\bigg)>\psi \quad\text{ in } M_{\delta}. \end{aligned} \end{align*} $$
From (3-4), there is a positive uniform constant
$\delta _3$
such that
$$ \begin{align*} \begin{aligned} \underset{M}\inf(u-\varphi)\geq 2\log\frac{\delta_3}{1+\delta_3}. \end{aligned} \end{align*} $$
Denote
$\delta _0=\min \{\delta _2,\delta _3\}$
and take
$$ \begin{align} \begin{aligned} \underline{w}=2\log \frac{\delta_0^2}{\delta_0^2 +\sigma}+\varphi. \end{aligned} \end{align} $$
Consequently, the comparison principle yields that
$$ \begin{align*} \begin{aligned} \inf_{M_{\delta_0}}(u - \underline{w}) = \inf_{\partial M_{\delta_0}} (u-\underline{w}) \geq 0. \end{aligned} \end{align*} $$
Thus,
$\underline {w}$
satisfies (3-10) and (3-9) in
$M_{\delta _0}.$
From (3-5) and (3-10),
$$ \begin{align} \begin{aligned} |\nabla u|\leq C \quad\text{on } \partial M. \end{aligned} \end{align} $$
3.4 Boundary estimate for second derivatives
Now, we review some notation.
For
$x_0\in \partial M$
, we choose a smooth local orthonormal frame
$e_1,\ldots , e_n$
around
$x_0$
such that
$e_n$
is the unit inner normal vector field when restricted to
$\partial M$
. Denote
$ g_{ij} =g(e_i,e_j)$
and
$\{g^{ij} \} = \{g_{ij} \}^{-1}.$
Under the Levi–Civita connection of g,
$\nabla _{e_i}e_j=\Gamma _{ij}^k e_k$
and
$\Gamma _{ij}^k$
denote the Christoffel symbols. For simplicity, we write
$$ \begin{align*}\nabla_i=\nabla_{e_i}, \nabla_{ij}=\nabla_i\nabla_j-\Gamma_{ij}^k\nabla_k,\quad \nabla_{ijk}=\nabla_i\nabla_{jk}-\Gamma_{ij}^l\nabla_{lk}-\Gamma^l_{ik}\nabla_{jl},\ldots, \text{ and so forth}.\end{align*} $$
The linearized operator of (1-1) at the solution u, say
$\mathcal {L}$
, is locally given by
$$ \begin{align} \mathcal{L} w=F^{i j} \nabla_{i j} w, \end{align} $$
where
$\mathfrak {g}_{i j}=\chi _{i j}+\nabla _{i j} u$
and
$F^{i j}=({\partial F(\mathfrak {g})}/{\partial \mathfrak {g}_{i j}})$
.
Let
$\underline {w}$
be as in (3-11), a local
$\mathring {{\Gamma }}_{\mathcal {G}}^{{f}}$
-admissible function obeying (3-10) and (3-9).
Applying such a local barrier function
$\underline {w}$
to Lemma A.2, there is a positive constant
$\varepsilon $
(independent of
$\delta _{\psi ,f}$
) such that either
$$ \begin{align} \begin{aligned} F^{i j} (\nabla_{ij}\underline{w} -\nabla_{ij}u ) \geq \varepsilon F^{i j}g_{i j} +\varepsilon \end{aligned} \end{align} $$
or
$$ \begin{align} \begin{aligned} F^{i j}\geq \varepsilon (1+F^{p q}g_{p q})g^{i j}. \end{aligned} \end{align} $$
Proposition 3.3. Let
$\delta _{\psi ,f}$
be as in (3-8). In addition to (1-3), (1-4), and (1-9), we assume
$(-\delta _0,0,\ldots ,0,1)\in \mathring {\Gamma }_{\mathcal {G}}^f$
for some
$\delta _0>0$
. Let
$u\in C^4(M)\cap C^2(\overline {M})$
be an admissible solution to the Dirichlet problem (1-1)–(1-2). Then,
$$ \begin{align} \begin{aligned} \sup_{\partial M}|\nabla^2u|\leq C \quad\text{independent of } \delta_{\psi,f}^{-1}, \end{aligned} \end{align} $$
where C is a positive constant depending only on
$\delta _0^{-1}$
,
$|u|_{C^0(M)}$
,
$|\nabla u|_{C^0(\partial M)}$
,
$|\underline {w}|_{C^2(M_{\delta _0})}$
,
$|\varphi |_{C^3(\partial M)}$
, and other known data.
Proof. The proof of the boundary estimate for pure tangential derivatives
$$ \begin{align} \begin{aligned} |\nabla_{\alpha \beta}u(x_0)|\leq C_5 \quad \text{for all } \alpha, \beta\leq n-1 \end{aligned} \end{align} $$
is standard by the boundary value condition. Here,
$C_5$
depends only on
$|\varphi |_{C^2(\partial M)}$
,
$|u|_{C^1(\partial M)}$
, and geometric quantities of
$\partial M$
.
By a minor modification of [Reference Yuan28, Proposition 3.1], we can employ (3-14) and (3-15) to derive boundary estimates for mixed derivatives
$$ \begin{align*} \begin{aligned} |\nabla_{\alpha n}u(x_0)|\leq C_6 \Big(1+\sup_M |\nabla u|\Big) \quad \text{for all } 1\leq \alpha< n \quad\text{independent of } \delta_{\psi,f}^{-1}, \end{aligned} \end{align*} $$
where
$C_6$
depends on
$|u|_{C^0(\overline {M})}$
,
$|\varphi |_{C^3(\partial M)}$
,
$|\psi |_{C^1(\overline {M})}$
, and other known data.
Let
$\underline {v}$
be as in (3-1). From (2-3) and (3-3), we get
$$ \begin{align*} F^{n n} \nabla_{n n}u(x_0)\leq -\sum_{i+j<2n} F^{i j}\nabla_{i j} u(x_0)+\sum_{i,j=1}^nF^{i j} \nabla_{i j} \underline{v}(x_0). \end{align*} $$
By Lemma 2.2 or (2-4),
$F^{n n}\geq ({\delta _0}/{n}) \sum _{i=1}^n F^{i i}$
at
$x_0$
. Thus,
$$ \begin{align*} \begin{aligned} u_{n n}(x_0)\leq \frac{n}{\delta_0} \sum_{i+j<2n} |\nabla_{i j} u(x_0)|+ \frac{n}{\delta_0}\sum_{i,j=1}^n |\nabla_{i j} \underline{v} (x_0)|. \end{aligned} \end{align*} $$
Finally, by (3-17) and
$\Gamma \subseteq \Gamma _1$
, we see
$\nabla _{nn}u(x_0)\geq -C$
.
3.5 Completion of the proof of Theorem 1.4
Let
$\underline {v}$
be a smooth function as constructed in (3-1), satisfying (3-2) and (3-3). For each
$t\in [0,1]$
, we consider
$$ \begin{align} \begin{cases} f(\lambda[\nabla^2u^t+\chi])= t\psi+(1-t) f(\lambda[\nabla^2 \underline{v}+\chi]) & \text{in } \overline{M}, \\ u^t= t\varphi+(1-t)\underline{v} & \text{on } \partial M. \end{cases} \end{align} $$
Denote
$$ \begin{align*} \begin{aligned} \mathrm{S}=\{t\in [0,1]: \text{ (3-18) has a unique smooth admissible solution for } t \}. \end{aligned} \end{align*} $$
Obviously, for
$t=0$
, the Dirichlet problem has a unique admissible solution
$\underline {v}$
. By the implicit function theorem and the theory of linear elliptic equations,
$\mathrm {S}$
is an open subset of
$[0,1]$
. Together with the Evans–Krylov theorem and Schauder theory, (3-4), (3-5), (3-7), (3-12), and (3-16) imply that
$\mathrm {S}$
is closed. Thus,
$\mathrm {S}=[0,1]$
.
We remark that the subsolution assumption is not needed in the present proof.
3.6 Proof of Theorem 1.6
By Theorem 1.4, for each
$\delta>0$
, there is a unique smooth admissible function
$u^\delta $
satisfying
$$ \begin{align*} \begin{aligned} f(\lambda[\nabla^2 u^\delta+\chi]) =\psi+\delta \quad\text{in } \overline{M}, \quad u^\delta =\varphi \quad\text{on } \partial M. \end{aligned} \end{align*} $$
From (3-4), (3-5), (3-7), (3-12), and (3-16), we deduce that
$$ \begin{align*} \begin{aligned} \sup_{\overline{M}}|\nabla^2 u^\delta|\leq C \quad\textrm{independent of } \delta^{-1}. \end{aligned} \end{align*} $$
Take
$\delta \to 0^+$
. By passing to a proper subsequence, we may obtain a weak
$C^{1,1}$
-solution to the Dirichlet problem for a degenerate equation.
4 Quantitative second-order estimate
We establish a real analogue of [Reference Yuan34, Theorem 1.7] in the unbounded case.
Proposition 4.1. In addition to (1-3), (1-4), (1-8), and (1-9), we assume that the Dirichlet problem (1-1)–(1-2) admits an admissible subsolution
$\underline {u}\in C^2(\overline {M})$
. For any admissible solution
$u\in C^4(M)\cap C^2(\overline {M})$
to the Dirichlet problem, we have a second estimate of the quantitative form
$$ \begin{align} \begin{aligned} \sup_{\overline{M}}|\nabla^2 u| \leq C\Big(1+\sup_{ M}|\nabla u|^2\Big), \end{aligned} \end{align} $$
and the Hessian estimate (1-10).
We first summarize some notation. Fix a point
$x_0\in \partial M$
. We choose a smooth local orthonormal frame
$e_1,\ldots , e_n$
around
$x_0$
such that
$e_n$
is the unit inner normal vector field when restricted to
$\partial M$
. Denote
$$ \begin{align*}\Gamma_\infty=\{(\lambda_{1},\ldots, \lambda_{n-1}): (\lambda_{1},\ldots, \lambda_{n-1},R)\in\Gamma\text{ for some } R>0\}.\end{align*} $$
Throughout this section, the Greek letters
$\alpha , \beta $
run from
$1$
to
$n-1$
.
The main issue is to derive (4-1). Following an estimate presented in [Reference Guan10, Section 3], we can prove that
$$ \begin{align*} \begin{aligned} \sup_{\overline{M}}|\nabla^2 u| \leq C\Big(1+\sup_{M}|\nabla u|^2+\sup_{\partial M}|\nabla^2 u|\Big). \end{aligned} \end{align*} $$
To derive (4-1), it is only required to prove the quantitative boundary estimate
$$ \begin{align*} \begin{aligned} \sup_{\partial M}|\nabla^2 u| \leq C\Big(1+\sup_{ M}|\nabla u|^2\Big). \end{aligned} \end{align*} $$
From [Reference Yuan28, Proposition 3.1], we have boundary estimate for mixed derivatives
$$ \begin{align*} \begin{aligned} |\nabla_{\alpha n}(x_0)|\leq C \Big(1+\sup_M|\nabla u|\Big) \quad \text{for all } 1\leq \alpha\leq n-1. \end{aligned} \end{align*} $$
By (3-17) and
$\Gamma \subseteq \Gamma _1$
, we see
$\nabla _{nn}u(x_0)\geq -C$
. So, it suffices to prove the following lemma. This is a real analogue of [Reference Yuan34, Proposition 7.1].
Lemma 4.2. In the setting of Proposition 4.1,
$$ \begin{align*} \begin{aligned} \mathfrak{g}_{nn}(x_0) \leq C\bigg(1+\sum_{\alpha=1}^{n-1} |\mathfrak{g}_{\alpha n}(x_0)|^2\bigg). \end{aligned} \end{align*} $$
Let
$\sigma $
be as in Subsection 3.3, the distance to the boundary. At
$x_0\in \partial M$
, the boundary value condition yields
$$ \begin{align} \begin{aligned} \mathfrak{g}_{\alpha \beta}= (1-t)\underline{\mathfrak{g}}_{\alpha \beta} +\{t\underline{\mathfrak{g}}_{\alpha \beta}+ \nabla_n(u-\underline{u}) \nabla_{\alpha \beta} \sigma \}. \end{aligned} \end{align} $$
Let
$t_0$
be the first t as we decrease t from
$1$
such that
$$ \begin{align*} \begin{aligned} \lambda(t\underline{\mathfrak{g}}_{\alpha \beta}+ \nabla_n(u-\underline{u}) \nabla_{\alpha \beta} \sigma)\in\partial\Gamma_\infty \end{aligned} \end{align*} $$
at
$x_0$
, where
$\partial \Gamma _\infty $
is the boundary of
$\Gamma _\infty $
. Obviously,
$$ \begin{align*} \begin{aligned} -T_0< t_0<1 \end{aligned} \end{align*} $$
for some uniform positive constant
$T_0$
under control. Following closely the outline of [Reference Caffarelli, Nirenberg and Spruck2] (see also [Reference Li19] in the complex case), one may obtain
$$ \begin{align} \begin{aligned} (1-t_0)^{-1}\leq C,\quad\text{possibly depending on } \delta_{\psi,f}^{-1}, \end{aligned} \end{align} $$
where C is a uniform positive constant depending on
$|u|_{C^0(M)}$
,
$|\nabla u|_{C^0(\partial M)}$
,
$|\underline {u}|_{C^2(M)}$
,
$\sup _M\psi $
,
$\partial M$
up to third derivatives, and other known data.
Denote
$$ \begin{align} \begin{aligned} \underline{\lambda}'=(\underline{\lambda}^{\prime}_1,\ldots,\underline{\lambda}^{\prime}_{n-1}) \equiv \lambda( \underline{\mathfrak{g}}_{\alpha \beta}), \end{aligned} \end{align} $$
$$ \begin{align} \begin{aligned} (A_{t_0})_{\alpha \beta}= t_0\underline{\mathfrak{g}}_{\alpha \beta}+ \nabla_n(u-\underline{u}) \nabla_{\alpha \beta} \sigma, \,\, \tilde{\lambda}'=(\tilde{\lambda}_1',\ldots,\tilde{\lambda}_{n-1}') \equiv \lambda [(A_{t_0})_{\alpha \beta}]. \end{aligned} \end{align} $$
For
$R>0$
, define the matrix
$$ \begin{align*} \begin{aligned} {A}(R) = \begin{pmatrix} \mathfrak{{g}}_{\alpha \beta} &\mathfrak{g}_{\alpha n}\\ \mathfrak{g}_{n \beta}& R \end{pmatrix}. \end{aligned} \end{align*} $$
By (4-2), we can decompose
$A(R)$
as follows:
$$ \begin{align*} \begin{aligned} {A}(R) = A'(R)+A"(R), \end{aligned} \end{align*} $$
where
$$ \begin{align*}A'(R)=\begin{pmatrix} (1-t_0)(\mathfrak{\underline{g}}_{\alpha \beta}-\dfrac{\varepsilon_0}{4} \delta_{\alpha\beta}) &\mathfrak{g}_{\alpha n}\\ \mathfrak{g}_{n \beta}& R/2 \end{pmatrix}, \quad A"(R)=\begin{pmatrix} (A_{t_0})_{\alpha \beta}+\dfrac{(1-t_0)\varepsilon_0}{4} \delta_{\alpha\beta} &0\\ 0& R/2 \end{pmatrix}.\end{align*} $$
Moreover, we need the following lemma, which was proved by [Reference Yuan34] in the complex case.
Lemma 4.3. There exist uniform constants
$R_3$
and
$C_{A"}$
depending on
$(1-t_0)^{-1}$
,
$\varepsilon _0^{-1}$
, f, and other known data under control, such that
$$ \begin{align} \begin{aligned} f(2\lambda[A"(R_3)] )\geq -C_{A"}. \end{aligned} \end{align} $$
Proof. We include the proof, following [Reference Yuan34]. Let
$\underline {\lambda }'$
and
$\tilde {\lambda }'$
be as in (4-4) and (4-5), respectively. One can see that there is a uniform constant
$C_0>0$
depending on
$|t_0|$
,
$\sup _{\partial M}|\nabla u|$
, and other known data, such that
$|\tilde {\lambda }'|\leq C_0$
, that is,
$$ \begin{align*} \begin{aligned} \tilde{\lambda}'\in {K}\equiv \{\lambda'\in \overline{\Gamma}_\infty: |\lambda'|\leq C_0\}. \end{aligned} \end{align*} $$
There is a positive constant
$R_2$
depending on
$((1-t_0)\varepsilon _0)^{-1}$
, K, and other known data, such that
$$ \begin{align*} \begin{aligned} \lambda \left[\begin{pmatrix} (A_{t_0})_{\alpha \beta}+\dfrac{(1-t_0)\varepsilon_0}{8} \delta_{\alpha\beta} &0\\ 0& R_2/2 \end{pmatrix}\right] \in \Gamma. \end{aligned} \end{align*} $$
Let
$R_3=R_2+ ({(1-t_0)\varepsilon _0}/{4})$
. So,
$$ \begin{align} \begin{aligned} \lambda[A"(R_3)] \in \Gamma+\frac{(1-t_0)\varepsilon_0}{8} \vec{\mathbf{1}}. \end{aligned} \end{align} $$
Notice that
$A"(R_3)$
is bounded. There exists a uniform constant
$C_{A"}$
depending on
$(1-t_0)^{-1}$
,
$\varepsilon _0^{-1}$
,
$R_3$
and f, such that (4-6) holds.
Proof of Lemma 4.2
We follow closely [Reference Yuan29, Reference Yuan30, Reference Yuan34]. By (1-8) and (4-3), there is a uniform positive constant
$R_1$
such that
$$ \begin{align} \begin{aligned} f(2(1-t_0)(\underline{\lambda}^{\prime}_{1}-{\varepsilon_0}/2),\ldots, 2(1-t_0)(\underline{\lambda}^{\prime}_{n-1} -{\varepsilon_0}/2), 2R_1) \geq 2f(\underline{\lambda})+ {C_{A"}} \end{aligned} \end{align} $$
and
$(\underline {\lambda }^{\prime }_1-\varepsilon _{0}/2,\ldots , \underline {\lambda }^{\prime }_{n-1}-\varepsilon _0/2, {(1-t_0)^{-1}}{R_1})\in \Gamma .$
Let us pick
$\epsilon = ({(1-t_0)\varepsilon _0}/{4})$
in Lemma A.1. Then, we set
$$ \begin{align*} \begin{aligned} R_c\,&\equiv \frac{8(2n-3)}{(1-t_0)\varepsilon_0}\sum_{\alpha=1}^{n-1} | \mathfrak{g}_{\alpha n}|^2 + 2(n-1) (1-t_0) \sum_{\alpha=1}^{n-1}|{\underline{\lambda}_\alpha'}|\\ \,&\quad +\frac{n(n-1)(1-t_0)\varepsilon_0}{2} +2R_1+2R_2, \end{aligned} \end{align*} $$
where
$\varepsilon _0$
,
$R_1$
, and
$R_2$
are the constants we fixed in (4-8) and (4-7). According to Lemma A.1, the eigenvalues
$\lambda [{A}'(R_c)]$
(possibly with an appropriate order) behave like
$$ \begin{align} \begin{aligned} \lambda_{\alpha}[{A}'(R_c)]\geq (1-t_0) (\underline{\lambda}^{\prime}_{\alpha}-{\varepsilon_0}/{2}), \, 1\leq \alpha\leq n-1; \,\, \lambda_{n}[{A}'(R_c)]\geq R_c/2. \end{aligned} \end{align} $$
In particular,
$\lambda [{A}'(R_c)]\in \Gamma $
. So,
$\lambda [A(R_c)]\in \Gamma $
.
Using the concavity of f, Lemma 4.3, and (4-8), (4-9), we deduce that
$$ \begin{align*} \begin{aligned} f(\lambda[A(R_c)])\geq \tfrac{1}{2}f(2\lambda[A'(R_c)])+\tfrac{1}{2}f(2\lambda[A"(R_c)]) \geq f(\underline{\lambda})\geq \psi. \end{aligned} \end{align*} $$
Thus,
$\mathfrak {g}_{n n} \leq R_c.$
By (4-3),
$(1-t_0)^{-1}$
is bounded. The proof is now complete.
Proof of Proposition 4.1
By (3-5) and the subsolution assumption,
$$ \begin{align*} \sup_M|u|+\sup_{\partial M} |\nabla u|\leq C. \end{align*} $$
Based on (4-1), without using the assumption (1-7), we use the blow-up argument developed in [Reference Yuan34], extending that of [Reference Dinew and Kołodziej4, Reference Székelyhidi23] to the optimal setting, to derive the gradient estimate
$$ \begin{align*} \sup_M|\nabla u|\leq C. \end{align*} $$
(The details can also be found in [Reference Yuan28, Section 5].) In conclusion, we get (1-10).
5 A continuity method and proof of Theorem 1.5
We now give a continuity method for the Dirichlet problem (1-1)–(1-2). Our method is based on Theorems 1.4 and 2.5. Let
$(f,\Gamma )$
be the pair corresponding to (1-1). If
$(0,\ldots ,0,1)\in \mathring {\Gamma }_{\mathcal {G}}^f$
, then we are done as in Theorem 1.4. From now on, we assume
$(0,\ldots ,0,1)\in \partial {\Gamma }_{\mathcal {G}}^f$
and then take the pair
$(f^t,\Gamma ^t)$
defined as in (2-5).
For each
$t\in [1,2]$
, consider
$$ \begin{align} \begin{aligned} {f}^t (\lambda[\nabla^2 u+\chi]) =\psi \quad\text{in } \overline{M}, \quad u=\varphi \quad\text{on } \partial M. \end{aligned} \end{align} $$
When
$t>1$
,
$(f^t,\Gamma ^t)$
is of type 2 according to Theorem 2.5. Then, by Theorem 1.4, for each
$t>1$
, (5-1) has a unique smooth solution
$u^t$
with
$\lambda [\nabla ^2u^t+\chi ]\in {\Gamma }^t$
.
Let
$\underline {u}$
be an admissible subsolution to the Dirichlet problem (1-1)–(1-2). By items (3) and (1) of Theorem 2.5,
$\underline {u}$
is also an admissible subsolution to (5-1) for all
$1\leq t\leq 2$
,
$$ \begin{align*} \begin{aligned} {f}^t(\lambda[\nabla^2\underline{u}+\chi]) \geq \psi, \quad \lambda[\nabla^2\underline{u}+\chi]\in \Gamma^t \quad\text{in } \overline{M}. \end{aligned} \end{align*} $$
By (1-8), Lemma 2.2, and Theorem 2.5 (
$t>1$
),
$(f^t,\Gamma ^t)$
uniformly obeys (1-8) for
${t\in [1,2]}$
. By Proposition 4.1, for
$t\in (1,2]$
, there is some
$\alpha \in (0,1)$
such that
$$ \begin{align*} \begin{aligned} |u^t|_{C^{4,\alpha}(\overline{M})} \leq C \quad\text{independently of } t. \end{aligned} \end{align*} $$
Passing to a proper subsequence as
$t\to 1^+$
, we obtain a
$C^{4,\alpha }$
-admissible solution u to the Dirichlet problem (1-1)–(1-2). The Schauder theory implies that u is indeed smooth. The uniqueness follows from the maximum principle.
Appendix A Useful lemmas
The following lemma is a key ingredient for the proof of Proposition 4.1.
Lemma A.1 [Reference Yuan29, Reference Yuan30, Reference Yuan34]
Let A be an
$n\times n$
symmetric matrix
$$ \begin{align*}\begin{pmatrix} d_1&& &&a_{1}\\ &d_2&& &a_2\\&&\ddots&&\vdots \\ && & d_{n-1}& a_{n-1}\\ a_1& a_2&\cdots& a_{n-1}& {\mathbf{a}} \end{pmatrix} \end{align*} $$
with
$d_1,\ldots , d_{n-1}, a_1,\ldots , a_{n-1}$
fixed, and with
${\mathbf {a}}$
variable. Denote the eigenvalues of A by
$\lambda =(\lambda _1,\ldots , \lambda _n)$
. Let
$\epsilon>0$
be a fixed constant. Suppose that the parameter
${\mathbf {a}}$
in A satisfies the quadratic growth condition
$$ \begin{align*} \begin{aligned} {\mathbf{a}}\geq \frac{2n-3}{\epsilon}\sum_{i=1}^{n-1}|a_i|^2 +(n-1)\sum_{i=1}^{n-1} |d_i|+ \frac{(n-2)\epsilon}{2n-3}. \end{aligned} \end{align*} $$
Then, the eigenvalues (possibly with a proper permutation) behave like
$$ \begin{align*} \begin{aligned} |d_{\alpha}-\lambda_{\alpha}| < \epsilon \quad\text{ for all } 1\leq \alpha\leq n-1;\quad 0\leq \lambda_{n}-{\mathbf{a}} < (n-1)\epsilon. \end{aligned} \end{align*} $$
Lemma A.1 was proposed in [Reference Yuan29, Reference Yuan30]. We also refer the interested reader to [Reference Yuan34] for a detailed proof.
In the following, we also present a lemma regarding a subsolution. Denote
$\partial \Gamma ^\sigma :=\{\lambda \in \Gamma : f(\lambda )=\sigma \}.$
Lemma A.2 [Reference Yuan34]
Assume (1-3) and (1-4) hold. Fix
$\sigma \in (\sup _{\partial \Gamma }f,\sup _\Gamma f)$
. Suppose
$\mu \in \mathbb {R}^{n}$
obeys for some
$\delta $
,
$R>0$
that
$$ \begin{align} \begin{aligned} (\mu-2\delta \vec{\mathbf{1}}+\Gamma_n) \cap \partial\Gamma^\sigma \subset B_R(0). \end{aligned} \end{align} $$
Then, there is a positive constant
$\varepsilon $
depending on
$\sigma $
,
$\mu $
,
$\delta $
, R such that if
$\lambda \in \partial \Gamma ^\sigma $
, then either
$$ \begin{align} \begin{aligned} \sum_{i=1}^n f_i(\lambda) (\mu_i-\lambda_i) \geq \varepsilon \bigg(1+\sum_{i=1}^n f_i(\lambda)\bigg) \end{aligned} \end{align} $$
or
$$ \begin{align} \begin{aligned} f_j(\lambda) \geq \varepsilon \bigg(1+\sum_{i=1}^n f_i(\lambda) \bigg) \quad \text{for all } 1\leq j\leq n. \end{aligned} \end{align} $$
Proof. For self-sufficiency, we include the details, following the proof of [Reference Yuan34]. Denote
$\tilde {\mu }=\mu -\delta \vec {\mathbf {1}}$
. By (A.1), there is a positive constant
$R_0$
depending only on
$\tilde {\mu }$
and R such that
$$ \begin{align*} \begin{aligned} f(\tilde{\mu}+R_0\vec{\mathbf{e}}_i)>\sigma, \quad \tilde{\mu}+R_0\vec{\mathbf{e}}_i\in\Gamma \quad \text{for all } 1\leq i\leq n, \end{aligned} \end{align*} $$
where
$\vec {\mathbf {e}}_i$
is the i th standard basis vector in
$\mathbb {R}^n$
. Denote
$\delta _0=\min _{1\leq i\leq n}\{f(\tilde {\mu }+R_0\vec {\mathbf {e}}_i)-\sigma \}$
. In the following, we prove the conclusion for
$\varepsilon = \min \{ ({\delta }/{2}), ({\delta }/{2R_0}), ({\delta _0}/{2}), ({\delta _0}/{2R_0})\}.$
Without loss of generality, we assume
$\lambda _1\leq \cdots \leq \lambda _n$
. Then,
$f_1(\lambda )\geq \cdots \geq f_n(\lambda ).$
We get (A.3) if
$f_n(\lambda )>({\delta _0}/{2R_0}) + ({\delta }/{2R_0}) \sum _{i=1}^n f_i(\lambda ). $
From now on, we assume
$({\delta _0}/{2}) + ({\delta }/{2}) \sum _{i=1}^n f_i(\lambda ) \geq R_0f_n(\lambda ). $
Combining this assumption with (2-3),
$$ \begin{align*} \begin{aligned} \sum_{i=1}^n f_i(\lambda) (\mu_i-\lambda_i) =\,& \sum_{i=1}^n f_i(\lambda) (\tilde{\mu}_i-\lambda_i) +\delta\sum_{i=1}^n f_i(\lambda)\\ \geq \,& f(\tilde{\mu}+R_0\vec{\mathbf{e}}_n)-f(\lambda)-R_0f_n(\lambda) +\delta \sum_{i=1}^n f_i(\lambda)\\ \geq \,& \frac{\delta_0}{2}+ \frac{\delta}{2} \sum_{i=1}^n f_i(\lambda). \end{aligned}\\[-47pt] \end{align*} $$
Remark A.3. According to [Reference Yuan34, Lemma 3.1], the right-hand sides of both inequalities (A.2) and (A.3) can be strengthened to
$\epsilon (1+\sum _{i=1}^n f_i(\lambda )+|\sum _{i=1}^nf_i(\lambda )\lambda _i|).$
Acknowledgement
The author appreciates the referee’s careful reading and useful suggestions.
