2 Boundary coordinates systems and Green’s functions

2.1 Boundary coordinate systems

We begin this discussion on the compact Riemannian manifold with boundary $(M,\partial M , \bar g )$ . For $\delta>0 $ small, we define the manifold $M_\delta $ by

$$ \begin{align*} M_\delta := \{x\in M : \rho(x)>\delta\}. \end{align*} $$

On $M\setminus M_\delta $ , we have the coordinate system given by $x\mapsto (\rho (x),\theta (x))$ , where $\theta (x) \in \partial M $ is the closest point on the boundary to $x\in M\setminus M_\delta $ . Observe that if $v\in T_{(0,\theta (x))}M$ with $|v|_g = 1 $ and $v\perp _{\bar g }T_{(0,\theta (x))}\partial M $ , then $x = \exp _{(0,\theta (x))}(\rho (x) v)$ . The metric in this coordinate systems is thus given by

(2-1) $$ \begin{align} \bar g = {d} \rho^2 + \tilde \beta(\rho,\theta)\,{d} \theta^2 \end{align} $$

for some smooth function $\tilde \beta $ . Since $\dim M = 2 $ , all Riemannian metrics are conformally related [Reference Lee24, Ch. 7]. In particular, $\bar g $ is conformally related to a metric $\bar g_0 $ such that $(M\setminus M_\delta ,\bar g_0 )$ is isometric to the annulus $ (\mathbb {D}\setminus \mathbb {D}_{1-\bar \delta },g_{\mathrm {Euc}})$ with inner radius $1-\bar \delta $ and outer radius 1 for some $0<\bar \delta <1$ under some isometry $\Phi : M\setminus M_\delta \to \mathbb D \setminus \mathbb {D}_{1- \bar \delta }$ , with $\Phi (\partial M) = \partial \mathbb {D} $ .

In particular, since $\bar g $ and $\bar g_0 $ are conformally related, there is a smooth function $b(x) \in C^\infty (M)$ and a positive smooth function $\beta (x) = e^{b(x)}\in C^\infty (M)$ such that

$$ \begin{align*} \bar g = \beta\cdot \bar g_0 = e^{-b}\cdot \bar g_0. \end{align*} $$

Definition 2.1. We define the boundary defining function $\rho _0 = \Phi ^*(\varrho ) \in C^\infty (M\setminus M_\delta )$ to be the pullback under the isometric diffeomorphism $\Phi $ of the boundary defining function $\varrho \in C^\infty (\mathbb {D}\setminus \mathbb {D}_{1-\bar \delta })$ given by

$$ \begin{align*} \varrho:\mathbb{D}\setminus \mathbb{D}_{1-\bar \delta}\to \mathbb{R}_+: (r,\vartheta)\mapsto \frac{1}{2}(1-r^2) , \end{align*} $$

where $(r,\vartheta )$ are polar coordinates on $\mathbb {D} $ .

Lemma 2.2. The boundary defining function $\varrho $ vanishes on the unit circle ${\partial \mathbb {D} = \{(r,\vartheta ):r = 1 \}}$ . Furthermore, the Riemannian manifold $(\mathbb {D}\setminus \mathbb {D}_{1-\bar \delta }, g_{\mathrm {Euc}}/\varrho ^2 )$ has constant scalar curvature of $-2 $ .

Proof. The first statement of the lemma follows directly from the definition of $\varrho $ .

For the second statement, we compute the scalar curvature of conformally related surfaces using [Reference Lee24, Theorem 7.30]. We have

$$ \begin{align*} R_{g_{\mathrm{Euc}}/\varrho^2} = e^{2\log \varrho} (R_{g_{\mathrm{Euc}}} +2 \Delta_{g_{\mathrm{Euc}}}\log \varrho ) = \varrho^2\bigg(0-\frac{2}{\varrho^2}\bigg) = -2 \end{align*} $$

since the scalar curvature $R_{g_{\mathrm {Euc}}} = 0 $ .

Lemma 2.3. The metric $g_{\mathrm {Euc} }$ on $\mathbb {D}\setminus \mathbb {D}_{1-\bar \delta }$ can be expressed as

$$ \begin{align*} \begin{aligned} g_{\mathrm{Euc}} &= {d} x^2 + {d} y^2 = {d} r^2 +r ^2\,{d} \vartheta^2 \\ &= \frac{1}{1-2\varrho}{d} \varrho^2 + (1-2\varrho)\,{d} \vartheta^2 = \bigg(\sum_{n=0}^\infty (2\varrho)^n \bigg)\, {d} \varrho^2 + (1-2\varrho)\,{d} \vartheta^2. \end{aligned} \end{align*} $$

Proof. The first equality gives the Euclidean metric in polar coordinates. The second equality then follows by the chain rule for differentiation. The final equation follows by the geometric series, since the image of $\varrho $ is contained in the interval $[0,\bar \delta -\frac {1}{2}\bar \delta ^2]\subset [0,1-\delta ')$ for some $\delta '>0 $ for sufficiently small $\bar \delta $ .

Remark 2.4. The manifold $(\mathbb {D}\setminus \mathbb {D}_{1-\bar \delta }, g_{\mathrm {Euc}}/\varrho ^2)$ is our model manifold throughout this paper. We can explicitly compute on this manifold and, using the isometric diffeomorphism $\Phi $ , we can therefore compute on $M\setminus M_\delta $ .

Lemma 2.5. Near the boundary, $\bar g_0 $ has the form

$$ \begin{align*} \bar g_0 =(1+2\rho_0 + \mathcal{O}(\rho_0^2)) \,{d} \rho_0^2 +(1-2\rho_0) \,{d} \theta^2 , \end{align*} $$

where $\rho _0 $ is the boundary defining function from Definition 2.1.

Proof. Since $\Phi (\partial M) = \partial \mathbb {D}$ , the pullback $\Phi ^*({d} \vartheta ^2 )$ of part of the metric $g_{\mathrm {Euc} }$ tangent to the boundary $\partial \mathbb {D} $ remains tangent to the pullback $\Phi ^*\partial \mathbb {D} = \partial M $ of the boundary under the isometric diffeomorphism $\Phi $ . Hence,

$$ \begin{align*} \bar g_0 = \Phi^* g_{\mathrm{Euc}}& = \Phi^*\bigg(\frac{1}{1-2\varrho}\,{d} \varrho^2 + (1-2\varrho)\,{d} \vartheta^2\bigg) \\ &= (1+2\rho_0 + \mathcal{O}(\rho_0^2)) \,{d} \rho_0^2 + (1-2\rho_0)\,{d} \theta^2. \\[-35pt] \end{align*} $$

Lemma 2.6. There is a positive function $c(\theta ) \in C^\infty (\partial M) $ such that near the boundary $\partial M $ , the boundary defining functions are related by

$$ \begin{align*} \rho_0 = c(\theta)\rho + \mathcal{O}(\rho^2). \end{align*} $$

Furthermore, the factor $c^2(\theta )\beta (0,\theta ) = 1 $ for each point $\theta \in \partial M $ .

Proof. From the local description of the metrics $\bar g $ in (2-1) and $\bar g_0 $ in Lemma 2.5 near the boundary $\partial M $ , it follows that the vector fields

$$ \begin{align*} \partial\rho |_{\rho=\rho_0=0}:= \frac{\partial}{\partial \rho}\bigg|_{\rho=\rho_0=0}\quad \text{and}\quad \partial\rho_0 |_{\rho=\rho_0=0}:=\frac{\partial}{\partial \rho_0}\bigg|_{\rho=\rho_0=0} \end{align*} $$

are both perpendicular to the boundary $\partial M $ . Hence, since at every point the smooth vector fields are colinear, it follows that there is a nonzero smooth function ${c(\theta ) \in C^\infty (\partial M)} $ such that

$$ \begin{align*} \partial\rho_0 = c(\theta)\partial \rho \end{align*} $$

for every point on the boundary $\partial M $ . Positivity follows from the fact that both smooth vector fields are inward pointing. Restricted to the boundary $\partial M $ , from the local descriptions,

$$ \begin{align*} 1 = |\partial \rho|_{\bar g} = |\partial \rho_0|_{\bar g_0}. \end{align*} $$

Thus, we now compute restricted to the boundary $\partial M $ ,

$$ \begin{align*} 1 = |\partial \rho_0|_{\bar g_0} = |c(\theta) \partial \rho|_{\bar g_0} = c(\theta) \beta(0,\theta)^{1/2}|\partial \rho|_{\bar g} = c(\theta) \beta(0,\theta)^{1/2} , \end{align*} $$

from which $c^2(\theta )\beta (0,\theta ) = 1 $ follows.

Hence, the boundary value problem (1-1) simplifies to

(2-2) $$ \begin{align} \Delta_{\bar g_0 } u_\varepsilon = -\frac{c^2\beta(\rho_0,\theta)}{\rho_0^2} = -\frac{1 + \beta^{(1)}(\theta)\rho_0 + \beta^{(2)}(\rho_0,\theta)}{\rho_0^2}. \end{align} $$

2.2 Green’s functions

In this subsection, we provide, for the metric $\bar {g}_0 $ , the Green’s functions, and give a decomposition into principal and remainder parts near the boundary $\partial M $ .

There is a unique Green’s Dirichlet function $G_0(x;y) \in \mathcal {D}'(M\times M ) $ with the properties [Reference Aubin3, Theorem 4.17]

(2-3) $$ \begin{align} \begin{cases} \Delta_{\bar{g }_0,y} G_0(x;y) = \delta_x(y) \quad &\text{for } x,y\in M^\circ,\\ G_0(x;y)|_{y\in \partial M} = 0\\ G_0(x;y) \in \mathbb{R} \quad &\text{for } x,y\in \overline M,\\ G_0(x;y) = G_0(y;x). \end{cases} \end{align} $$

Since away from the diagonal $\mathrm {diag}(M\times M )$ the Green’s function $G_0(x;y)$ is smooth, we have the following lemma.

Lemma 2.7. Assume that $\delta>0 $ is chosen sufficiently small. For each fixed $x_0\in M^\circ $ , there is a positive constant $C_{x_0} $ such that

$$ \begin{align*} |G_0(x_0;y)| \leq C_{x_0} \rho_0(y) \end{align*} $$

for $y\in M^\circ $ such that $\rho _0(y) < \min \{\delta ,\frac {1}{2}\rho _0(x_0)\}$ . Moreover, for any $K\Subset M^\circ $ compact, there is a positive constant $C_K $ such that

$$ \begin{align*} |G_0(x;y)|\leq C_K \rho_0(y) \end{align*} $$

for $ x\in K , y \in M ^\circ $ with $\rho _0(y) < \min \{\delta ,\frac {1}{2}\min _{x\in K }\rho _0(x)\}$ .

Proof. For each fixed $x_0\in M^\circ $ and $y\in \partial M $ ,

$$ \begin{align*} G_0(x_0,y)|_{y\in \partial M} = 0. \end{align*} $$

Now, because $G_0(x_0;y) $ is smooth for each fixed $x_0\in M ^\circ $ and $y\in \partial M $ , by applying Taylor’s theorem in its integral form to the function $G_0(x_0;\cdot )$ , it follows that at ${y = (\rho _0(y),\theta (y) )}$ ,

$$ \begin{align*} G_0(x_0;y) = \rho_0(y) \int_0^1 \frac{\partial G_0(x_0;t\rho_0(y),\theta(y))}{\partial \rho_0(y)}\,{d} t. \end{align*} $$

Hence, there is a positive constant $C_{x_0}$ such that

$$ \begin{align*} |G_0(x_0;y)| \leq C_{x_0} \rho_0(y) \end{align*} $$

for $y\in M^\circ $ such that $\rho _0(y) < \min \{\delta ,\frac {1}{2}\rho _0(x_0)\}$ . Since the Green’s function is smooth away from the diagonal, it follows that the function

$$ \begin{align*} C_x = \int_0^1 \frac{\partial G_0(x;t\rho_0(y),\theta(y))}{\partial \rho_0(y)}\,{d} t \end{align*} $$

is continuous for $x \in K \Subset M ^\circ $ . Hence, if $K \Subset M^\circ $ is compact, we can take the maximum $C_K = \max _{x\in K } C_x $ , then

$$ \begin{align*} |G_0(x;y)|\leq C_K \rho_0(y) \end{align*} $$

for $ x\in K , y \in M ^\circ $ with $\rho _0(y) < \min \{ \delta ,\frac {1}{2}\min _{x\in K }\rho _0(x)\}$ .

We now give a decomposition of the Green’s function $G_0(x;y) $ into a sum of a principal Green’s function $G_{0,\mathrm {prin}}(x;y)$ and a remainder Green’s function $G_{0,\mathrm {rem}}(x;y)$ .

Definition 2.8. Assume without loss of generality that $(M\setminus M_\delta ,\bar g_0) $ can be extended to some manifold $(\widetilde M\setminus M_\delta ,\bar g_0 )$ isometric to $(\mathbb {R}^2\setminus \mathbb {D}_{1-\bar \delta },g_{\mathrm {Euc}})$ and let $\Phi :\widetilde M\setminus M_\delta \to \mathbb {R}^2\setminus \mathbb {D}_{1-\bar \delta }$ be this isometry.

For $x,y\in M\setminus M_\delta $ , denote

$$ \begin{align*} |x|: = |\Phi(x)| , \ |x-y|:= |\Phi(x)-\Phi(y)|{,} \quad \text{and}\quad \frac{x}{|x|^2} := \Phi^{-1}\bigg(\frac{\Phi(x)}{|\Phi(x)|^2}\bigg)\in \widetilde M\setminus M_\delta , \end{align*} $$

then define the distribution $G_{0,\mathrm {prin}}(x;y)\in \mathcal {D}'(M\times M) $ by

$$ \begin{align*} G_{0,\mathrm{prin}}(x;y) = \frac{\kappa(x;y)}{2\pi}\bigg[\!\log |x-y| - \log\bigg||x| \cdot \bigg(y-\frac{x }{|x|^2} \bigg)\bigg| \bigg] , \end{align*} $$

where $\kappa (x;y)= \kappa _1(x)\cdot \kappa _1(y)\cdot \kappa _0(x;y)$ and $\kappa _1\in C^\infty (M),\kappa _0(x;y)\in C^\infty (M\times M) $ are smooth cutoff functions on $M $ with the properties that

(2-4) $$ \begin{align} \begin{cases} \kappa_1(x) \equiv 1 \quad &\text{if } \rho_0(x)\leq \delta/4,\\ \kappa_1(x) \equiv 0 &\text{if } \rho_0(x) \geq \delta/3, \end{cases} \quad \text{and}\quad \begin{cases} \kappa_0(x;y) \equiv 1 \quad &\text{if } d_{\bar g_0}(x;y)<\delta/4,\\ \kappa_0(x;y) \equiv 0 &\text{if } d_{\bar g_0}(x;y)>\delta/3,\\ \kappa_0(x;y)\equiv \kappa_0(y;x)&\text{for all } x,y\in M. \end{cases} \end{align} $$

Remark 2.9. The distribution

$$ \begin{align*} G_{0,\mathrm{prin},\mathcal{A}}(z_1;z_2) = \frac{1}{2\pi}\bigg[\log |z_1-z_2| - \log\bigg| |z_1|\cdot \bigg( z_2-\frac{z_1}{|z_1|^2}\bigg)\bigg|\bigg] \end{align*} $$

is the symmetric Green’s function on the unit disc, with the properties of (2-3) for the Euclidean metric $g_{\mathrm {Euc}}$ [Reference Evans13, Section 2.2 and (41)]. Therefore,

$$ \begin{align*} \begin{cases} \Delta_{g_{\mathrm{Euc}},z_2} G_{0,\mathrm{prin},\mathcal{A}} (z_1,z_2) = \delta_{z_1}(z_2),\\\kern27.5pt G_{0,\mathrm{prin},\mathcal{A}}(z_1,z_2) = G_{0,\mathrm{prin},\mathcal{A}}(z_2,z_1),\\\kern8.5pt G_{0,\mathrm{prin},\mathcal{A}}(z_1,z_2)|_{|z_2| = 1} = 0. \end{cases} \end{align*} $$

Lemma 2.10. We can write

$$ \begin{align*} G_{0,\mathrm{prin}}(x;y) =\kappa(x;y) (\Phi^*G_{0,\mathrm{prin},\mathcal{A}})(x;y) = \kappa(x;y) G_{0,\mathrm{prin},\mathcal{A}}(\Phi(x);\Phi(y)). \end{align*} $$

Proof. In Definition 2.8, $G_{0,\mathrm {prin}}$ is exactly defined using the distribution $G_{0,\mathrm {prin},\mathcal {A}}$ .

Lemma 2.11. The distribution $G_{0,\mathrm {prin}}(x;y) $ is symmetric, $G_{0,\mathrm {prin}}(y;x) = G_{0,\mathrm {prin}}(x;y) $ .

Proof. The smooth cut-off function $\kappa (x;y) $ is symmetric by definition. On the support $\mathrm {supp}\ \kappa (x;y)$ , we therefore have

$$ \begin{align*} G_{0,\mathrm{prin}}(y;x) &= {\kappa(y;x)} G_{0,\mathrm{prin},\mathcal{A}}(\Phi(y);\Phi(x)) \\[4pt] &= {\kappa(x;y)} G_{0,\mathrm{prin},\mathcal{A}}(\Phi(x);\Phi(y)) = G_{0,\mathrm{prin}}(x;y). \\[-38pt] \end{align*} $$

Proposition 2.12. We have the following decomposition of the Green’s function $G_0(x;y)$ :

$$ \begin{align*} G_0(x;y) = G_{0,\mathrm{prin} }(x;y) + G_{0,\mathrm{rem} }(x;y) , \end{align*} $$

where $G_{0,\mathrm {rem} }(x;y) $ satisfies:

1. (1) $G_{0,\mathrm {rem} }(x;y) $ is a smooth symmetric function for $x,y\in M\setminus M_{\delta /4}$ , with $G_{0,\mathrm {rem}}(x;y)= 0 $ whenever $y\in \partial M $ ; and

2. (2) if $0<\varepsilon \ll \delta $ , then there there are constants $C_0,C_1,C_2$ such that

   $$ \begin{align*} \sup_{z\in \partial M_{\varepsilon}}|\partial_{\nu_z} G_{0,\mathrm{rem}}(z;y)| \leq C_0 \cdot \rho_0(y) + C_1 (\varepsilon + \rho(y))^2+ C_2 (\varepsilon + \rho(y))^3 \end{align*} $$
   holds for all $y \in M\setminus M_{\delta /4}$ and $\varepsilon>0$ .

Proof. We have

(2-5) $$ \begin{align} \begin{aligned}\kern-6pt \Delta_{\bar g_0,y}G_{0,\mathrm{prin}}(x;y) &= \frac{1}{2\pi}\bigg[ \kappa(x;y) \Delta_{\bar g_0,y}\bigg(\log |x-y| - \log \bigg||x| \cdot \bigg(y-\frac{x }{|x|^2}\bigg) \bigg|\bigg)+ F(x;y) \bigg] \end{aligned} \end{align} $$

where

$$ \begin{align*} \begin{aligned} F(x;y) &:= \Delta_{\bar g_0,y}\kappa(x;y)\bigg[\log |x-y| - \log\bigg||x| \cdot \bigg(y-\frac{x }{|x|^2}\bigg)\bigg| \bigg] \\[4pt] &\qquad + 2\bigg\langle{\nabla_y\kappa(x;y)},{\nabla_y\bigg(\log |x-y| - \log \bigg||x| \cdot \bigg(y-\frac{x }{|x|^2}\bigg)\bigg| \bigg)}\bigg\rangle_{\bar g_0} \end{aligned} \end{align*} $$

and $\nabla _y $ denotes the gradient in $y $ . By Lemma 2.10, it follows that

$$ \begin{align*} \begin{aligned} &\frac{\kappa(x;y)}{2\pi}\Delta_{\bar g_0,y}\bigg(\log |x-y| -\log \bigg||x| \cdot \bigg( y-\frac{x }{|x|^2} \bigg) \bigg|\bigg) \\[4pt]& \qquad = \kappa(x;y)\Delta_{\Phi^*g_{\mathrm{Euc}}}(\Phi^*G_{0,\mathrm{prin},\mathcal{A}})(x;y) \\[4pt]& \qquad =\kappa(x;y)\Phi^*( \Delta_{g_{\mathrm{Euc}}}G_{0,\mathrm{prin},\mathcal{A}})(x;y) = \kappa(x;y)\delta_{x}(y) , \end{aligned} \end{align*} $$

since we assumed that $\bar g_0 $ is isometric to $g_{\mathrm {Euc}} $ . So, (2-5) becomes

(2-6) $$ \begin{align} \Delta_{\bar g_0,y}G_{0,\mathrm{prin}}(x;y) = \kappa(x;y)\delta_x(y) + F(x;y). \end{align} $$

By construction, we have that the supports of $d_y \kappa (x;y) $ and $\Delta _{\bar g_0} \kappa (x;y)$ are contained in the union

$$ \begin{align*} \begin{aligned} &\mathrm{supp}\ d_{y}\kappa(x;y), \ \mathrm{supp}\ \Delta_{\bar g_0,y} \kappa(x;y) \\& \qquad \subset \{(M_{\delta/4}\setminus M_{\delta/2})\times (M_{\delta/4}\setminus M_{\delta/2}) \} \cup \{\delta/4 <d_{\bar g_0}(x;y) <\delta/2\}. \end{aligned} \end{align*} $$

So, we have that $F(x;y)$ is smooth for $x,y\in M\setminus M_{\delta /4}$ . If we now let

$$ \begin{align*} G_{0,\mathrm{rem}} (x;y) := G_{0}(x;y)- G_{0,\mathrm{prin}}(x;y), \end{align*} $$

then for $x,y\in M\setminus M_{\delta /4 }$ , we have that (2-6) becomes

$$ \begin{align*} \Delta_{\bar g_0,y} G_{0,\mathrm{rem}}(x;y) = \delta_x(y) - \kappa(x;y) \delta_x(y) - F(x;y) = -F(x;y), \end{align*} $$

since $\kappa (x;y) \equiv 1 $ for $x,y\in M \setminus M_{\delta /4 }$ . By elliptic regularity [Reference Taylor39], it therefore follows that $G_{0,\mathrm {rem}}(x;y) $ is smooth on $M\setminus M_{\delta /4}$ . Furthermore, since both $G_0(x;y) $ and $G_{0,\mathrm {prin}}(x;y) $ are symmetric and zero on the boundary $\partial M $ , it follows that $G_{0,\mathrm {rem}}(y;x)=G_{0,\mathrm {rem}}(x;y)$ and $G(x;y)|_{y\in \partial M}=0$ . This concludes Item (1).

Finally, since $G_{0,\mathrm {rem}} (x;y)$ is jointly smooth near the boundary, we can take a joint Taylor polynomial around $\rho _0(x) = \rho _0(y) = 0 $ such that there are smooth functions $A_{ij }(\theta (x),\theta (y))$ such that

$$ \begin{align*} \begin{aligned} G_{0,\mathrm{rem}}(x;y) &= A_{00}(\theta(x),\theta(y))+ A_{10}(\theta(x),\theta(y))\rho_0(x)+ A_{01}(\theta(x),\theta(y))\rho_0(y) \\&\qquad + A_{11}(\theta(x),\theta(y))\rho_0(x)\rho_0(y) + A_{20}(\theta(x),\theta(y) ) \rho_0(x)^2 \\&\qquad + A_{02}(\theta(x),\theta(y) ) \rho_0(y)^2 + R_3(x;y) \end{aligned} \end{align*} $$

with the remainder $R_3 (x;y) $ given by

$$ \begin{align*} \begin{aligned} R_3(x;y) &= \sum_{i+j = 3} \frac{3(\rho_0(x))^i (\rho_0(y))^j}{i!j!}\\ &\hspace{1cm}{\cdot}\, \bigg[\int_0^1 (1-t)^{2}\frac{\partial^3 G_{0,\mathrm{rem}}(t\rho_0(x),\theta(x),t\rho_0(y),\theta(y))}{(\partial \rho_0(x))^i (\partial \rho_0(y))^j}\,{d} t \bigg]. \end{aligned} \end{align*} $$

Since $G_{0,\mathrm {rem}}(x;y) |_{y\in \partial M}\equiv 0 $ , it follows that the functions $A_{00}, A_{10} $ are identically 0. Furthermore, also because $G_{0,\mathrm {rem} }(x;y)|_{y\in \partial M }\equiv 0 $ for all $x \in M^\circ $ , we find that the normal derivative $\partial _{\nu _z}G_{0,\mathrm {rem} }(z;y)|_{\substack {y\in \partial M\\ z\in \partial M_\varepsilon }}\equiv 0$ , and that the functions $ A_{10}= A_{20} $ are identically 0.

Then, since the normal derivative $\partial _\nu $ at the boundary $\partial M_\varepsilon $ in the local coordinates $(\rho _0,\theta ) $ satisfies the relation $\partial _\nu = -\partial _{\rho _0}|_{\rho _0 = \varepsilon }$ , we find that

$$ \begin{align*} \partial_{\nu_z}G_{0,\mathrm{rem}}(z;y)|_{z\in \partial M_\varepsilon} = -A_{11}(\theta(z),\theta(y)) \rho_0(y) + \partial_{\nu_z}R_3(z;y)|_{z\in \partial M_\varepsilon}. \end{align*} $$

We can now bound the function $A_{11}(\theta (x),\theta (y))$ uniformly on $\partial M \times \partial M $ by a constant $C_0>0 $ such that

$$ \begin{align*} \sup_{\theta(x),\theta(y)\in \partial M }|A_{11}(\theta(x),\theta(y))|\leq C_0. \end{align*} $$

Furthermore,

$$ \begin{align*} \bigg|\partial_{\nu_z} R_3(z;y)|_{z\in \partial M_\varepsilon}\bigg| &\leq \sum_{i+j= 3}\bigg|\frac{3 i (\varepsilon)^{i-1} (\rho_0(y))^j}{i!j!}\int_0^1 (1-t)^{2}\frac{\partial^3 G_{0,\mathrm{rem}}(t\varepsilon,\theta(z),t\rho_0(y),\theta(y))}{(\partial \rho_0(x))^i (\partial \rho_0(y))^j}\,{d} t \bigg|\\&\qquad+\sum_{i+j=3}\bigg|\frac{3(\varepsilon)^i (\rho_0(y))^j}{i!j!}\int_0^1 (1-t)^{2}\frac{\partial^4 G_{0,\mathrm{rem}}(t\varepsilon,\theta(z),t\rho_0(y),\theta(y))}{(\partial \rho_0(x))^{i+1} (\partial \rho_0(y))^j}\,{d} t \bigg|\\&\leq C_1 (\varepsilon + \rho(y))^2 + C_2 (\varepsilon + \rho(y))^3 \end{align*} $$

by bounding the derivatives uniformly on the right-hand side over $M\setminus M_{\delta /4}\times M\setminus M_{\delta /4}$ due to the joint smoothness of the function $G_{0,\mathrm {rem} }(x;y)$ . Then, it follows that there are constants $C_0,C_1,C_2$ such that

$$ \begin{align*} \sup_{z\in \partial M_{\varepsilon}}|\partial_{\nu_z} G_{0,\mathrm{rem}}(z;y)| \leq C_0 \cdot \rho_0(y) + C_1 (\varepsilon + \rho(y))^2+ C_2 (\varepsilon + \rho(y))^3 \end{align*} $$

holds for all $y \in M\setminus M_{\delta /4}$ and $\varepsilon>0$ . This concludes Item (2).

In Section 3, we use these decompositions to explicitly compute some integrals near the boundary.

2.3 Sobolev spaces

In this subsection, we give definitions of the Sobolev spaces of interest in this paper.

Definition 2.13. For $k\in \mathbb {Z}^+,(\Omega ,\bar g)\subset ( \overline M ,\bar g)$ an open domain and $\bar g $ a smooth metric on $\overline \Omega $ , we define the Sobolev spaces as the completions with respect to the $H^k(\Omega , \bar g)$ -norm

$$ \begin{align*} \begin{aligned} H^{k}(\Omega,\bar g) &:= \overline{\{u\in C^\infty(\Omega)\}}^{\|\cdot\|_{H^k(\Omega, \bar g )}},\\ H^{k,p}_0(\Omega,\bar g) &:= \overline{\{u\in C^\infty_0(\Omega)\}}^{\|\cdot\|_{H^k(\Omega, \bar g )}} , \end{aligned} \end{align*} $$

where the $H^k(\Omega , \bar g )$ norm is given by

$$ \begin{align*} \|f\|_{H^k(\Omega, \bar g )} = \bigg(\sum_{i=0}^k \|D^i f \|_{L^2(\Omega, \bar g )}^2\bigg)^{1/2}, \end{align*} $$

where we interpret the $D^i f $ as the ith weak derivative of $f\in C^\infty (\Omega )$ .

We define the localised Sobolev space

$$ \begin{align*} H^{k}_{\mathrm{loc}}(\Omega,\bar g):= \{u \in \mathcal{D}'(\overline{\Omega}): \varphi u \in H^{k}(\Omega, \bar g ) \, \text{for all } \varphi\in C^\infty_0(\Omega)\}. \end{align*} $$

For $k\in \mathbb {Z}^+ $ , the space $H^{-k}(\Omega , \bar g )$ is defined as the formal dual to $H^{k}_0(\Omega , \bar g )$ .

For non-integer values $l>0 $ , we define the Sobolev spaces $H^{l}(\Omega , \bar g )$ using the complex interpolation method [Reference Bergh and Löfström5, Theorem 6.4.5 part (7)].

This definition depends implicitly on the metric (and therefore also on the $L^2 $ -norms). By [Reference Hebey18, Proposition 2.2], it follows that if $\bar g,\bar h $ are two smooth metrics for $\overline {\Omega } $ , then the Sobolev space $H^{l}(\Omega ,\bar g) = H^{l}(\Omega ,\bar h) $ , since the metrics $\bar g $ and $\bar h $ provide equivalent norms on $\Omega $ .

Next, the Poincaré inequality also provides an equivalent norm for $H^{1}_0(\Omega , \bar g )$ . Indeed, we see for $\varphi \in H^{1}_0(\Omega , \bar g )$ that

$$ \begin{align*} \|D \varphi \|_{L^2(\Omega, \bar g )} \leq \Big(\|\varphi\|_{L^2(\Omega, \bar g )}^2 + \|D \varphi\|_{L^2(\Omega, \bar g )}^2 \Big)^{1/2} \leq C(\Omega) \|D\varphi\|_{L^2(\Omega, \bar g )}. \end{align*} $$

The second inequality is the Poincaré inequality. Since, by definition, $\|\varphi \|_{H^1(\Omega ,\bar g)}:= (\|\varphi \|_{L^2(\Omega , \bar g )}^2 + \|D\varphi \|^2_{L^2(\Omega , \bar g )})^{1/2}$ , the conclusion follows.

The following lemmas show that (weak) solutions to elliptic boundary value problems have the ‘expected’ regularity.

Proposition 2.14. Let $\Omega \subset \overline M $ be a bounded domain with smooth boundary and let $\bar g $ be a smooth metric on $\overline {\Omega }$ . Then, if $f \in H^{-1}(\Omega ,\bar g)$ , the boundary value problem

$$ \begin{align*} \begin{cases} \Delta_{\bar g } u = f, \\ u |_{\partial \Omega} = 0 \end{cases} \end{align*} $$

has a unique solution $u \in H^{1}_0(\Omega ,\bar g)$ and there is a constant $C> 0 $ such that

$$ \begin{align*} \| u \|_{H^{1}(\Omega,\bar g)} \leq C \|f\|_{H^{-1}(\Omega,\bar g)}. \end{align*} $$

This is a consequence of the boundary elliptic regularity [Reference McLean27, Theorem 4.10] together with the maximum principle for the Laplacian.

Proposition 2.15. Let $\Omega \subset \overline M $ be a bounded domain with smooth boundary, then for $s\in \mathbb {R} $ , if $f\in H^{s-2}_{\mathrm {loc }}(\Omega , \bar g )$ , and if $u $ satisfies $\Delta _{\bar g} u = f$ for $x\in \Omega $ , then $u \in H^{s}_{\mathrm {loc }}(\Omega , \bar g )$ .

This is [Reference Folland14, Theorem 6.33] for smooth manifolds.

Proposition 2.16 (Sobolev embedding theorem)

Let $\overline M $ be a compact smooth Riemannian manifold of dimension $n $ with smooth boundary $\partial M $ . Then, the embedding of $H^{l}(M, \bar g ) \hookrightarrow C^s(\overline M )$ is continuous if $l-n/2> s\geq 0 $ for integer s.

For a proof, see for example, [Reference Aubin3, Theorem 2.30].

We finish this section with Hardy’s inequality. For a discussion of the best constants, see [Reference Marcus, Mizel and Pinchover25].

Proposition 2.17. Let $\Omega \subset M $ be an open domain and let $\bar g $ be a smooth metric on $\overline {\Omega }$ . Then, there exists a constant $c $ such that for $u \in H^{1}_0(\Omega , \bar g )$ , the inequality

$$ \begin{align*} \int_{\Omega} \bigg(\frac{|u(x)|}{\mathrm{dist}_{\bar g} (x,\partial \Omega) }\bigg)^2 \ \mathrm{dVol}_{\bar g }( x) \leq c \int_{\Omega}|D u |^2 \ \mathrm{dVol}_{\bar g}( x) \end{align*} $$

holds.

3 Proof of Theorem 1.1

Now, to continue, assume that $\varepsilon $ is sufficiently small. Fix some $x_0\in M_\varepsilon ^\circ $ and let $\delta>\varepsilon >0 $ such that $\rho _0(x_0)> \delta $ . Similarly to before, set $M_\delta = \{x\in M : \rho _0(x)> \delta \}$ . We further assume that the estimates from Lemma 2.7 hold for $y\in M\setminus M_\delta $ . This can be achieved by choosing $\delta $ sufficiently small. On the neighbourhood $M_\varepsilon \setminus M_\delta $ , we can define the function

(3-1) $$ \begin{align} v_\varepsilon(x) = \log \rho_0(x) -\log \varepsilon. \end{align} $$

On the model asymptotically hyperbolic surface $(\mathbb {D}\setminus \mathbb {D}_{1-\overline \delta },g_{\mathrm {Euc}}/\varrho ^2 )$ , the function $v_\varepsilon (x) $ solves the boundary value problem (1-1) exactly. By multiplying $v_\varepsilon $ with a smooth cut-off function $\chi \in C^\infty (\overline {M}) $ , depending only on $\rho _0(x) $ , which has the properties $\chi |_{\rho _0(x)<\delta /2 }\equiv 1 $ and $\mathrm {supp}\ \chi \subset \overline {M}\setminus M_\delta $ , we see that the function $\chi v_\varepsilon \in C^\infty (M_\varepsilon ) $ .

Proof. We simply calculate the Laplacian applied to $w_\varepsilon $ and use (3-1) for $v_\varepsilon $ to get

(3-4) $$ \begin{align} \begin{aligned} \Delta_{\bar{g}_0} w_\varepsilon(x) &= -\frac{1-\chi(x) + \beta^{(1)}(\theta)\rho_0(x) + \beta^{(2)}(\rho_0,\theta)}{\rho_0(x)^2 }\\&\qquad -\Delta_{\bar{g}_0}(\chi(x)) v_\varepsilon(x) -2\langle{d\chi(x)},{d v_\varepsilon(x)}\rangle_{\bar{g}_0} \\&= f_1(x)\log\varepsilon + f_2(x) , \end{aligned} \end{align} $$

where $f_1 = \Delta _{\bar {g}_0}\chi $ and

$$ \begin{align*} f_2 = -\frac{(1-\chi) + \beta^{(1)}(\theta)\rho_0 + \beta^{(2)}(\rho_0,\theta)}{\rho_0^2} -\Delta_{\bar{g}_0}\chi\cdot \log \rho_0- 2 \langle{d\chi},{d v_\varepsilon}\rangle_{\bar{g}_0}. \end{align*} $$

Then by construction, $f_1,f_2 \in C^\infty (M_\varepsilon ) $ are smooth. This gives part (1).

Now, we note that $f_1 $ is identically 0 on a neighbourhood of the boundary $\partial M_\varepsilon $ , so that we can extend $f_1 $ to a smooth function on $\overline {M }$ with $f_1|_{\overline {M}\setminus M_\varepsilon } \equiv 0 $ .

Given these expressions, we can compute

$$ \begin{align*} \begin{aligned} I_1(x) &= \int_{M_\varepsilon}G_0(x;y)f_1(y) \ \mathrm{dVol}_{\bar{g}_0}(x) =\int_{M_\varepsilon}G_0(x;y)\Delta_{\bar{g}_0}\chi (y) \ \mathrm{dVol}_{\bar{g}_0}(y) \\ &= \chi(x) + \int_{\partial M_\varepsilon}G_0(x;y)\partial_{\nu} \chi(y) - \partial_{\nu_y} G_0(x;y) \chi(y) \ \mathrm{dVol}_h(y) = \chi(x)-1 , \end{aligned} \end{align*} $$

due to the fact that $\int _{\partial M_\varepsilon }\partial _\nu G_0(x;y)\ \mathrm {dVol}_h(y) = 1 $ by integration by parts and, hence, $I_1(x) \in C^\infty (\overline {M })$ . This gives part (2).

We now show that for each $x_0\in M^\circ $ , the integral $I_2(x_0) $ is $\mathcal {O}(1)$ as $\varepsilon \to 0 $ . The functions

$$ \begin{align*} -\frac{1-\chi}{\rho_0^2}{,}\quad \Delta_{\bar{g}_0}\chi \cdot \log \rho_0{,}\quad \langle{d \chi},{d v_\varepsilon}\rangle_{\bar{g}_0} \end{align*} $$

are all identically 0 in a collar neighbourhood containing $\partial M_\varepsilon $ of the boundary $\partial M $ . Similarly, the function $\beta ^{(2)}(\rho _0,\theta )/\rho _0^2$ is a bounded smooth function on this collar neighbourhood. Hence, for each $x_0\in M_\varepsilon $ , the function

$$ \begin{align*} G_0(x_0;y) \bigg[ -\frac{1-\chi+ \beta^{(2)}(\rho_0,\theta)}{\rho_0^2}-\Delta_{\bar{g}_0}\chi \cdot \log \rho_0-\langle{d \chi},{d v_\varepsilon}\rangle_{\bar{g}_0} \bigg] \end{align*} $$

is absolutely integrable over $\overline {M }$ . Finally, by Lemma 2.7, we notice that for each ${x_0\in M^\circ} $ , the function $G_0(x_0;y)/\rho _0(y) $ is also absolutely integrable over $\overline {M }$ . Then, the integral

$$ \begin{align*} \int_{M_\varepsilon}G_0(x;y) f_2(y) \ \mathrm{dVol}_{\bar g_0}(y) = \int_{\overline M}G_0(x;y) f_2(y)\mathbf{1}_{\rho_0>\varepsilon} (y)\ \mathrm{dVol}_{\bar g_0}(y) \end{align*} $$

converges to

$$ \begin{align*} \int_{\overline M}G_0(x;y) f_2(y) \ \mathrm{dVol}_{\bar g_0}(y) = \mathcal O (1), \end{align*} $$

by the dominated convergence theorem. This gives part (3).

We can extend the smooth function $f_2 $ to a distribution in the space $H^{-1}(M,\bar g_0)$ by setting for each $\varphi \in H^{1}_0(M,\bar g_0)$ , the pairing

$$ \begin{align*} \begin{aligned} \langle{\varphi},{f_2 }\rangle = \int_{\overline M} \overline{\varphi(x) } \bigg(&-\frac{(1-\chi) + \beta^{(1)}(\theta)\rho_0 + \beta^{(2)}(\rho_0,\theta)}{\rho_0^2} \\&-\Delta_{\bar{g}_0}\chi\cdot \log \rho_0- 2 \langle{d\chi},{d v_\varepsilon}\rangle_{\bar{g}_0}\bigg) \ \mathrm{dVol}_{\bar g_0}(x). \end{aligned} \end{align*} $$

Note that $\varphi \mapsto \langle \varphi , f_2\rangle $ is a bounded linear function on $H^1_0(M, \bar g_0)$ thanks to Hardy’s inequality, Proposition 2.17. Indeed, near the boundary, the function

$$ \begin{align*}f_2(x) =\frac{\beta^{(1)}(\theta)}{\rho_0(x)} + \mathcal{O}(1) = \frac{\beta^{(1)}(\theta)}{d_{\bar g}(\partial M , x)}+ \mathcal{O}(1),\end{align*} $$

such that by the Cauchy–Schwarz and Hardy inequalities, there are constants $\hat {c}, \hat {C}$ ,

$$ \begin{align*} \begin{aligned} |\langle{\varphi},{f_2 }\rangle|&\leq \bigg\|\frac{\varphi}{d_{\bar g}(\partial M , x)}\bigg\|_{L^2({M}, \bar g )}\cdot \|{f_2}\cdot {d_{\bar g}(\partial M , x)}\|_{L^2({M}, \bar g )}\leq \hat{c}\|D\varphi\|_{L^2({M}, \overline{g})}\leq \hat{C}\|\varphi\|_{H^1(M,\bar{g})}. \end{aligned} \end{align*} $$

Given this, we observe then that the function $\tilde u_\varepsilon (x): M \to \mathbb {R} $ defined by

$$ \begin{align*} \tilde u_\varepsilon(x) =I_2(x)= \int_{M_\varepsilon}G_0(x;y) f_2(y) \ \mathrm{dVol}_{\bar g_0}(y) = \int_M G_0(x;y) \mathbf{1}_{\rho_0>\varepsilon}(y)f_2\ \mathrm{dVol}_{\bar g_0}(y) \end{align*} $$

solves the partial differential equation

$$ \begin{align*} \begin{cases} \Delta_{\bar g_0} \tilde u_\varepsilon(x) =\mathbf{1}_{\rho_0>\varepsilon}(x) f_2(x) \quad &\text{for } x\in M^\circ,\\ \tilde u_\varepsilon|_{\partial M} = 0. \end{cases} \end{align*} $$

For any $\varepsilon>0 $ , Proposition 2.14 implies that the function $\tilde u_\varepsilon (x) \in H^{1}_0(M,\bar g_0)$ , since the right-hand side $\mathbf {1}_{\rho _0>\varepsilon }f_2 \in H^{-1}(M,\bar g_0)$ .

Notice further that $\tilde u (x) $ given by (3-3) solves the distributional partial differential equation

(3-5) $$ \begin{align} \begin{cases} \Delta_{\bar g_0} \tilde u (x) = f_2(x) \quad &\text{for } x\in M^\circ,\\ \tilde u|_{\partial M} = 0. \end{cases} \end{align} $$

We now show that $\tilde u_\varepsilon (x) $ converges to $\tilde u(x) $ in $H^{1}_0(M,\bar g_0)$ . By Proposition 2.14, it follows that solutions to (3-5) are unique. This shows that $\tilde u (x)\in H^{1}_0(M, \bar g_0) $ is the unique solution to (3-5).

Now, we have by the Cauchy–Schwarz inequality and by Proposition 2.17, for each $\varepsilon>0 $ and all $\varphi \in H^{1}_0(M,\bar g_0)$ , that there are constants $c,d,d'>0 $ such that

$$ \begin{align*} \begin{aligned} \int_{M}&|(\mathbf{1}_{\rho_0>\varepsilon}(x) - 1) f_2(x) \varphi(x) |\ \mathrm{dVol}_{\bar g_0} x \\ &\quad\leq c \bigg(\int_{M}\bigg|\frac{\varphi}{\rho_0}\bigg|^{2} \ \mathrm{dVol}_{\bar g_0}(x)\bigg)^{1/2} \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^{2}(M,\bar g_0 )}\\ &\quad\leq d\bigg( \int_M |D\varphi(x)|^{2} \ \mathrm{dVol}_{\bar g_0} x \bigg)^{1/2}\cdot \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^{2}(M,\bar g_0)} \\ &\quad= d\|D\varphi\|_{L^{2}(M)} \cdot \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^{2}(M,\bar g_0)}\\ &\quad \leq d' \|\varphi\|_{H^{1}(M,\bar g_0)}\cdot \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^{2}(M,\bar g_0)}. \end{aligned} \end{align*} $$

Hence, we have that in the dual space topology on $H^{-1}(M,\bar g_0)$ ,

$$ \begin{align*} \begin{aligned} \|\Delta_{\bar g_0} (\tilde u_\varepsilon - \tilde u)\|_{H^{-1}(M,\bar g_0)} &= \sup_{\substack{\varphi\in H^{1}_0(M,\bar g_0)\\ \|\varphi\|_{H^{1}(M,\bar g_0)}=1}}|\langle{\Delta_{\bar g_0} (\tilde u_\varepsilon - \tilde u)},{\varphi}\rangle|\\ &= \sup_{\substack{\varphi\in H^{1}_0(M,\bar g_0)\\ \|\varphi\|_{H^{1}(M,\bar g_0)}=1}}|\langle{(\mathbf{1}_{\rho_0>\varepsilon}(x) - 1) f_2(x)},{\varphi}\rangle|\\ &\leq d' \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^{2}(M,\bar g_0)} \xrightarrow{\varepsilon\to 0 } 0. \end{aligned} \end{align*} $$

Thus, by Proposition 2.14, it follows that there is a constant C such that

$$ \begin{align*} \|\tilde u_\varepsilon - \tilde u\|_{H^{1}(M,\bar g_0)} \leq C \|\mathbf{1}_{\rho_0>\varepsilon}-1\|_{L^2(M,\bar g_0)} \xrightarrow{\varepsilon\to 0 }0; \end{align*} $$

hence, $\tilde u_\varepsilon (x) $ converges to $\tilde u(x) $ in $H^{1}_0$ .

Now, for any compact set $K \Subset M^\circ $ , we can then take $\varepsilon>0 $ sufficiently small such that the set $K \subset \overline {\{\rho _0> 2 \varepsilon \}}$ . Because $f_2 \in H^{k-2}_{\mathrm {loc }}(M,\bar g_0)$ for any $k\in \mathbb {Z}^+ $ , it follows that $\tilde u_\varepsilon , \tilde u \in H^{k}_{\mathrm {loc }}(M,\bar g_0)$ for any $k\in \mathbb {Z}^+ $ by Proposition 2.15. Furthermore, for any $\varphi \in C^\infty _0(M) $ such that $\mathrm {supp}\ \varphi \subset K $ ,

$$ \begin{align*} \varphi \Delta_{\bar g_0 } \tilde u_\varepsilon = \varphi f_2 = \varphi \Delta_{\bar g_0} \tilde u, \text{ which implies that } \varphi \Delta_{\bar g_0} (\tilde u_\varepsilon - \tilde u) = 0. \end{align*} $$

This means that $\tilde u_\varepsilon - \tilde u \in H^{k}(K,\bar g_0) $ for any $k \in \mathbb {Z}^+$ . Now, by the complex interpolation method [Reference Bergh and Löfström5, Theorem 6.4.5 part (7)], we find that for any $l = (1-\theta ')k +\theta ', \theta '\in (0,1 ) $ ,

$$ \begin{align*} \begin{aligned} \|\tilde u_\varepsilon - \tilde u\|_{H^{l}(K,\bar g_0)} &\leq C(k) \| \tilde u_\varepsilon - \tilde u \|^{1-\theta'}_{H^{k}(K,\bar g_0)} \cdot \|\tilde u_\varepsilon - \tilde u\|_{H^{1}(K,\bar g_0)}^{\theta'}\\ &\leq C(k) (\|\tilde u_\varepsilon\|^{1-\theta'}_{H^{k}(K,\bar g_0)} + \| \tilde u \|^{1-\theta'}_{H^{k}(K,\bar g_0)} ) \cdot\|\tilde u_\varepsilon - \tilde u\|_{H^{1}(K,\bar g_0)}^{\theta'} \xrightarrow{\varepsilon \to 0 }0. \end{aligned} \end{align*} $$

Hence, by Proposition 2.16, it follows that for any integer $s $ , we can find l sufficiently large such that

$$ \begin{align*} \|\tilde u_\varepsilon - \tilde u\|_{C^s(K)} \leq C(s,k')\|\tilde u_\varepsilon - \tilde u \|_{H^{l}(M,\bar g_0)} \xrightarrow{\varepsilon\to 0 }0. \end{align*} $$

This shows that for any compact subsets $K \Subset M^\circ $ , the function $\tilde u \in C^\infty (K) $ and $\tilde u_\varepsilon $ converges uniformly to $\tilde u $ on $K $ . This finishes the proof of the proposition.

We now give one more estimate for the functions $f_1 $ and $f_2 $ in the following proposition.

Proposition 3.2. Let $f_1,f_2 \in C^\infty ({M_\varepsilon })$ be the functions from Proposition 3.1. Then, for $z\in \partial M_\varepsilon $ , there are functions $g_1, g_2:\partial M_\varepsilon \to \mathbb {R} $ such that

(3-6) $$ \begin{align} \int_{M_\varepsilon} \partial_{\nu_z}G_0(z;y)(f_1(y)\log \varepsilon + f_2(y))\ \mathrm{dVol}_{\bar{g}_0}(y) = g_1(z)\log\varepsilon + g_2(z) , \end{align} $$

with $\sup _{z\in \partial M_\varepsilon }| g_i (z)|= \mathcal {O}(1)$ as $\varepsilon \to 0 $ .

Define the family of functions $\tilde g_1, \tilde g_2: \partial M_\varepsilon \to \mathbb {R} $ by

(3-7) $$ \begin{align} \tilde g_1(z) := \int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y)f_1(y) \ \mathrm{dVol}_{\bar{g}_0}(y) \end{align} $$

and

(3-8) $$ \begin{align} \tilde g_2 (z):=\int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y)\bigg[\frac{1-\chi}{\rho_0^2}+\Delta_{\bar{g}_0}\chi \cdot \log \rho_0+\langle{d\chi},{d v_\varepsilon}\rangle_{\bar{g}_0}\bigg]\ \mathrm{dVol}_{\bar{g}_0}(y). \end{align} $$

The proof to Lemma 3.3 is given in Appendix A. We continue with the proof of Proposition 3.2.

Proof of Proposition 3.2

We recall that $f_2 $ is given by

$$ \begin{align*} f_2 (y) = -\frac{(1-\chi) + \beta^{(1)}(\theta)\rho_0 + \beta^{(2)}(\rho_0,\theta)}{\rho_0^2} - \Delta_{\bar{g}_0}\chi \cdot \log \rho_0 - 2\langle{d\chi},{dv_{\varepsilon}}\rangle_{\bar{g}_0}. \end{align*} $$

Hence, by the definition of $\tilde {g}_2(z) $ and Lemma 3.3, we may rewrite the left-hand side of (3-6), for $z\in \partial M_\varepsilon $ , to

$$ \begin{align*} \begin{aligned} &\int_{M_\varepsilon} \partial_{\nu_z}G_0(z;y)(f_1(y)\log \varepsilon + f_2(y))\ \mathrm{dVol}_{\bar{g}_0}(y) \\&\qquad =\tilde g_2(z) - \int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y) \frac{\beta^{(1)}(\theta(y))\rho_0(y)+ \beta^{(2)}(\rho_0,\theta)}{\rho_0(y)^2}\ \mathrm{dVol}_{\bar{g}_0}(y) \end{aligned} \end{align*} $$

with $\tilde g_2(z)$ uniformly bounded on $\partial M_\varepsilon $ as $\varepsilon \to 0 $ as shown in Appendix A.

To complete the proof, it remains to analyse the integral

$$ \begin{align*} \int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y) \frac{\beta^{(1)}(\theta(y))\rho_0(y)+ \beta^{(2)}(\rho_0,\theta)}{\rho_0(y)^2}\ \mathrm{dVol}_{\bar{g}_0}(y). \end{align*} $$

We show that the absolute integral is bounded uniformly by a function that is a multiple of $\log \varepsilon $ for $\varepsilon>0 $ . In Appendix A, we show that there are families of functions ${h_1,h_2: \partial M_\varepsilon \to \mathbb {R} }$ such that

$$ \begin{align*} \int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y) \frac{\beta^{(1)}(\theta(y))\rho_0(y)+ \beta^{(2)}(\rho_0,\theta)}{\rho_0(y)^2}\ \mathrm{dVol}_{\bar{g}_0}(y) = h_1(z) + h_2(z), \end{align*} $$

where $\sup _{z\in \partial M_\varepsilon }|h_1(z) | = \mathcal {O}(-\log \varepsilon )$ and $\sup _{z\in \partial M_\varepsilon }|h_2(z)|= \mathcal O(1)$ as $\varepsilon \to 0 $ ; see Proposition A.1.

Hence, it follows that the functions

$$ \begin{align*} g_1(z) = \frac{h_1(z)}{\log \varepsilon }\quad \text{and}\quad g_2(z) = \tilde g_2 (z) + h_2(z) \end{align*} $$

satisfy the estimates $\sup _{z\in \partial M_\varepsilon }|g_i(z)| =\mathcal {O}(1)$ as $\varepsilon \to 0 $ . Combining these facts, we conclude that

$$ \begin{align*} \int_{M_\varepsilon} \partial_{\nu_z}G_0(z;y)(f_1(y)\log \varepsilon + f_2(y))\ \mathrm{dVol}_{\bar{g}_0}(y) = g_1(z)\log\varepsilon + g_2(z) \end{align*} $$

with $\sup _{z\in \partial M_\varepsilon }| g_i (z)|= \mathcal {O}(1)$ as $\varepsilon \to 0 $ .

Proposition 3.4. Let $w_\varepsilon (x)= u_\varepsilon (x) -\chi (x) v_\varepsilon (x) $ from Proposition 3.1. The following estimate holds:

$$ \begin{align*} \|\partial_\nu w_\varepsilon(z) \|_{L^2(\partial M_\varepsilon)} = \mathcal{O}( -\log \varepsilon) \quad \text{as } \varepsilon\to 0. \end{align*} $$

We prove this proposition using layer potential theory [Reference Taylor39, Ch. 7, Section 11].

Definition 3.5. Let $N_\varepsilon ^\# $ be the double-layer potential

(3-9) $$ \begin{align} (N^\#_\varepsilon f)(z) = 2\int_{\partial M_\varepsilon}\partial_{\nu_z} G_0(z;y)f(y)\ \mathrm{dVol}_{h}(y). \end{align} $$

Then, $N_\varepsilon ^\#$ is a pseudodifferential operator of order $-1$ on $\partial M_\varepsilon $ [Reference Taylor39, Ch. 7, Proposition 11.3].

For $f \in C^\infty (\partial M_\varepsilon )$ , we write

$$ \begin{align*} (N_\varepsilon^\# f)(z) = (N_{\varepsilon,\mathrm{prin}}^\# f)(z) + (N^\#_{\varepsilon,\mathrm{rem}} f)(z), \end{align*} $$

with

$$ \begin{align*} (N_{\varepsilon,\mathrm{prin}}^\# f)(z) = 2\int_{\partial M_\varepsilon} \partial_{\nu_z} G_{0,\mathrm{prin}}(z;y) f(y)\ \mathrm{dVol}_h(y) \end{align*} $$

and

$$ \begin{align*} (N_{\varepsilon,\mathrm{rem}}^\# f)(z) = 2\int_{\partial M_\varepsilon} \partial_{\nu_z} G_{0,\mathrm{rem}}(z;y) f(y)\ \mathrm{dVol}_h(y) , \end{align*} $$

where $G_{0,\mathrm {prin}}$ and $G_{0,\mathrm {rem}}$ are the principal and remainder parts of the Green’s function from Definition 2.8.

Definition 3.6. For $f\in L^2(\partial M_\varepsilon )$ , we define the Fourier transform $\hat {f}\in L^2(\mathbb {Z})$ by the isometry $\Phi $ from Definition 2.1. In particular,

$$ \begin{align*} \hat{f} = \mathcal{F} \circ (\Phi^{-1})^* f , \end{align*} $$

where $\mathcal F $ is the Fourier transform on the circle of radius $\sqrt {1-2\varepsilon }$ given by

$$ \begin{align*} \hat{g}(n) = \frac{1}{2\pi \sqrt{1-2\varepsilon}} \int_{-\pi}^\pi e^{-in \vartheta} g(\vartheta) \sqrt{1-2\varepsilon}\,{d} \vartheta \end{align*} $$

for $g(\vartheta )\in L^2(\partial \mathbb {D}_{\sqrt {1-2\varepsilon }})$ .

The proof to Proposition 3.7 is given in Appendix B. We continue with the proof of Proposition 3.4.

Proof of Proposition 3.4

We begin with Green’s formula

$$ \begin{align*} w_\varepsilon(x) + \int_{\partial M_\varepsilon}G_0(x;y) \partial_{\nu_y} w_\varepsilon(y) \ \mathrm{dVol}_h(y) = \int_{M_\varepsilon}{G_0(x;y)}(\Delta_{\bar{g}_0}w_\varepsilon)(y) \ \mathrm{dVol}_{\bar{g}_0}(y), \end{align*} $$

take $x \in M_\varepsilon $ to some point $z \in \partial M_\varepsilon $ and apply the outward normal derivative $\partial _{\nu }$ at that $z\in \partial M_\varepsilon $ . Using [Reference Taylor39, Ch. 7, Proposition 11.3], we find with (3-4) that

$$ \begin{align*} \begin{aligned} &\partial_{\nu} w_\varepsilon(z) + \frac{1}{2}( -\partial_{\nu}w_\varepsilon(z) + N_\varepsilon^\# (\partial_{\nu}w)(z)) \\&\qquad = \partial_{\nu}\int_{M_\varepsilon}{G_0(z;y)}(f_1(y)\log \varepsilon + f_2(y))\ \mathrm{dVol}_{\bar{g }_0}(y). \end{aligned} \end{align*} $$

On the right-hand side, we pull the normal derivative into the integral, as the integral is absolutely convergent, and then apply Proposition 3.2. The conclusion is that for $\varepsilon $ sufficiently small,

(3-10) $$ \begin{align} \frac{1}{2}\big(\partial_{\nu} w_\varepsilon(z) + N_\varepsilon^\# (\partial_\nu w_\varepsilon)(z)\big) = g_1(z) \log \varepsilon + g_2(z). \end{align} $$

Now, since we know by Item (1) of Proposition 3.7 that $({N}_{\varepsilon ,\mathrm {prin}}^\# f)^\wedge (n) = (1-2\varepsilon )^{|n|}\hat {f}(n)$ , it follows that the inverse of the operator $A_\varepsilon := \frac {1}{2}(I + N_{\varepsilon ,\mathrm {prin}}^\#)$ satisfies the bound

$$ \begin{align*} \begin{aligned} \|A_\varepsilon^{-1} f \|_{L^2(\partial M_\varepsilon)}^2&= \bigg\|\sum_{k\in \mathbb{Z}}\frac{2e^{in\theta}}{1+\hat{N}_{\varepsilon,\mathrm{prin}}^\#(n)} \hat{f}(n)\bigg\|_{L^2(\partial M_\varepsilon)}^2\\ &\leq |\partial M_\varepsilon|_h \sum_{k\in \mathbb{Z}}\frac{4|\hat{f}(n)|^2}{(1+(1-2\varepsilon)^{|n|})^2}\leq 4|\partial M_\varepsilon|_h \sum_{k\in \mathbb{Z}}|\hat{f}(n)|^2\\ & = 4 |\partial M_\varepsilon|_h \|\hat{f}\|_{L^2(\mathbb{Z})}^2 = 4 \|f\|_{L^2(\partial M_\varepsilon)}^2 , \end{aligned} \end{align*} $$

by Parseval’s identity: $\|f \|_{L^2(\partial M_\varepsilon )}^2 =|\partial M_\varepsilon |_h \|\hat {f }\|_{L^2(\mathbb {Z})}^2 $ . Thus, by the Neumann series for operators,

$$ \begin{align*} \bigg(\frac{1}{2}(I + N_\varepsilon^\# )\bigg)^{-1} = \bigg(A_\varepsilon + \frac{1}{2} N_{\varepsilon,\mathrm{rem}}^\#\bigg)^{-1} = A_\varepsilon^{-1}\sum_{k=0}^\infty \bigg(-\frac{1}{2}N_{\varepsilon,\mathrm{rem}}^\# A_\varepsilon^{-1}\bigg)^k , \end{align*} $$

with

$$ \begin{align*} \begin{aligned} \bigg\| \bigg(\frac{1}{2}(I + N_\varepsilon^\# )\bigg)^{-1} \bigg\|_{L^2\to L^2} &\leq \|A_\varepsilon^{-1}\|_{L^2\to L^2} \cdot \bigg\|\sum_{k=0}^\infty \bigg(-\frac{1}{2}N_{\varepsilon,\mathrm{rem}}^\# A_\varepsilon^{-1}\bigg)^k \bigg\|_{L^2\to L^2}\\ &\leq 2 + \mathcal{O}(\varepsilon) \leq 3 \end{aligned}, \end{align*} $$

for sufficiently small $\varepsilon $ , because $\|N_{\varepsilon ,\mathrm {rem}}^\#\|_{L^2\to L^2} = \mathcal {O}(\varepsilon )$ as $\varepsilon \to 0$ by Item (2) of Proposition 3.7.

To find asymptotics of $\partial _\nu w_\varepsilon $ , we apply the inverse $( \frac {1}{2} I + N_\varepsilon ^\# )^{-1} $ to both sides of (3-10). We find that

$$ \begin{align*} \begin{aligned} \|\partial_\nu w_\varepsilon(z)\|_{L^2(\partial M_\varepsilon)} &= \bigg\|\bigg(\frac{1}{2}(I+N_\varepsilon^\#)\bigg)^{-1}(g_1(z)\log\varepsilon + g_2(z))\bigg\|_{L^2(\partial M_\varepsilon)}\\ &\leq \bigg\|\bigg(\frac{1}{2}(I + N_\varepsilon^\#)\bigg)^{-1}\bigg\|_{L^2\to L^2}\cdot \|g_1(z)\log\varepsilon + g_2(z)\|_{L^2(\partial M_\varepsilon)}\\ &\leq 3 \| g_1(z) \log \varepsilon + g_2 (z) \|_{L^2(\partial M_\varepsilon )} = \mathcal{O}(-\log \varepsilon ) \quad \text{as } \varepsilon \to 0, \end{aligned} \end{align*} $$

because $\|g_1\log \varepsilon + g_2\|_{L^2(\partial M_\varepsilon )}\leq |\partial M_\varepsilon |_h\|g_1\log \varepsilon + g_2\|_{L^\infty (\partial M_\varepsilon )} = \mathcal O(-\log \varepsilon ) $ as $\varepsilon \to 0 $ by Proposition 3.2.

We now only need one more proposition to prove Theorem 1.1.

Proposition 3.8. Let $\tilde u_\varepsilon (x) $ be the function from Item (4) of Proposition 3.1. For $x\in M^\circ $ , there is a function $r_\varepsilon (x) $ such that

$$ \begin{align*} w_\varepsilon(x) = (\chi(x)-1) \log \varepsilon + \tilde u_\varepsilon(x) + r_\varepsilon(x) , \end{align*} $$

with a positive constant $D_x $ such that $|r_\varepsilon (x)|\leq -D_x \varepsilon \log \varepsilon $ for sufficiently small $\varepsilon $ . If ${K \Subset M^\circ} $ is compact, then there is positive constant $D_K $ such that $\sup _{x\in K }|r_\varepsilon (x)| \leq -D_K \varepsilon \log \varepsilon $ .

Proof. By Green’s formula,

(3-11) $$ \begin{align} \begin{aligned} w_\varepsilon(x) = \int_{M_\varepsilon}G_0(x;y)\Delta_{\bar g_0 ,y}w_\varepsilon(y)\ \mathrm{dVol}_{\bar g_0}(y)-\int_{\partial M_\varepsilon}G_0(x;y) \partial_\nu w_\varepsilon(y) \ \mathrm{dVol}_h(y). \end{aligned} \end{align} $$

By Item (1) of Proposition 3.1,

(3-12) $$ \begin{align} \int_{M_\varepsilon} G_0(x;y) \Delta_{\bar g_0, y }w_\varepsilon(y) \ \mathrm{dVol}_{\bar g_0} (y) = \int_{M_\varepsilon} G_0(x;y) (f_1(y) \log\varepsilon + f_2(y)) \ \mathrm{dVol}_{\bar g_0}(y). \end{align} $$

By Item (2) of Proposition 3.1,

(3-13) $$ \begin{align} \int_{M_\varepsilon} G_0(x;y) f_1(y)\log \varepsilon \ \mathrm{dVol}_{\bar{g}_0}(y) = (\chi(x) -1)\log \varepsilon. \end{align} $$

By Item (3) of Proposition 3.1, we have that for each $x\in M^\circ $ , the function

(3-14) $$ \begin{align} \tilde u_\varepsilon(x) = \int_{M_\varepsilon} G_0(x;y) f_2(y) \ \mathrm{dVol}_{\bar g_0 }(y) = \mathcal{O}(1) \quad \text{as } \varepsilon\to 0. \end{align} $$

If $K\Subset M^\circ $ , then $\tilde u _\varepsilon (x) $ converges uniformly on $K $ to $\tilde u(x) \in C^\infty (K) $ by Item (4) of Proposition 3.1.

Setting $r_\varepsilon (x) $ to be

(3-15) $$ \begin{align} r_\varepsilon(x):= - \int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_\varepsilon(y)\ \mathrm{dVol}_h(y) , \end{align} $$

we find by the Cauchy–Schwarz inequality that

$$ \begin{align*} \begin{aligned} |r_\varepsilon(x) | = \bigg|\int_{\partial M_\varepsilon}G_0(x;y)\partial_\nu w_\varepsilon(y) \ \mathrm{dVol}_h(y) \bigg|&\leq \|G_0(x;\cdot)\|_{L^2(\partial M_\varepsilon)} \cdot \| \partial w_\varepsilon\|_{L^2(\partial M_\varepsilon)}. \end{aligned} \end{align*} $$

Because $\|\partial _\nu w_\varepsilon (z)\|_{L^2(\partial M_\varepsilon )} = \mathcal {O}(-\log \varepsilon )$ as $\varepsilon \to 0 $ and because $\sup _{y\in \partial M_\varepsilon }|G_0(x;y)| \leq C_x \cdot \varepsilon $ as $\varepsilon \to 0 $ by Lemma 2.7, it follows that there is a positive constant $D_x $ such that

(3-16) $$ \begin{align} |r_\varepsilon(x) | \leq -D_x \varepsilon\log \varepsilon \quad \text{for sufficiently small } \varepsilon. \end{align} $$

If $K \Subset M^\circ $ is compact, then, by the same argument as in the proof of Lemma 2.7, it follows that $D_x $ is continuous for $x \in K \Subset M^\circ $ and $y\in \partial M_\varepsilon $ if $\varepsilon < \frac {1}{2}\min _{x\in K}\rho _0(x)$ . Thus, taking $ D_K = \max _{x\in K }D_x $ , we conclude that $\sup _{x\in K } |r_\varepsilon (x) | \leq -D_K \varepsilon \log \varepsilon $ .

Plugging (3-12)–(3-14) into the first integral on the right-hand side of (3-11), and (3-15) into the second integral in (3-11),

$$ \begin{align*} \begin{aligned} w_\varepsilon(x) &= \int_{M_\varepsilon}G_0(x;y) (f_1(y) \log \varepsilon+f_2(y))\ \mathrm{dVol}_{\bar{g}_0}(y) \\&\qquad-\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_\varepsilon(y) \ \mathrm{dVol}_h(y)\\& = (\chi(x)-1) \log \varepsilon + \tilde{u}_\varepsilon(x) +r_\varepsilon(x) , \end{aligned} \end{align*} $$

with $|r_\varepsilon (x) |\leq -D_x \varepsilon \log \varepsilon $ by (3-16). Furthermore, for $K \Subset M ^\circ $ , there is a constant $D_K $ such that $|r_\varepsilon (x)|\leq -D_K \varepsilon \log \varepsilon $ and we have that $\tilde {u}_\varepsilon $ converges uniformly to some smooth bounded function $\tilde {u}(x) $ on $K $ .

Proof of Theorem 1.1

We have by the definition of $w_\varepsilon (x) $ in Proposition 3.1 that

$$ \begin{align*} u_\varepsilon(x) = \chi(x) v_\varepsilon(x) + w_\varepsilon(x). \end{align*} $$

Thus, by (3-1) and Proposition 3.8, we find for each $x\in M^\circ $ that

$$ \begin{align*} u_\varepsilon(x) = -\log \varepsilon +\chi(x) \log \rho_0(x)+ \tilde{u}_\varepsilon(x)+ r_\varepsilon(x) \end{align*} $$

with $|r_\varepsilon (x)|\leq -D_x \varepsilon \log \varepsilon $ and $\tilde u_\varepsilon (x) = \mathcal O (1) $ as $\varepsilon \to 0 $ by Proposition 3.8.

Finally, if $K \Subset M^\circ $ is compact, then there is a constant $D_K $ such that $\sup _{x\in K } |r_\varepsilon (x)| \leq -D_K \varepsilon \log \varepsilon $ and the function

$$ \begin{align*} \tilde U_\varepsilon : K \to \mathbb{R}: x \mapsto \chi(x) \log \rho_0(x) + \tilde u_\varepsilon (x) \end{align*} $$

converges uniformly to the smooth and bounded function

$$ \begin{align*} \tilde{U}: K \to \mathbb{R}: x\mapsto \chi(x) \log \rho_0(x) + \tilde u(x) \end{align*} $$

by applying Item (4) of Proposition 3.1 to $\tilde u_\varepsilon $ , and noting that $\chi (x)\log \rho _0(x) $ is bounded and smooth on $K $ .

4 Gas giant geometries

In the following sections, we turn to gas giant geometries introduced in [Reference de Hoop, Ilmavirta, Kykkänen and Mazzeo22]. A Riemannian manifold $(M,\partial M , g )$ has a gas giant metric $g $ of order $\alpha \in (0,2) $ if it can be written in the form

$$ \begin{align*} g = \frac{\bar{g }}{\rho^\alpha} \end{align*} $$

for some boundary defining function $\rho $ and some smooth nondegenerate metric $\bar {g }$ . For $\varepsilon>0 $ , we again set $M_\varepsilon $ to be the Riemannian manifold with boundary ${\{x\in M : \rho (x)> \varepsilon \}}$ . We once again wish to study the behaviour of the mean first escape time $\mathbb {E}(\tau _\varepsilon ^x)$ of the Brownian motion $(X_t^x,\mathbb {P}^x_t)$ on $(M,g)$ starting at $x\in M $ as $\varepsilon \to 0 $ . Once again, by [Reference Nursultanov, Tzou and Tzou31, Appendix A], $u_\varepsilon (x)= \mathbb {E}(\tau _\varepsilon ^x )$ satisfies the boundary value problem (1-1),

$$ \begin{align*} \Delta_g u_\varepsilon(x)|_{x\in M_\varepsilon^\circ }= -1 {,} \quad u_\varepsilon(x)|_{x\in \partial M_\varepsilon} = 0 \quad \text{for } u\in H^2(M_\varepsilon)\cap H^1_0(M_\varepsilon). \end{align*} $$

The following theorem shows that the behaviour of Brownian motion on gas giant surfaces differs significantly from Brownian motion on asymptotically hyperbolic surfaces. We show that the mean first escape time on a gas giant surface $u_{\varepsilon }(x) = \mathcal O(1) $ as $\varepsilon \to 0 $ .

Theorem 4.1. Assume that $(M,g) $ is a gas giant surface of order $\alpha $ and that $u_\varepsilon (x) $ satisfies the boundary value problem (1-1). Then, there are functions $\tilde U_{\varepsilon ,\alpha }(x),r_{\varepsilon ,\alpha }(x) $ such that

$$ \begin{align*} u_\varepsilon(x) =\tilde U_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) , \end{align*} $$

where for each fixed $x\in M^\circ $ , the function $\tilde U_{\varepsilon ,\alpha }(x) = \mathcal O(1)$ as $\varepsilon \to 0 $ and there is a constant $D_{x}$ such that $|r_{\varepsilon ,\alpha }(x)|\leq D_x \varepsilon $ for sufficiently small $\varepsilon $ . Finally, if $K \Subset M^\circ $ is compact, then there is a smooth function $\tilde U_\alpha \in C^\infty (K)$ and a constant $D_K $ such that $\tilde U_{\varepsilon ,\alpha }(x)$ converges uniformly on $K $ to $\tilde U_\alpha $ and $\sup _{x\in K }|r_\varepsilon (x)| \leq D_K \varepsilon $ .

The proof of Theorem 4.1 follows the same steps as the proof of Theorem 1.1. Notice first that by Lemma 2.6, we can once again reduce the boundary value problem (1-1) to

$$ \begin{align*} \Delta_{\bar{g}_0}u_\varepsilon(x) = -\frac{1+ \beta^{(1)}(\theta)\rho_0(x) + \beta^{(2)}(\rho_0,\theta)}{\rho_0^\alpha(x)}. \end{align*} $$

We define the functions $v_{\varepsilon ,\alpha }$ by

(4-1) $$ \begin{align} v_{\varepsilon,\alpha}(x):= \begin{cases} \displaystyle\frac{1}{(\alpha -1) (2-\alpha )}\bigg[\rho_0 ^{2-\alpha } \, _2F_1(1,2-\alpha ;3-\alpha ;2 \rho_0 ) \\\qquad\qquad -{\varepsilon }^{2-\alpha } \, _2F_1(1,2-\alpha ;3-\alpha ;2 \varepsilon )\bigg] &\text{if } \alpha\neq 1 ,\\\qquad +\displaystyle \frac{2^{\alpha -2} (\log (1-2 \rho_0 )-\log (1-2 \varepsilon ))}{\alpha -1} \\\displaystyle\frac{1}{2} (\text{Li}_2(1-2 \varepsilon )-\text{Li}_2(1-2\rho_0))\quad &\text{if } \alpha = 1, \end{cases} \end{align} $$

where $\, _2F_1$ is the Hypergeometric function [Reference Abramowitz and Stegun1, Ch. 15] and $\text {Li}_2$ is the Dilogarithm function [Reference Abramowitz and Stegun1, Section 27.7]. On the model gas giant surface $(\mathbb {D}\setminus \mathbb {D}_{1- \bar \delta },g_{\mathrm {Euc}}/\varrho ^\alpha )$ , the functions $v_{\varepsilon ,\alpha }(x) $ solve the boundary value problem (1-1) exactly.

Using the power series expansion of the Hypergeometric function

$$ \begin{align*} \, _2F_1(a,b,c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n }{(c)_n } \frac{z^n }{n!} \quad \text{where} \, (d)_n = d \cdot (d+1) \cdots (d+n-1) \end{align*} $$

around $0 $ [Reference Abramowitz and Stegun1, 15.1.1], and the Dilogarithm

$$ \begin{align*} \text{Li}_2(z) = \sum_{k=1}^\infty (-1)^k \frac{(z-1)^k }{k^2} \end{align*} $$

around 1 [Reference Abramowitz and Stegun1, 27.7.2], we find that the functions $v_{\varepsilon ,\alpha }$ satisfy the expansions around $\varepsilon ,\rho _0 = 0 $ ,

(4-2) $$ \begin{align} v_{\varepsilon,\alpha}(x)= \frac{1}{\alpha-1}\bigg(\frac{1}{2-\alpha}( \rho_0^{2-\alpha}-\varepsilon^{2-\alpha})+{2^{\alpha -1} \varepsilon }-{2^{\alpha -1} \rho_0}\bigg) + \mathcal{O}(\rho_0^2) + \mathcal{O}(\varepsilon^2) \end{align} $$

if $\alpha \neq 1$ , and

$$ \begin{align*} v_{\varepsilon,1} = \rho_0 (-\log \rho_0+1-\log 2)+\varepsilon (\log \varepsilon -1+\log 2) + \mathcal{O}(\rho_0^2\log \rho_0 )+ \mathcal{O}(\varepsilon^2 \log \varepsilon). \end{align*} $$

Now, if $\chi (x) \in C^\infty (\overline M )$ is a cut-off function depending only on $\rho _0(x) $ , with the properties that $\chi |_{\rho _0(x) < \delta /2 }\equiv 1 $ and $\mathrm {supp}\ \chi \subset \overline M \setminus M_\delta $ , then $\chi v_{\varepsilon ,\alpha } \in C^\infty (M_\varepsilon )$ and we have the following lemma, similar to Proposition 3.1.

Lemma 4.2. Let $w_{\varepsilon ,\alpha }(x) = u_{\varepsilon ,\alpha }(x)-\chi (x)\cdot v_{\varepsilon ,\alpha }(x)$ . Then, there are smooth functions $f_{1,\alpha },f_{2,\alpha }\in C^\infty (M_\varepsilon ) $ such that

(4-3) $$ \begin{align} \Delta_{\bar{g}_0}w_{\varepsilon,\alpha}(x) = \begin{cases} \displaystyle\frac{1}{(\alpha-1)(2-\alpha)}f_{1,\alpha}(x) \varepsilon^{2-\alpha} +f_{2,\alpha}(x) \quad &\text{if } \alpha> 1,\\ \displaystyle -f_{1,1}(x)\varepsilon(\log \varepsilon-1+\log 2) +f_{2,1}(x) \quad &\text{if } \alpha = 1,\\ \displaystyle \frac{-2^{\alpha-1}}{\alpha-1}f_{1,\alpha} (x)\varepsilon + f_{2,\alpha} (x)&\text{if } \alpha< 1. \end{cases} \end{align} $$

Furthermore, $f_{1,\alpha } $ extends to a smooth function that is 0 on $\overline {M}\setminus M_\varepsilon $ . The integral

(4-4) $$ \begin{align} I_{1,\alpha}(x)=\int_{M_\varepsilon} G_0(x;y) f_{1,\alpha}(y) \ \mathrm{dVol}_{\bar{g}_0}(y) &= \chi(x) -1, \end{align} $$

and for each $x\in M^\circ $ , the integral satisfies the estimate

(4-5) $$ \begin{align} I_{2,\alpha}(x)=\int_{M_\varepsilon}G_0(x;y) f_{2,\alpha}(y) \ \mathrm{dVol}_{\bar{g}_0}(y) &= \mathcal{O}(1), \end{align} $$

as $\varepsilon \to 0 $ .

Finally, if $K\Subset M^\circ $ is a compact set, then the function $\tilde u_{\varepsilon ,\alpha }:= I_{2,\alpha }(x) $ converges uniformly on $K $ to some bounded smooth function $\tilde u_\alpha (x) \in C^\infty (K)$ given by

$$ \begin{align*} \tilde u_{\alpha}(x) : = \int_{\overline M }G_0(x;y) f_{2,\alpha}(y) \ \mathrm{dVol}_{\bar g_0}(y). \end{align*} $$

Proof. The proof is identical to the proof of Proposition 3.1, by replacing $v_\varepsilon $ from (3-1) by $v_{\varepsilon ,\alpha }$ from (4-1), since the integral

$$ \begin{align*} \int_{M_\varepsilon}G_0(x;y) \rho_0^{1-\alpha}(y) \ \mathrm{dVol}_{\bar g_0} (y ) \end{align*} $$

converges to the integral

$$ \begin{align*} \int_{\overline{M}} G_0(x;y) \rho_0^{1-\alpha}(y) \ \mathrm{dVol}_{\bar g_0}( y) \end{align*} $$

as $\varepsilon \to 0 $ , as for every fixed $x \in M^\circ $ , we may bound $|G_0(x;y)\rho _0(y)^{1-\alpha }| \leq C_x \rho _0(y)^{2-\alpha }$ for some constant $C_x $ by Lemma 2.7.

Proposition 3.2 can also be modified for the gas giant geometries.

Lemma 4.3. Let $f_{1,\alpha },f_{2,\alpha }\in C^\infty (\overline {M })$ be the functions from Lemma 4.2. Then, there are functions $g_{i,\alpha }:\partial M_\varepsilon \to \mathbb {R}$ such that for $z\in \partial M_\varepsilon , \alpha \in (0,2)$ , the following estimates hold. If $\alpha> 1 $ , then

$$ \begin{align*} \int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y)(f_{1,\alpha}(y)\varepsilon^{2-\alpha}+f_{2,\alpha}(y))\ \mathrm{dVol}_{\bar{g}_0}(y) = g_{1,\alpha}(z) \varepsilon^{2-\alpha} + g_{2,\alpha} (z), \end{align*} $$

if $\alpha = 1$ , then

$$ \begin{align*} \begin{aligned} &\int_{M_\varepsilon}\partial_{\nu_x}G_0(z;y)(-f_{1,1}(y)\varepsilon(\log \varepsilon-1+\log 2 )+f_{2,1}(y))\ \mathrm{dVol}_{\bar{g}_0}(y) \\& \qquad= g_{1,1}(z) \varepsilon + g_{2,1}(z) \end{aligned} \end{align*} $$

and if $\alpha < 1 $ , then

$$ \begin{align*} \int_{M_\varepsilon}\partial_{\nu_z} G_0(z;y)(f_{1,\alpha}(y)\varepsilon + f_{2,\alpha})\ \mathrm{dVol}_{\bar g_0}(y) = g_{1,\alpha}(z) \varepsilon^{2-\alpha} + g_{2,\alpha}(z) \end{align*} $$

with $\sup _{z\in \partial M_\varepsilon }|g_{i,\alpha }(z)| = \mathcal {O}(1) $ , as $\varepsilon \to 0 $ .

Proof. The proof is identical to the proof of Proposition 3.2 with the change that the integral

$$ \begin{align*} \sup_{z\in \partial M_\varepsilon}\bigg|\int_{M_\varepsilon}\partial_{\nu_z}G_0(z;y) \beta^{(1)}(\theta(y)) \rho_0^{1-\alpha}(y)\ \mathrm{dVol}_{\bar{g}_0}(y)\bigg| = \mathcal{O}(1) \end{align*} $$

as $\varepsilon \to 0 $ .

A consequence of these lemmas is the following proposition, similar to Proposition 3.4.

Proposition 4.4. The following estimate holds:

$$ \begin{align*} \|\partial_{\nu} w_{\varepsilon,\alpha}(z)\|_{L^2(\partial M_\varepsilon)} = \mathcal{O}(1) \quad \text{as } \varepsilon\to 0. \end{align*} $$

Proof. The proof is once again identical to the proof of Proposition 3.4 with the change that since the limit $ \lim _{\varepsilon \to 0 } \varepsilon ^{2-\alpha } = 0$ ,

$$ \begin{align*} \begin{aligned} \|\partial_{\nu}w_{\varepsilon,\alpha}(z)\|_{L^2(\partial M_\varepsilon)} &\leq \bigg\|\bigg(\frac{1}{2} (I+N_\varepsilon^\#)\bigg)^{-1}\bigg\|_{L^2\to L^2}\cdot \|g_{1,\alpha}(z)\varepsilon^{2-\alpha} + g_{2,\alpha}(z)\|_{L^2(\partial M_\varepsilon)} \\ &= \mathcal{O}(1) \end{aligned} \end{align*} $$

by Proposition 3.7 and Lemma 4.3.

To finish the proof of Theorem 4.1, we modify Proposition 3.8.

Proposition 4.5. Let $\tilde u_{\varepsilon ,\alpha }(x) $ be the function defined in Lemma 4.2. For $x\in M^\circ $ , there is a function $r_{\varepsilon ,\alpha }(x) $ such that

$$ \begin{align*} w_{\varepsilon,\alpha}(x) = \begin{cases} \tilde u_{\varepsilon,\alpha}(x)+\displaystyle\frac{1}{(\alpha-1)(2-\alpha)}(\chi(x)-1) \varepsilon^{2-\alpha} + r_{\varepsilon,\alpha}(x)\quad &\text{if } \alpha> 1,\\ \tilde u_{\varepsilon,1}(x)+(\chi(x)-1) \varepsilon(\log\varepsilon-1+\log 2) + r_{\varepsilon,\alpha}(x)& \text{if } \alpha = 1,\\ \tilde u_{\varepsilon,\alpha}(x) -\displaystyle\frac{2^{\alpha-1}}{\alpha-1} (\chi(x)-1) \varepsilon + r_{\varepsilon,\alpha}(x)&\text{if } \alpha< 1, \end{cases} \end{align*} $$

with a positive constant $D_{x,\alpha } $ depending on $ x $ such that $|r_{\varepsilon ,\alpha }(x)|\leq D_{x,\alpha } \varepsilon $ for sufficiently small $\varepsilon $ . Furthermore, if $K\Subset M^\circ $ is compact, then there is a positive constant $D_{K,\alpha }$ such that $\sup _{x\in K }|r_{\varepsilon ,\alpha }(x)|\leq D_{K,\alpha }\varepsilon $ .

Proof. By Green’s formula,

$$ \begin{align*} \begin{aligned} w_{\varepsilon,\alpha}(x) &= \int_{M_\varepsilon}G_0(x;y)\Delta_{\bar g_0 ,y}w_{\varepsilon,\alpha}(y)\ \mathrm{dVol}_{\bar g_0}(y)-\int_{\partial M_\varepsilon}G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y). \end{aligned} \end{align*} $$

By (4-3), it follows that

$$ \begin{align*} \Delta_{\bar{g}_0}w_{\varepsilon,\alpha}(x) = \begin{cases} \displaystyle\frac{1}{(\alpha-1)(2-\alpha)}f_{1,\alpha}(x) \varepsilon^{2-\alpha} +f_{2,\alpha}(x) \quad &\text{if } \alpha> 1,\\ f_{1,1}(x)\varepsilon(\log\varepsilon-1+\log 2) +f_{2,1}(x) \quad &\text{if } \alpha = 1,\\ \displaystyle\frac{-2^{\alpha-1}}{\alpha-1}f_{1,\alpha} (x)\varepsilon + f_{2,\alpha} (x)&\text{if } \alpha< 1. \end{cases} \end{align*} $$

Furthermore, by (4-4),

$$ \begin{align*} \int_{M_\varepsilon} G_0(x;y) f_{1,\alpha}(y) \ \mathrm{dVol}_{\bar g_0}(y) = \chi(x) -1 , \end{align*} $$

and for each $x\in M^\circ $ , by (4-5),

$$ \begin{align*} \tilde u_{\varepsilon,\alpha}(x) = \int_{M_\varepsilon} G_0(x;y) f_{2,\alpha}(y) \ \mathrm{dVol}_{\bar g_0 }(y) = \mathcal{O}(1) \quad \text{as } \varepsilon\to 0. \end{align*} $$

Finally, if $K\Subset M^\circ $ is compact, then there is a smooth function $\tilde u_\alpha (x) \in C^\infty (K)$ such that $\tilde u_{\varepsilon ,\alpha }$ converges uniformly to $\tilde u_\varepsilon $ on $K $ .

We set

(4-6) $$ \begin{align} r_{\varepsilon,\alpha}(x): = -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y). \end{align} $$

By Proposition 4.4, it follows that $\|\partial _\nu w_\varepsilon (z)\|_{L^2(\partial M_\varepsilon )} = \mathcal {O}(1)$ as $\varepsilon \to 0 $ . By Lemma 2.7, it follows that there is a constant $C_x $ such that $\sup _{y\in \partial M_\varepsilon }|G_0(x;y)| \leq C_x \varepsilon $ . Thus, there is a constant $D_{x,\alpha } $ such that

$$ \begin{align*} \begin{aligned} |r_{\varepsilon,\alpha}(x)| &= \bigg|\int_{\partial M_\varepsilon}G_0(x;y)\partial_\nu w_\varepsilon(y) \ \mathrm{dVol}_h(y) \bigg| \\ &\leq \|G_0(x;\cdot)\|_{L^2(\partial M_\varepsilon)}\cdot \|\partial_\nu w_\varepsilon\|_{L^2(\partial M_\varepsilon)} \leq D_{x,\alpha} \varepsilon. \end{aligned} \end{align*} $$

If $K\Subset M^\circ $ is compact, then by taking $D_{K,\alpha } = \sup _{x\in K }D_{x,\alpha }$ , we can conclude that $\sup _{x\in K}|r_{\varepsilon ,\alpha }(x)| \leq D_{K,\alpha }\varepsilon $ . Combining these facts, if $\alpha> 1$ , then by (4-3)–(4-6),

$$ \begin{align*} \begin{aligned} w_{\varepsilon,\alpha}(x)&= \int_{M_\varepsilon}G_0(x;y) \Delta_{\bar g_0} w_{\varepsilon,\alpha}(x)\ \mathrm{dVol}_{\bar{g}_0}(y) -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y)\\&=\int_{M_\varepsilon}G_0(x;y) \bigg(\frac{\varepsilon^{2-\alpha}}{(\alpha-1)(2-\alpha)}f_{1,\alpha}(y) +f_{2,\alpha}(y)\bigg)\ \mathrm{dVol}_{\bar{g}_0}(y) \\&\qquad -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y)\\&= \frac{1}{(\alpha-1)(2-\alpha)}(\chi(x)-1) \varepsilon^{2-\alpha} + \tilde u_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) , \end{aligned} \end{align*} $$

if $\alpha =1 $ , then

$$ \begin{align*} \begin{aligned} w_{\varepsilon,1} (x)&= \int_{M_\varepsilon}G_0(x;y) \Delta_{\bar g_0} w_{\varepsilon,1}(x)\ \mathrm{dVol}_{\bar{g}_0}(y) -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y)\\&= \int_{M_\varepsilon}G_0(x;y) (f_{1,1}(y) \varepsilon(\log\varepsilon-1+\log 2) +f_{2,1}(y))\ \mathrm{dVol}_{\bar{g}_0}(y) \\&\qquad -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,1}(y) \ \mathrm{dVol}_h(y)\\& = (\chi(x)-1) \varepsilon(\log\varepsilon-1+\log 2) + \tilde u_{\varepsilon,1}(x) + r_{\varepsilon,\alpha}(x) , \end{aligned} \end{align*} $$

if $\alpha < 1 $ , then

$$ \begin{align*} \begin{aligned} w_{\varepsilon,\alpha} (x)&= \int_{M_\varepsilon}G_0(x;y) \Delta_{\bar g_0} w_{\varepsilon,\alpha}(x)\ \mathrm{dVol}_{\bar{g}_0}(y) -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y)\\&= \int_{M_\varepsilon}G_0(x;y) \bigg(\frac{-2^{\alpha-1}}{\alpha-1}f_{1,\alpha}(y) \varepsilon+f_{2,\alpha}(y)\bigg)\ \mathrm{dVol}_{\bar{g}_0}(y) \\&\qquad -\int_{\partial M_\varepsilon} G_0(x;y) \partial_\nu w_{\varepsilon,\alpha}(y) \ \mathrm{dVol}_h(y)\\&= -\frac{2^{\alpha-1}}{\alpha-1}(\chi(x)-1) \varepsilon + \tilde u_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) \end{aligned} \end{align*} $$

with $\tilde u_{\varepsilon ,\alpha } = \mathcal O(1) $ as $\varepsilon \to 0 $ , and a constant $D_{x,\alpha }$ such that $|r_{\varepsilon ,\alpha }(x)| \leq D_{x,\alpha }\varepsilon $ for sufficiently small $\varepsilon $ . If $K\Subset M^\circ $ , then there is a function $\tilde u_{\alpha }$ such that $\tilde u_{\varepsilon ,\alpha }$ converges uniformly to $\tilde u_\alpha $ on $K $ and a constant $D_{K,\alpha }$ such that $\sup _{x\in K }|r_{\varepsilon ,\alpha }(x)| \leq D_{K,\alpha }\varepsilon $ .

Proof of Theorem 4.1

Let $\tilde u_{\varepsilon ,\alpha }$ be the function defined in Lemma 4.2 and set

$$ \begin{align*} \hat{U}_{\varepsilon,\alpha}(x) = \begin{cases} \tilde{u}_{\varepsilon,\alpha}(x) + \chi(x) \bigg(v_{\varepsilon,\alpha}(x) - \displaystyle\frac{\varepsilon^{2-\alpha}}{(\alpha-1)(2-\alpha)}\bigg) &\text{if } \alpha> 1,\\[4pt]\tilde u_{\varepsilon,1}(x) + \chi(x) (v_{\varepsilon,1}(x) - \varepsilon(\log \varepsilon-1 + \log 2 ))\quad &\text{if } \alpha = 1,\\[4pt]\tilde u_{\varepsilon,\alpha}(x) + \chi(x) \bigg(v_{\varepsilon,\alpha}(x) - \displaystyle\frac{2^{\alpha-1}}{\alpha-1}\varepsilon\bigg) & \text{if } \alpha< 1. \end{cases} \end{align*} $$

We have by the definition of $w_{\varepsilon ,\alpha }$ in Lemma 4.2 that

$$ \begin{align*} u_{\varepsilon,\alpha}(x) =w_{\varepsilon,\alpha}(x)+\chi(x)v_{\varepsilon,\alpha}(x). \end{align*} $$

Thus, by the definition of $v_{\varepsilon ,\alpha }$ in (4-1), their expansions in (4-2) and the expansions of $w_{\varepsilon ,\alpha }$ in Proposition 4.5, we find that if $\alpha>1 $ , then

$$ \begin{align*} \begin{aligned} u_{\varepsilon,\alpha}(x) &= \frac{1}{(\alpha-1)(2-\alpha)}(\chi(x)-1) \varepsilon^{2-\alpha} + \tilde u_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) + \chi(x) v_{\varepsilon,\alpha}(x)\\ &= -\frac{1}{(\alpha-1)(2-\alpha)} \varepsilon^{2-\alpha} + \hat U_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) = \tilde U_{\varepsilon,\alpha} (x) + r_{\varepsilon,\alpha}(x) , \end{aligned} \end{align*} $$

if $\alpha = 1 $ , then

$$ \begin{align*} \begin{aligned} u_{\varepsilon,1}(x) &= -(\chi(x)-1)\varepsilon(\log\varepsilon-1+\log 2) + \tilde u_{\varepsilon,1}(x) + r_{\varepsilon,1}(x) + \chi(x) v_{\varepsilon,1}(x)\\ &= \varepsilon(\log\varepsilon-1+\log 2) + \hat U_{\varepsilon,1}(x) + r_{\varepsilon,1}(x) = \tilde U_{\varepsilon,1}(x) + r_{\varepsilon,1}(x) \end{aligned} \end{align*} $$

and if $\alpha <1 $ , then

$$ \begin{align*} \begin{aligned} u_{\varepsilon,\alpha}(x) &= -\frac{2^{\alpha-1}}{\alpha-1}(\chi(x)-1) \varepsilon + \tilde u_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) + \chi(x) v_{\varepsilon,\alpha}(x)\\ &= \frac{2^{\alpha-1}}{\alpha-1} \varepsilon + \hat U_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x) =\tilde U_{\varepsilon,\alpha}(x) + r_{\varepsilon,\alpha}(x)\ , \end{aligned} \end{align*} $$

where for each fixed $x\in M^\circ $ , the function $\tilde U_{\varepsilon ,\alpha }(x) = \mathcal O(1)$ as $\varepsilon \to 0 $ and for each fixed $x\in M^\circ $ , there is a constant $D_{x}$ such that $|r(x)|\leq D_x \varepsilon $ . Finally, if $K \Subset M^\circ $ is compact, then there is a smooth function $\tilde U_\alpha \in C^\infty (K)$ and a constant $D_K $ such that $\tilde U_{\varepsilon ,\alpha }(x)$ converges uniformly on $K $ to $\tilde U_\alpha $ and $\sup _{x\in K }|r_\varepsilon (x)| \leq D_K \varepsilon $ .

5 Limiting behaviour of MFET on gas giant surfaces to asymptotically hyperbolic surfaces

In this section, we examine the behaviour of the mean first escape times of Brownian motion on the unit disc $\mathbb {D} $ with a family of gas giant metrics

(5-1) $$ \begin{align} g_\alpha = \frac{2^\alpha ({d} x _1^2 + {d} x_2^2 )}{(1-x_1^2-x_2^2 )^\alpha} \end{align} $$

as $\alpha \to 2$ , that is, as the metric $g_\alpha $ tends from gas giant to the Poincaré metric

$$ \begin{align*} g_{PC} = \frac{4 ({d} x_1^2 + {d} x_2 ^2 )}{(1-x_1^2 -x_2^2)^2}. \end{align*} $$

We proved in Theorem 4.1 that if $\alpha \in (0,2)$ , then the mean first escape time of the Brownian motion starting at $x \in M^\circ $ satisfies $u_{\varepsilon ,\alpha }(x)= \mathcal {O}(1)$ as $\varepsilon \to 0 $ , whereas in Theorem 1.1, we proved that the mean first escape time of the Brownian motion starting at $x \in M^\circ $ satisfies $u_{\varepsilon ,2}(x)= \mathcal {O}(-\log \varepsilon )$ as $\varepsilon \to 0 $ . In this section, we examine for $x = (x_1,x_2) \in \mathbb {D} $ the behaviour of $u_{\varepsilon ,\alpha }(x)$ , when $\varepsilon \to 0 $ , $\alpha \to 2 $ and $\rho _0(x) \to \varepsilon $ simultaneously. We show here that this behaviour can be described completely by localising the three variables $\rho _0(x), \varepsilon $ and $\alpha $ into various regimes, such that the function $u_{\varepsilon ,\alpha }$ is nice with respect to the natural coordinate systems defining said regimes. By ‘nice’, we mean that it admits an asymptotic expansion in those coordinates or, in more technical (and accurate) language, that it is polyhomogeneous conormal. We describe this in detail after doing a few calculations.

In (4-1), the solutions $u_{\varepsilon ,\alpha }(x)$ to the boundary value problem (1-1) on the unit disc $\mathbb {D} $ with the family of metrics $g_\alpha $ from (5-1) are given by

$$ \begin{align*} \begin{aligned} u_{\varepsilon,\alpha}(x) &= \frac{\rho_0 ^{2-\alpha } \, _2F_1(1,2-\alpha ;3-\alpha ;2 \rho_0 ) - {\varepsilon }^{2-\alpha } \, _2F_1(1,2-\alpha ;3-\alpha ;2 \varepsilon )}{(\alpha -1) (2-\alpha )}\\&\qquad +\frac{2^{\alpha -2} (\log (1-2 \rho_0 )-\log (1-2 \varepsilon ))}{\alpha -1} , \end{aligned} \end{align*} $$

where $1 < \alpha < 2$ and $\rho _0 =\rho _0(x) = \frac {1}{2}(1-x_1^2-x_2^2)$ . We can rewrite this as

$$ \begin{align*} \begin{aligned} u_{\varepsilon,\alpha}(x)&= \frac{\rho_0^{2-\alpha}}{(2-\alpha)(\alpha-1)}\bigg(1-\bigg(\frac{\varepsilon}{\rho_0} \bigg)^{2-\alpha} \bigg)\\&\qquad +\frac{\rho_0^{2-\alpha}}{(\alpha-1)}\bigg(f_{\alpha}(2\rho_0) - \bigg(\frac{\varepsilon}{\rho_0}\bigg)^{2-\alpha} f_{\alpha}\bigg(2\frac{\varepsilon}{\rho_0}\rho_0\bigg)\bigg) \\&\qquad + \frac{2^{\alpha -2} \bigg(\log (1-2 \rho_0 )-\log \bigg(1-2 \frac{\varepsilon}{\rho_0}\rho_0\bigg)\bigg)}{\alpha -1} , \end{aligned} \end{align*} $$

where

$$ \begin{align*}f_{\alpha}(z) = \frac{\, _2F_1(1,2-\alpha ;3-\alpha ;z) -1 }{(2-\alpha)} \in C^{\infty}\bigg(\bigg[0,\frac 12\bigg)_{z}\times (1,2]_\alpha\bigg) ,\end{align*} $$

such that $f_\alpha (0) = 0 $ and $\sup _{z\in [0,1/2)}|f_{\alpha }(z)| \leq C$ for some $C>0 $ uniformly as $\alpha \to 2 $ . There are now three limits we wish to understand:

1. (1) the limit as $\varepsilon \to 0 $ ;

2. (2) the limit as $\rho _0 \to 0 $ , note that this also requires $\varepsilon \to 0 $ simultaneously; and

3. (3) the limit as $\alpha \to 2^{-}$ .

We intend to describe this limiting behaviour using the theory of polyhomogeneous conormal distributions on manifolds with corners. Despite the technical-sounding jargon, this means that we find coordinate patches, valid in various regimes of $(x, \varepsilon , \alpha )$ space, such that $u_{\varepsilon , \alpha }(x)$ admits an asymptotic expansion when written in those coordinates. However, both to discover these coordinates and to prove the corresponding regularity statements, we use the theory of radial blow-ups developed by Melrose. For an introduction to the subject and a comprehensive overview, we refer the reader to [Reference Grieser16, Reference Melrose28, Reference Melrose29, Reference Schulze33]. In general, given some manifold with corners $M $ and a submanifold $N \subset M $ (satisfying some conditions), we construct a new manifold with corners $M' :=[M,N] $ together with a surjective map $\pi : [M,N] \to M $ called the blow-down map, such that $\pi : [M,N]^\circ \to M^\circ $ is a diffeomorphism, by replacing the submanifold $N $ by a local product of a sphere $S^{n-k-1}$ of the right dimension and $\mathbb {R}^k $ (in the language of the b-calculus, such a map is a b-map, see [Reference Grieser16, Section 2.3.2] for more details). At every step, the blow-up produces new systems of coordinates valid on different subsets of the space; here, we specify at each step what these new valid coordinates are. Thus, the reader can take away simply that the function is nice in the sense that nice coordinates can be found according to a breakdown of the space into coordinate patches produced by our blow-up process. All of this is visualised using the blow-up picture in Figure 1.

Figure 1

Required blow-up space to resolve the function $ U(y,\rho _0,\beta )$ from (5-2).

Figure 1 Long description

The diagram is anchored by three axes. The vertical axis points upward and is labeled beta = 2 minus alpha. The horizontal axis extending to the bottom-left is labeled xi = 1 all over log(1 all over rho sub 0). The horizontal axis extending to the bottom-right is labeled eta = 1 all over log(rho sub 0 all over epsilon). At the center of the diagram is a dome-like structure divided into regions by dashed lines.* Region I is at the top of the dome* Regions II a and II b are on the left side of the dome.* Regions III a and III b are on the right side of the dome. Two cylindrical surfaces extend from the dome along the horizontal axes.* The structure divides the three-dimensional space into regions. The region surrounding the beta axis is labelled A. The region surrounding the left cylinder is labelled B1 at the bottom and B2 at the top of the cylinder. The region surrounding the right cylinder is labelled C1 at the bottom and C2 at the top of the cylinder. Dotted lines indicate the internal axes.

First, we begin by considering the manifold with corners

$$ \begin{align*} X_0 = [0,1)_y \times \bigg[0,\frac{1}{2}\bigg)_{\rho_0}\times [0,1)_\beta , \end{align*} $$

where, for $\rho _0 \neq 0$ , we set

$$ \begin{align*} y = \varepsilon /\rho_0, \quad \rho_0 = \rho_0, \quad \beta = 2-\alpha. \end{align*} $$

The functions $\{ y, \rho _0, \beta \}$ still provide global boundary defining functions for the space we are analysing. This is because $0\leq y = \varepsilon /\rho _0 \leq 1 $ is bounded. On this manifold with corners, we analyse the function

$$ \begin{align*} U(y,\rho_0,\beta) := u_{y\rho_0,2-\beta}(\rho_0). \end{align*} $$

It turns out that $X_0 $ together with the map

$$ \begin{align*} \varpi_0: X_0 \to \bigg[0,\frac 12\bigg)_\varepsilon \times \bigg[\varepsilon,\frac 12\bigg)_{\rho_0} \times (1,2]_\alpha :(y,\rho_0,\beta)\mapsto (y\rho_0,\rho_0,2-\beta ) \end{align*} $$

is a blow-up space of the original space $[0,\frac {1}{2})_{\varepsilon }\times [\varepsilon ,\frac 12)_{\rho _0}\times (1,2]_\alpha $ .

In these coordinates, we now find that

(5-2) $$ \begin{align} U(y,\rho_0,\beta)&= \frac{\rho_0^\beta(1-y^\beta)}{\beta(1-\beta)} + \frac{\rho_0^{\beta}(\overline f(2\rho_0,\beta) - y^{\beta}\overline f (2y \rho_0,\beta)) }{(1-\beta)} \nonumber\\&\qquad + \frac{2^\beta (\log (1-2\rho_0)- \log (1-2y\rho_0))}{1-\beta}, \end{align} $$

where $\overline f (z,\beta ) = f_{2-\beta }(z) \in C^\infty ([0,\frac 12)_z\times [0,1)_\beta )$ . By [Reference Grieser16, Remark 2.4], we note that this function is not a polyhomogeneous conormal function; indeed, such functions cannot have coordinate-dependent exponents. To start to resolve this, we introduce the coordinate changes

$$ \begin{align*} \eta = -1/{\log} (y)\quad\text{and}\quad {\xi = -1/{\log}(\rho_0)}. \end{align*} $$

This amounts to changing the smooth structure of $X_0$ , and is equivalent to performing logarithmic blow-ups $[X_0, B_0^1]_{\log } $ and $[X_0,B_0^2]_{\log }$ of the boundary hypersurfaces ${B_0^1 = \{y = 0\}\subset X_0}$ and ${B_0^2 = \{\rho _0 = 0\} \subset X_0 }$ with the property that the iterated logarithmic blow-up commutes [Reference Melrose29, page V.28], that is, the space $X_1 $ ,

$$ \begin{align*} X_1 = [[X_0,B_0^1]_{\log},B_0^2]_{\log}= [[X_0,B_0^2]_{\log},B_0^1]_{\log} , \end{align*} $$

is a well-defined blow-up of $X_0 $ with a surjective map $\varpi _1: X_1 \to X_0 $ satisfying the properties of a blow-down map and with a $C^\infty $ structure of a manifold with corners (see [Reference Melrose29, page V.28] for more information on this coordinate change—one of the characteristics of this coordinate change is that finite order vanishing turns into infinite order vanishing).

On the space $X_1 $ with global boundary defining functions $\{ \eta ,\xi ,\beta \}$ , the function $\varpi ^*_1 U$ is given by

(5-3) $$ \begin{align} (\varpi_1^* U)(\eta,\xi,\beta) &= \frac{e^{-\beta/\xi}}{\beta(1-\beta)}\bigg(1-e^{-\beta/\eta}\bigg) \nonumber\\&\qquad+ \frac{e^{-\beta/\xi}}{(1-\beta)}\bigg(\overline f(2e^{-1/\xi},\beta) + e^{-\beta/\eta}\overline f (2e^{-1/\xi\eta},\beta)\bigg) \nonumber\\&\qquad + \frac{2^{\beta}}{1-\beta}{\bigg(\log (1-2e^{-1/\xi}) - \log (1-2e^{-1/\xi\eta})\bigg)}. \end{align} $$

We show that there is a blow-up space $X $ of $X_1 $ in which $\varpi ^*_2 (\varpi _1 ^*U)$ is polyhomogeneous conormal.

We do this by first considering the third term in the sum of (5-3). The function

$$ \begin{align*} \frac{2^\beta (\log (1-2e^{-1/\xi})- \log (1-2e^{-1/\xi\eta}))}{1-\beta} \end{align*} $$

is smooth on $[0,1)_\eta \times [0,1)_\xi \times [0,1)_{\beta }$ and infinitely vanishing at the boundary hypersurfaces given by $B_1^1=\{ \eta = 0\} \subset X_1$ and $B_1^2= \{ \xi = 0 \}\subset X_1$ .

Next, we consider the function

$$ \begin{align*} \overline{U}(\eta,\xi,\beta) = \frac{e^{-\beta/\xi}}{\beta(1-\beta)}(1-e^{-\beta/\eta}). \end{align*} $$

We first analyse the limits of this function to the three different coordinate axes away from the origin. In Figure 1, these regions are denoted by (A), (B) and (C).

1. (A): $\beta \gg 0 $ fixed. The function $\overline U(\eta ,\xi ,\beta )$ is already smooth around here. This is clear from (5-3), as $e^{-1/\eta }$ and $e^{-1/\xi }$ are smooth functions on $[0,1)_\eta $ and $[0,1)_\xi $ , respectively. We further see infinite order vanishing as $\xi \to 0 $ and a 0 th order vanishing as $\eta \to 0 $ .

2. (B): $\xi \gg 0 $ fixed. Whenever $\xi \gg 0 $ , we can analyse the function

   $$ \begin{align*} \overline U(\eta,\xi,\beta) = \frac{e^{-\beta/\xi}}{\beta(1-\beta)}\big(1-e^{-\beta/\eta}\big) = \frac{e^{-\eta(\beta/\eta\xi)}}{\eta(\beta /\eta)(1-\eta (\beta/\eta))}\big(1-e^{-\beta/\eta}\big) \end{align*} $$
   around the $\xi $ axis. We do a standard blow-up of this axis away from $\xi = 0 $ . We recognise that there are two regimes:

   1. (i) where $\eta> \beta $ and

      $$ \begin{align*}\{\eta,\quad s := \beta/\eta\}\end{align*} $$
      provide boundary defining functions;

   2. (ii) where $\beta> \eta $ and

      $$ \begin{align*}\{\beta, \quad t := \eta/\beta \}\end{align*} $$
      provide boundary defining functions.

   For Case (i), we recognise that

   (5-4) $$ \begin{align} \overline U (\eta,\xi,s) = \frac{e^{-\eta s /\xi}}{\eta(1-\eta s )}\bigg(\frac{1-e^{-s}}{s }\bigg) \end{align} $$
   is polyhomogeneous conormal on $[0,1)_\eta \times [0,1)_s$ , since $(1-e^{-s})/s \in C^\infty [0,1)_s $ . Notice further that (5-4) returns the asymptotics from Theorem 1.1. By the definition of the derivative, we find that the limit $\lim _{s\to 0}(1-e^{-s})/s =1$ , so that $ \overline U(\eta ,\xi ,0)= 1/\eta = \log \rho _0/\varepsilon $ , which is what was found in Theorem 1.1.

   For Case (ii), we find that

   $$ \begin{align*} \overline U(t,\xi,\beta)= \frac{e^{-\beta/\xi}}{\beta(1-\beta)} \big(1-e^{-1/t}\big) \end{align*} $$
   is polyhomogeneous conormal on $[0,1)_{\beta } \times [0,1)_t$ , since $e^{-1/t }\in C^\infty [0,1)_t $ is smooth all the way up to the boundary and rapidly decaying.

3. (C): $\eta \gg 0 $ fixed. We once again rewrite the function $\overline U(\eta ,\xi ,\beta )$

   $$ \begin{align*} \overline U(\eta,\xi,\beta)= \frac{e^{-\beta/\xi}}{\beta(1-\beta)}\big(1-e^{-\beta/\eta}\big) = \frac{e^{-\beta/\xi}}{\xi (\beta/\xi)(1-\xi(\beta/\xi))}\big(1-e^{-\xi(\beta/\xi)/\eta}\big). \end{align*} $$
   We now do a standard blow-up of the $\eta $ -axis away from $\eta = 0 $ . We once again get two regimes:

   1. (i) where $\xi>\beta $ and

      $$ \begin{align*}\{\xi,\quad q: = \beta/\xi\}\end{align*} $$
      provide boundary defining functions;

   2. (ii) where $\beta> \xi $ and

      $$ \begin{align*}\{\beta, \quad r := \xi /\beta \}\end{align*} $$
      provide boundary defining functions.

   For Case (i), we see that

   $$ \begin{align*} \begin{aligned} \overline U(\eta,\xi,q) &= \frac{e^{- q}}{\xi q(1-\xi q )}\big(1-e^{-\xi q/\eta}\big) \\ &= \frac{e^{-q}}{\eta(1-\xi q )}\sum_{k=0}^\infty \frac{1}{(k+1)!}\bigg(\frac{-\xi q}{\eta}\bigg)^k \in C^\infty[0,1)_\xi\times [0,1)_q. \end{aligned} \end{align*} $$
   We can also once again recover $\overline U(\eta ,\xi ,0) = 1/\eta = \log \rho _0/\varepsilon $ here by noting that $q = 0 $ if and only if $\beta = 0$ . For Case (ii), we get that
   $$ \begin{align*} \begin{aligned} \overline U(\eta,r,\beta)&= \frac{e^{-1/r}}{\beta(1-\beta)}\big( 1-e^{-\beta/\eta}\big) \\ &= \frac{e^{-1/r}}{\eta(1-\beta)}\sum_{k=0}^\infty \frac{1}{(k+1)!} \bigg(\frac{-\beta}{\eta}\bigg)^k \in C^\infty[0,1)_\beta \times [0,1)_r \end{aligned} \end{align*} $$
   from which we see that $\overline U(\nu ,\xi ,\mu )$ is polyhomogeneous conormal. We still need this blow-up as the limits from the two directions are different.

Finally, we have the origin to analyse. We consider three regimes and see that the blow-ups of the coordinate axes as above are required. The different regimes are drawn in Figure 1.

1. (I): $\beta> \xi ,\eta $ , where the coordinates

   $$ \begin{align*}\{\gamma:=\eta/\beta,\quad\vartheta:=\xi/\beta ,\quad \beta \}\end{align*} $$
   provide boundary defining functions. Here,
   $$ \begin{align*} \overline U(\gamma,\vartheta,\beta) = \frac{e^{-1/\vartheta }}{\beta(1-\beta)}\big( 1-e^{-1/\gamma } \big) , \end{align*} $$
   which is polyhomogeneous conormal on $[0,1)_\gamma \times [0,1)_\vartheta \times [0,1)_\beta $ .

2. (II): $\xi> \beta ,\eta $ , where the coordinates

   $$ \begin{align*}\{\nu:=\eta/\xi,\quad\xi,\quad\mu:=\beta/\xi \}\end{align*} $$
   provide boundary defining functions. Here,
   $$ \begin{align*} \overline U(\nu,\xi,\mu) = \frac{e^{-\mu}}{ \xi\mu(1-\xi \mu)}\big(1-e^{-\mu/\nu}\big) = \frac{1}{\xi}\bigg( \frac{e^{-\mu}}{\mu(1-\xi \mu)}\big(1-e^{-\mu/\nu}\big)\bigg) , \end{align*} $$
   which is not polyhomogeneous conormal, but we recognise that there are two regimes:

   1. (IIa): $\mu>\nu $ , where

      $$ \begin{align*}\{\sigma := \nu/\mu= \eta/\beta,\quad \xi,\quad \mu= \beta/\xi \}\end{align*} $$
      provide boundary defining coordinates. Here,
      $$ \begin{align*} \overline U(\sigma,\xi,\mu) = \frac{1}{\xi}\bigg(\frac{e^{-\mu}}{\mu(1-\xi \mu)}\big( 1-e^{-1/\sigma}\big) \bigg), \end{align*} $$
      which is polyhomogeneous conormal on $[0,1)_\sigma \times [0,1)_\xi \times [0,1)_\mu $ .

   2. (IIb): $\nu>\mu $ , where

      $$ \begin{align*}\{\nu = \eta/\xi,\quad\xi,\quad \tau:=\mu/\nu = \beta/\eta \}\end{align*} $$
      provide boundary defining functions. Here,
      $$ \begin{align*} \overline U(\nu,\xi,\tau) = \frac{1}{\xi\nu}\bigg(\frac{e^{-\nu \tau}}{\tau(1-\xi\nu\tau)}\big(1-e^{-\tau}\big) \bigg), \end{align*} $$
      which is polyhomogeneous conormal on $[0,1)_\nu \times [0,1)_\xi \times [0,1)_\tau $ . Once again, we recover Theorem 1.1 here, because $\xi \nu = \xi \eta /\xi = \eta $ . Because $\tau = 0 $ if and only if $\beta = 0 $ , we find that $\overline U(\nu ,\xi ,0) = 1/(\xi \nu ) = \log \rho _0/\varepsilon $ in this regime.

3. (III): $\eta> \beta ,\xi $ , where

   $$ \begin{align*}\{\eta , \quad \chi:=\xi/\eta,\quad \phi:=\beta/\eta \}\end{align*} $$
   provide boundary defining functions. Here,
   $$ \begin{align*} \overline U(\eta,\chi,\phi) = \frac{1}{\eta}\bigg(\frac{e^{-\phi/\chi}}{\phi(1-\eta\phi)}(1-e^{-\phi})\bigg), \end{align*} $$
   which is not polyhomogeneous conormal on $[0,1)_\eta \times [0,1)_\phi \times [0,1)_\chi $ , but we once again recognise that there are two regimes:

   1. (IIIa): $\phi>\chi $ , where

      $$ \begin{align*}\{\eta,\quad \psi:=\chi/\phi=\xi/\beta,\quad \phi = \beta/\eta\}\end{align*} $$
      provide boundary defining functions. Here,
      $$ \begin{align*} \overline U(\eta,\psi,\phi) = \frac{1}{\eta}\bigg(\frac{e^{-1/\psi}}{\phi(1-\eta\phi)}\big(1-e^{-\phi}\big) \bigg), \end{align*} $$
      which is polyhomogeneous conormal on $[0,1)_\eta \times [0,1)_\psi \times [0,1)_\phi $ .

   2. (IIIb): $\chi>\phi $ , where

      $$ \begin{align*}\{\eta,\quad \chi=\xi/\eta,\omega:=\phi/\chi =\beta/\xi\}\end{align*} $$

   3. provide boundary defining functions. Here,

      $$ \begin{align*} \overline U(\eta,\chi,\omega)= \frac{1}{\eta}\bigg(\frac{e^{-\omega}}{\chi\omega(1-\eta\chi\omega)}\big( 1-e^{\chi\omega}\big)\bigg), \end{align*} $$
      which is polyhomogeneous conormal on $[0,1)_\eta \times [0,1)_\chi \times [0,1)_\omega $ . Finally, we also recover Theorem 1.1 here. Since $\omega = 0 $ if and only if $\beta = 0 $ , we find that $\overline u_{\varepsilon ,0} = 1/\eta = \log \rho _0/\varepsilon $ in this regime.

We conclude that the function $\overline U(\eta ,\xi ,\beta )$ resolves to a polyhomogeneous conormal function on the iterated blow-up space

(5-5) $$ \begin{align} X = [[X_1 ;\{0\}];\{ 0 \}_\eta\times[0,1)_\xi\times \{0\}_\beta\cup [ 0,1 )_\eta\times\{0\}_\xi\times \{0\}_\beta] , \end{align} $$

by blowing up the origin of the space $X_1 $ and then separately blowing up the coordinate axes, given by the sets ${\{0\}_\eta \times [0,1)_\xi \times \{0\}_\beta }$ and ${[0,1)_\eta \times \{0\}_\xi \times \{0\}_\beta }$ . Figure 1 gives an impression of the blow-up space $X $ as a blow-up from $X_1 $ . By [Reference Grieser16, Remark 3.13], it follows that this space comes with a surjective map $\varpi _{2}:X\to X_1 $ satisfying the properties of a blow-down map. Now, $\varpi = \varpi _2\circ \varpi _1 : X\to X_0$ is a well-defined blow-down map.

Since the functions $\overline f(2y,\beta ) $ and $\overline f(2\rho _0,\beta )$ from (5-2) are smooth on $X_0 $ , they remain smooth after blow-ups [Reference Grieser16] and thus by following the same procedure as before, one can show that the function

(5-6) $$ \begin{align} \frac{e^{-\beta/\xi}}{(1-\beta)}\Big( \overline f(2e^{-1/\xi},\beta)+e^{-\beta/\eta} \overline f(2e^{1/\xi\eta},\beta) \Big) \end{align} $$

resolves to a smooth function on the same space $X $ given by (5-5). Since the function $\overline f (z,\beta )$ vanishes for $z = 0 $ , by the discussion on logarithmic blow-ups, it follows that the order of vanishing of the functions $\overline f(2e^{-1/\xi },\beta )$ and $\overline f(2e^{-1/\xi \eta },\beta ) $ in (5-6) are infinite. We can therefore conclude that there is a manifold with corners $X $ and a surjective map $\varpi : X\to X_0 $ , which is a diffeomorphism on the interior, such that $\varpi ^* U $ is polyhomogeneous conormal on X.

Given the previous discussion about the mean first escape times of Brownian motion on a family of gas giant metrics on the unit disc, we conjecture the following.

Conjecture 5.1. Let $(M,g_\alpha )$ be a two-dimensional smooth manifold with a family of gas giant metrics $g_\alpha $ . Assume that the function $\beta ^{(1)}(\theta )$ from (2-2) is identically equal to 0 and let $U(x,\varepsilon ,\alpha ):= u_{\varepsilon ,\alpha }(x) $ be the solutions to the boundary value problems (1-1) on the gas giant surfaces $(M,g_\alpha )$ . Then, there exists a manifold with corners $Z $ obtained by the blow-up

$$ \begin{align*} Z = \big[[[[Z_0,B_0^1]_{\log},B_0^2]_{\log},\{0\}],\{0\}_\eta\times[0,1)_\xi\times \{0\}_\beta\cup [ 0,1 )_\eta\times\{0\}_\xi\times \{0\}_\beta\big] , \end{align*} $$

where

$$ \begin{align*} Z_0 = \bigg[0,\frac 12\bigg)_{y}\times \partial M \times\bigg[0,\frac 12\bigg)_{\rho_0} \times [0,1)_\beta {,}\quad y =\frac{\varepsilon}{\rho_0}, \quad \beta = 2-\alpha , \end{align*} $$

the hypersurfaces $B_0^1 = \{y = 0\}, B_0^2 = \{\rho _0 = 0 \}\subset Z_0$ and the functions

$$ \begin{align*} \eta = -1/\log (y) \quad \text{and}\quad \xi = -1/\log (\rho_0 ) , \end{align*} $$

together with a surjective map

$$ \begin{align*} \varpi:Z \to \overline{M_\varepsilon}\times \bigg[0,\frac{1}{2}\bigg)_\varepsilon\times (1,2]_\alpha \end{align*} $$

such that $\varpi $ is a blow-down map and such that $\varpi ^* U$ is polyhomogeneous conormal on  $Z $ .

The manifold $Z_0 $ is the natural extension of the manifold $X_0 $ to the general case, by considering

$$ \begin{align*} \overline M_\varepsilon \setminus M_\delta\times \bigg[ 0,\frac{1}{2} \bigg)_\varepsilon \cong \partial M \times \bigg[ \varepsilon,\frac 12 \bigg)_{\rho_0} \times \bigg[ 0,\frac{1}{2} \bigg)_\varepsilon. \end{align*} $$

Then, $Z $ is obtained by performing the same blow-ups as before. The condition of the function $\beta ^{(1)}(\theta )$ being identically 0 is due to Lemma A.2. As a consequence of Lemma A.2, then we get estimates for $\partial _\nu w_\varepsilon = \mathcal O (\log \varepsilon )$ and, consequently, together with Lemma 2.7 only, we get pointwise estimates for $u_{\varepsilon }(x)$ as $\varepsilon \to 0 $ , whereas we require uniform estimates for Conjecture 5.1.

6 Numerical results

In this section, we compare the analytical results from the previous sections with simulations of the Brownian motion and with numerical partial differential equation solvers on the unit disc.

To generate the Brownian motion on the Riemannian manifolds, we use the algorithms described in [Reference Schwarz, Herrmann, Sturm and Wardetzky35]. Up to a rescaling of constants in the generator of the Brownian motion, Algorithms 1 and 2 provide an efficient method of generating Brownian motion on Riemannian manifolds. The resulting generator is a time rescaled version of the generator used in this paper. To account for this fact, we multiply the analytically found solutions by a factor of 2.

Figure 2 shows the mean first escape time of a Monte Carlo simulation of the Brownian motion starting at the origin on the unit disc $\mathbb {D} $ with the gas giant metrics from (5-1) of order $\alpha $ and the Poincaré disc as a function of $\varepsilon $ in blue based on 300 000 simulations.

Figure 2

The mean first escape time of a Monte Carlo simulation of the Brownian motion starting at the origin on the unit disc $\mathbb {D} $ with the gas giant metrics from (5-1) of order $\alpha $ and the Poincaré disc as a function of $\varepsilon $ in blue based on 300 000 simulations. The square marks denote the analytically found solutions in (3-1) and in the gas giant case, in (4-1). The dashed red lines denote the value of the numerically found solution to the boundary value problem (1-1).

Figure 2 Long description

A six-panel grid of line graphs. Each graph has a horizontal x-axis labelled epsilon with tick marks at 0.05, 0.1, and 0.3 in a logarithmic scale, and a vertical y-axis labelled t.* Panel a, alpha equals 0.1. The y-axis ranges from 0 to 2.5. A blue curve starts at t equals 0.5 and curves downward to 0 as epsilon increases. Square marks and a dashed red line overlap the curve.* Panel b, alpha equals 0.5. The y-axis ranges from 0 to 2.5. The curve starts higher, near t equals 0.8, and descends to 0.* Panel c, alpha equals 1.0. The y-axis ranges from 0 to 2.5. The curve starts near t equals 1.4 and descends to 0.* Panel d, alpha equals 1.5. The y-axis scale increases, ranging from 0 to 8. The curve starts near t equals 2.8 and descends almost linearly toward 0.* Panel e, alpha equals 1.9. The y-axis ranges from 0 to 8. The curve starts near t equals 4.8 and descends almost linearly to 0.* Panel f, labelled Asymptotically hyperbolic. The y-axis ranges from 0 to 8. The curve starts at its highest point, near t equals 5.8, and descends in a straight line to 0 at the far right of the x-axis. In all panels, blue circles represent Monte Carlo simulations, square marks represent analytical solutions, and dashed red lines represent numerical boundary value solutions. All three data types align closely in every panel.

The square marks denote the analytically found solutions in (3-1) and, in the gas giant case, in (4-1).

Using Wolfram’s built-in function ParametricNDSolve [Reference Research32], we solved the boundary value problem (1-1) on the disc. The value of this numerically found solution at the origin is given in Figure 2 as the dashed red line. We can see that both the Monte Carlo simulations of the Brownian motion, the analytical solution and the numerical solutions correspond, and are finite as $\varepsilon \to 0 $ if the metric is gas giant and $ -\log (\varepsilon )$ as $\varepsilon \to 0 $ if the metric is asymptotically hyperbolic.

We can also see that indeed as the gas giant parameter $\alpha $ approaches $2 $ , the mean first escape time of the Brownian motion on the gas giant of order $\alpha $ approaches the mean first escape time of the Brownian motion on the asymptotically hyperbolic disc.