Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad.

Denote $U=U_2 \backslash \bar{U_1}$, consider the Dirichlet problem \begin{equation} \Delta u=0 \text{ on }U, \end{equation} \begin{equation} u=1 \text{ on } \partial U_1, \quad u=0 \text{ on } \partial U_2. \end{equation} Let $$ U^r_1=\{x \in \mathbb{R}^n: d(x,U_1)\leqslant r\}, $$ $$ g_r=\max\{ 1-\frac{d(x,U_1)}{r}, 0\}, $$ an extension from $1,0$ function to a Lipschitz function defined on $\bar{U}$. By the standard argument, we can prove that there exists a function $u \in W^{1,2}(U)$ satisfying (0.1) and $u-g_r \in W^{1,2}_0(U)$. For different $r$ and different extensions, the solutions remain unchanged, i.e. the solution is unique.

Now suppose $U_2 \Subset U_3$, consider the Dirichlet problem \begin{equation} \Delta v=0 \text{ on } U_3 \backslash \bar{U}_1, \end{equation} \begin{equation} v=1 \text{ on } \partial U_1, \quad v=0 \text{ on } \partial U_3. \end{equation} Do we have $v\geqslant u$ on $U$?

If $\partial U_i$ are smooth for $i=1,2,3$, then $u\in C(\bar{U})$, $v\in C(\bar{U_3} \backslash U_1)$. $u, v|_{\partial U_1}=1$, $u|_{\partial U_2}=0$ and $v|_{\partial U_3}=0$. Then $v-u \geqslant 0$ on $\partial U_1$ and $\partial U_2$, by the maximum principle, we have $v\geqslant u$. However, the boundaries we considered here are not smooth.

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad.

Denote $U=U_2 \backslash \bar{U_1}$, consider the Dirichlet problem \begin{equation} \Delta u=0 \text{ on }U, \end{equation} \begin{equation} u=1 \text{ on } \partial U_1, \quad u=0 \text{ on } \partial U_2. \end{equation} Let $$ U^r_1=\{x \in \mathbb{R}^n: d(x,U_1)\leqslant r\}, $$ $$ g_r=\max\{ 1-\frac{d(x,U_1)}{r}, 0\}, $$ an extension from $1,0$ function to a Lipschitz function defined on $\bar{U}$. By the standard argument, we can prove that there exists a function $u \in W^{1,2}(U)$ satisfying the above boundary value problem and $u-g_r \in W^{1,2}_0(U)$. For different $r$ and different extensions, the solutions remain unchanged, i.e. the solution is unique.

Now suppose $U_2 \Subset U_3$, consider the Dirichlet problem \begin{equation} \Delta v=0 \text{ on } U_3 \backslash \bar{U}_1, \end{equation} \begin{equation} v=1 \text{ on } \partial U_1, \quad v=0 \text{ on } \partial U_3. \end{equation} Do we have $v\geqslant u$ on $U$?

If $\partial U_i$ are smooth for $i=1,2,3$, then $u\in C(\bar{U})$, $v\in C(\bar{U_3} \backslash U_1)$. $u, v|_{\partial U_1}=1$, $u|_{\partial U_2}=0$ and $v|_{\partial U_3}=0$. Then $v-u \geqslant 0$ on $\partial U_1$ and $\partial U_2$, by the maximum principle, we have $v\geqslant u$. However, the boundaries we considered here are not smooth.

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad.

Denote $U=U_2 \backslash \bar{U_1}$, consider the Dirichlet problem \begin{equation} \Delta u=0 \text{ on }U, \end{equation} \begin{equation} u=1 \text{ on } \partial U_1, \quad u=0 \text{ on } \partial U_2. \end{equation} Let $$ U^r_1=\{x \in \mathbb{R}^n: d(x,U_1)\leqslant r\}, $$ $$ g_r=\max\{ 1-\frac{d(x,U_1)}{r}, 0\}, $$ an extension from $1,0$ function to a Lipschitz function defined on $\bar{U}$. By the standard argument, we can prove that there exists a function $u \in W^{1,2}(U)$ satisfying (0.1) and $u-g_r \in W^{1,2}_0(U)$. For different $r$ and different extensions, the solutions remain unchanged, i.e. the solution is unique.

If $\partial U_i$ is smooth for $i=1,2$, then $u$ is continuous up to the boundary, $u|_{\partial U_1}=1$. However, the boundaries we considered here are not smooth. For a domain $V\subset \mathbb{R}^n$ such that $U_1\Subset V \Subset U_2$, do we have $$ \inf_{\bar{V}\backslash \bar{U_1} }u=\inf_{\partial V}u? $$

For a domain $W \Subset U_1$, consider $$ \Delta w =0 \text{ on } U_2 \backslash \bar{W}, $$ $$ w=1 \text{ on } \partial W, \quad w=0 \text{ on } \partial U_2. $$ Denote $$ \sup_{\partial U} f=\sup\{l: \max\{f-l, 0\} \in W^{1,2}_0(U)\}. $$ Do we have $\sup_{\partial U} (w-u) \leqslant 0$? If this holds, by maximum principle, we can get $w\leqslant u$ on $U$.

Let $U_1$ and $U_2$ be two bounded domains in $\mathbb{R}^n$ such that $U_1 \Subset U_2$. Note that we don't assume $\partial U_i$ to be smooth or Lipschitz, they may be very bad.

Denote $U=U_2 \backslash \bar{U_1}$, consider the Dirichlet problem \begin{equation} \Delta u=0 \text{ on }U, \end{equation} \begin{equation} u=1 \text{ on } \partial U_1, \quad u=0 \text{ on } \partial U_2. \end{equation} Let $$ U^r_1=\{x \in \mathbb{R}^n: d(x,U_1)\leqslant r\}, $$ $$ g_r=\max\{ 1-\frac{d(x,U_1)}{r}, 0\}, $$ an extension from $1,0$ function to a Lipschitz function defined on $\bar{U}$. By the standard argument, we can prove that there exists a function $u \in W^{1,2}(U)$ satisfying (0.1) and $u-g_r \in W^{1,2}_0(U)$. For different $r$ and different extensions, the solutions remain unchanged, i.e. the solution is unique.

Now suppose $U_2 \Subset U_3$, consider the Dirichlet problem \begin{equation} \Delta v=0 \text{ on } U_3 \backslash \bar{U}_1, \end{equation} \begin{equation} v=1 \text{ on } \partial U_1, \quad v=0 \text{ on } \partial U_3. \end{equation} Do we have $v\geqslant u$ on $U$?

If $\partial U_i$ are smooth for $i=1,2,3$, then $u\in C(\bar{U})$, $v\in C(\bar{U_3} \backslash U_1)$. $u, v|_{\partial U_1}=1$, $u|_{\partial U_2}=0$ and $v|_{\partial U_3}=0$. Then $v-u \geqslant 0$ on $\partial U_1$ and $\partial U_2$, by the maximum principle, we have $v\geqslant u$. However, the boundaries we considered here are not smooth.