I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition.
Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ and $r>0$ satisfy $B(x,r)\cap D \subsetneq D$. Here, $B(x,r)$ denotes the open ball in $\mathbb{R}^d$ centered at $x$ with radius $r>0$. Let $U$ be an open subset of $\mathbb{R}^d$ such that $B(x,r/2)\subset U \subset B(x,r)$ and $U \cap D$ is a bounded $C^{1,\alpha}$-domain.
Let $H^{1}(D)$ denote the first-order Sobolev space on $D$. That is, \begin{align*} H^{1}(D)=\{u \in L^2(D,dx) \mid \partial u/\partial x_i \in L^2(D,dx),\quad i=1,\ldots,d\}. \end{align*} Here, $\partial u/\partial x_i$ denotes the distributional derivative of $u$. Define $C^{1}(\overline{D})$ and $C^{1}_{U}(\overline{D})$ by \begin{align} C^{1}(\overline{D})&=\{u \in C^{1}(D) \mid \text{all of derivatives of $f$ and $f$ extend to $\overline{D}$ continuously}\}, \\ C^{1}_{U}(\overline{D})&=\{ u \in C^{1}(\overline{D}) \mid u=0 \text{ outside }U \cap \overline{D}\}. \end{align} Let $H_{U}^{1}(D)$ denote the closure of $C^{1}_{U}(\overline{D})$ in $H^{1}(D)$. Let $f \in C^{\infty}_c(\mathbb{R}^d)$ satisfy $f=0$ outside $U \cap \overline{D}.$ Suppose that $u \in H_{U}^{1}(D)$ satisfies \begin{align} \sum_{i,j=1}^d \int_{D}\frac{\partial u}{\partial x_i}(x)\frac{\partial v}{\partial x_j}(x)\,dx+\int_{D}u(x)v(x)\,dx=\int_{D}f(x) v(x)\,dx\quad \text{for any $v \in H_{U}^{1}(D)$}. \end{align}
Heuristically, $u$ satisfies $-\Delta u+u=f$ on $U \cap D$, and the following mixed boundary condition: \begin{align*} u=0\, \text{ on }\,\partial U \cap D\quad\text{ and }\quad \frac{\partial u}{\partial \nu}=0 \,\text{ on }\, U \cap \partial D, \end{align*} where $\nu$ denotes the inward unit normal vector on $\partial D.$
I know that $u$ has a continuous version on $\overline{D}$ which vanishes outside $U \cap \overline{D}$. Can we show that $u$ has a version which belongs to $C^{1}(\overline{D})$ ?
