Dirichlet boundary value problem in convex domains with discontinuous boundary values

Question

Consider $\Omega$ an open, bounded, and convex domain in $\mathbb{R}^n$. Let $g \in L^{2}(\partial \Omega)$ such that the problem

$$ \left\{ \begin{array}{ccccccc} \Delta u = 0, \ \text{in} \ \Omega, \\ u = g, \ \text{on} \ \partial \Omega \\ \end{array} \right. $$

has a unique solution $u \in H^{1}(\Omega).$

My questions are

1. Let $x_0 \in \partial \Omega$ with $g$ continuous at this point. Is it true that $$ \lim_{y \rightarrow x_0} u(y) = g(x_0) ? $$

2. Supoose that I have found a function $v \in H^{1}(\Omega)$ with $$ \lim_{y \rightarrow x} v(y) = g(x)$$ for all $x \in \partial \Omega.$ Then $v$ is the unique solution (in $ H^{1}(\Omega)$) of the problem that I said above?

I don't know the answer of these questions, but if they are true then I could understand a passage of an article that I am studying.

Could someone please point to me a reference that can help me with the questions above?

Answer 1

Quote from the article:

Let $C$ be the class of smooth, bounded and convex domains in $\mathbb R^n$ such that $K$ belongs to the boundary of the domain. Let $\Omega \in C$, we denote furthermore by $u_\Omega$ the function fulfilling $$\begin{align} \Delta u_\Omega &= 0, \\ u_\Omega &= 0 \text{ in }\partial \Omega \setminus K \\ u_\Omega &= 1 \text{ in } K \end{align}$$

First, we cannot expect $u_\Omega$ to be in $H^1(\Omega)$; the jump between $0$ and $1$ will have "ripple effect" inside the domain, making $|\nabla u|^2$ just large enough for the integral to diverge. A typical example of this behavior, in the plane, is $u(z) = \frac{1}{\pi}\arg z$ on the upper half-plane. Along the real axis, this function jumps from $0$ to $1$ at the origin. (I only consider the local behavior near $0$, which is representative of what happens in your case.) Since $|\nabla u(z)| \approx |z|^{-1}$, the $L^2$ norm of $|\nabla u|$ is infinite.

So, $u_\Omega$ is not a variational (Dirichlet-energy-minimizing) solution, since it has infinite energy. Sobolev spaces don't play a role in its existence. The existence and uniqueness are established with the help of potential theory. Key words: Perron solution, harmonic measure, Poisson kernel, Green function. One reference is section 2.8 of Gilbarg & Trudinger. The Perron solution is uniquely defined for every bounded function on the boundary. It satisfies $\lim_{y \rightarrow x_0} u(y) = g(x_0) $ whenever $g$ is continuous at $x_0$ and $x_0$ is a regular boundary point. In a convex domain, every boundary point is regular.