Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Question

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem

$$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), & q \in \partial \Omega\end{cases},$$ where $g: \partial \Omega \to \mathbb{R}\cup \{\pm \infty\}$ is finite and smooth almost everywhere on $\partial \Omega$ (but possibly infinite or discontinuous on a set of measure zero)? Is there a good references that discusses these types of boundary conditions?

Answer 1

Yes, there are existence results. One way of getting them is Perron's method: take the supremum of all subharmonic functions $u$ in $\Omega $ such that $\limsup_{p\to q}u(p)\le g(q)$ for all $q\in \Omega$. If it does not degenerate to $+\infty$, you have a harmonic function $H_g$. In what sense this harmonic function solves the Dirichlet problem is a tricky issue. It may overshoot or undershoot the boundary values in places. Generally, one is content with having $H_{g}= -H_{-g}$, in which case the boundary value problem is called resolutive. A satisfying result is that this happens if and only if $g$ is integrable with respect to the harmonic measure.

General PDE books do not go into these subtleties; the topic is classified as potential theory. See

1. Chapter 6 of Classical Potential Theory by Armitage and Gardiner
2. Chapter VIII of Classical Potential Theory and its probabilistic counterpart by Doob.

Unfortunately, Doob presents the results in extra-general-with-$h$-on-top setting. I strongly prefer the first reference.