Boundary value problems with $L^2$ boundary data

Question

Recently, I read the following result from "A Remark on the Regularity of Solutions of Maxwell’s Equations on Lipschitz Domains" by Martin Costabel:

Let $\Omega$ be a bounded Lipschitz domain, $u\in H^1(\Omega)$ and $\Delta u=0$. Then the following conditions are equivalent:

(a) $u\mid_{\partial \Omega}\in H^{1}(\partial \Omega)$

(b) $\frac{\partial u}{\partial n}\mid_{\partial \Omega}\in L^2(\partial \Omega)$

Moreover, each of them implies $u\in H^{3/2}(\Omega)$.

I have found a paper on this result. I'm interested in finding a book or lecture notes on this topic, i.e., boundary value problems for general elliptic equations with $L^2$ boundary data.

I also find an interesting result from Lions' classical book "Homogeneous Boundary Value Problems and Applications", which studies the case where both coefficients and domain are smooth. But the book seems to be too broad. I'm only interested in second-order equations. Does anyone know some brief material on this topic?

Answer 1

This is too long for a comment.

Something is wrong in your statement. Because $v\in H^1(\Omega)$, $v|_{\partial\Omega}$ is naturally in $H^{1/2}$, by the trace theorem. On the other hand, when in addition $\Delta u=0$, the normal derivative $\partial u/\partial\nu$ is only $H^{-1/2}(\partial\Omega)$. Actually, the following is true: given $g\in H^{-1/2}(\partial\Omega)$, with the compatibility condition $\langle g,{\bf1}\rangle_{H^{-1/2},H^{1/2}}=0$, there exists a $u\in H^1(\Omega)$ such that $\Delta u=0$ and $\partial u/\partial\nu=g$ ($u$ is unique up to an additive constant. Therefore your $\partial u/\partial\nu$ is not $L^2$ in general.