# Boundary value problems with $L^2$ boundary data

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?

• The author of the paper you cite, Martin Costabel from Rennes, is a user here. Maybe he'll be willing to help you. The subject matter strikes me as "hard" vs "soft" analysis, contrary to the basic Sobolev machinery (traces in $H^{\pm1/2}$) mentioned in Denis Serre's answer. Sep 10, 2017 at 13:43
• Thank you. But I thought that it is forbidden to invite authors. If it is ok, how should I do? Sep 10, 2017 at 13:48
• I don't know if it's OK on the site (he's been no longer active anyway), but there's probably nothing wrong if you email him to draw his attention to the question. Sep 11, 2017 at 7:39
• Could you refer us to the paper you've found on this problem? Oct 10, 2017 at 8:07
• @AmirSagiv D. S. Jerison, C. E. Kenig. The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. 4 (1981) 203–207. It relies on harmonic analyis which I'm not very familiar with Oct 11, 2017 at 9:00

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.
• It seems to me that the question is whether additional boundary regularity (either a or b) produces additional interior regularity ($H^{3/2}(\Omega)$). It's true that a and b generally fail for $u\in H^1(\Omega)$, but it does sound reasonable that if the boundary values are more regular, so is the function itself. Sep 12, 2017 at 10:46
• If $\Omega$ is a disc in the plane, which is always a useful toy model, this is certainly true by Fourier series. I would like to see the argument for the Lipschitz case. Jun 15, 2019 at 12:08