On the traces of harmonic functions $H^{1/2}$ and $H^{3/2}$ in Lipschitz domains

TL;DR

This paper investigates the behavior of harmonic functions in Lipschitz domains, revisiting classical estimates, extending solvability of boundary value problems in fractional Sobolev spaces, and proposing new functional spaces with trace properties.

Contribution

It completes the solvability of the Dirichlet problem in fractional Sobolev spaces for polygonal domains and introduces new functional spaces with trace properties.

Findings

Classical inequalities do not hold in general Lipschitz domains.

New functional space E(∇; Ω) embeds between H^{1/2}_{00}(Ω) and H^{1/2}(Ω).

Inequalities hold if the domain is of class C^{1,1}.

Abstract

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\textit{\textbf x}_0$ a fixed point in a bounded Lipschitz domain $\Omega$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in $\Omega$ and vanishes at $\textit{\textbf x}_0$, then \begin{equation*} C^{-1} \Vert u \Vert_{L^2(\Gamma)} \leq \Big(\int_\Omega \varrho\vert \nabla u \vert^2\Big)^{1/2} \leq C \Vert u \Vert_{L^2(\Gamma)}, \end{equation*} where $\varrho$ is the distance to the boundary of $\Omega$. Using Grisvard's work and interpolation theory for subspaces, we complete the solvability of the inhomogeneous Dirichlet problem: $$ (\mathscr{L}_D^0)\ \ \ \ -\Delta u = f\quad \ \mbox{in}\ \Omega \quad \mbox{and } \quad u = 0 \ \ \mbox{on }\Gamma, $$ in a framework of fractional Sobolev spaces $H^s(\Omega)$, when $\Omega$ is a polygon or a polyhedron domain and $1/2 \leq s \leq 2$. Thanks to these regularity results and an explicit function given by Ne$\mathrm{\check{c}}$as, we show that the above inequalities cannot be valid in their current form. On the other hand, we identify a functional space which satisfies the embeddings $H^{1/2}_{00}(\Omega)\hookrightarrow E(\nabla;\, \Omega) \hookrightarrow H^{1/2}(\Omega)$ and the trace operator $\gamma_0$ from $E(\nabla;\, \Omega)$ into $L^2(\Gamma)$ is well-defined and continuous. This leads to an alternative to the functions $H^{1/2}(\Omega)$, non necessarily harmonic, for having a trace in $L^2(\Gamma)$ and also to a new characterization of $H^{1/2}_{00}(\Omega)$ as the kernel of this operator. However, we show that if the domain $\Omega$ is of class $\mathscr{C}^{1, 1}$, then the above inequalities are valid.