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

Abstract

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $\mathbf{x}_0$ 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 $\mathbf{x}_0$, 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.