| Abstract: |
| In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let $ 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 $ 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. |
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