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

In this work, we revisit the following estimate due to Dahlberg \cite{Dahl}. Let ${\mathit{\text{x}}}_0$ a fixed point in a bounded Lipschitz domain $\mathrm{\Omega }$. Then there exists a constant $C>0$ such that if $u$ is a harmonic function in $\mathrm{\Omega }$ and vanishes at ${\mathit{\text{x}}}_0$, then \begin{equation*} C^{-1} \Vert u \Vert_{L^2(Γ)} \leq \Big(\int_Ω\varrho\vert \nabla u \vert^2\Big)^{1/2} \leq C \Vert u \Vert_{L^2(Γ)}, \end{equation*} where $\varrho$ is the distance to the boundary of $\mathrm{\Omega }$. Using Grisvard’s work and interpolation theory for subspaces, we complete the solvability of the inhomogeneous Dirichlet problem: $$({\mathcal{L}}_D^0)-\mathrm{\Delta }u=fin\mathrm{\Omega }andu=0on\mathrm{\Gamma },$$ in a framework of fractional Sobolev spaces $H^s(\mathrm{\Omega })$, when $\mathrm{\Omega }$ is a polygon or a polyhedron domain and $1\mathrm{/}2\le s\le 2$. Thanks to these regularity results and an explicit function given by Ne$\check{\mathrm{c}}$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 $H_{00}^{1\mathrm{/}2}(\mathrm{\Omega })\hookrightarrow E(\mathrm{\nabla };\mathrm{\Omega })\hookrightarrow H^{1\mathrm{/}2}(\mathrm{\Omega })$ and the trace operator ${\gamma }_0$ from $E(\mathrm{\nabla };\mathrm{\Omega })$ into $L^2(\mathrm{\Gamma })$ is well-defined and continuous. This leads to an alternative to the functions $H^{1\mathrm{/}2}(\mathrm{\Omega })$, non necessarily harmonic, for having a trace in $L^2(\mathrm{\Gamma })$ and also to a new characterization of $H_{00}^{1\mathrm{/}2}(\mathrm{\Omega })$ as the kernel of this operator. However, we show that if the domain $\mathrm{\Omega }$ is of class ${\mathcal{C}}^{1,1}$, then the above inequalities are valid.