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

TL;DR

This work investigates traces of harmonic functions in Lipschitz domains at the critical Sobolev levels $H^{1/2}(\Omega)$ and $H^{3/2}(\Omega)$. It shows that classical Dahlberg-type inequalities cannot hold in general and develops a refined trace framework using the space $E(\nabla;\Omega)$ and the subspace $H^{1/2}_{00}(\Omega)$, clarifying when traces into $L^2(\Gamma)$ (and into $H^1(\Gamma)$ for $H^{3/2}$) are meaningful. Interpolation of subspaces and Grisvard regularity are leveraged to characterize solvability of the inhomogeneous Dirichlet problem on polygonal/polyhedral domains and to describe the limiting behavior of traces via the spaces $M_{\theta}(\Omega)$, including sharp results in the $\,\mathscr{C}^{1,1}$ setting. A counterexample demonstrates the limitations of existing inequalities, while the $\,\mathscr{C}^{1,1}$ theory yields precise trace isomorphisms for harmonic functions at these limit regularities, bridging boundary traces with interior Sobolev structure. The findings provide a robust framework for trace analysis and Dirichlet problem solvability in nonsmooth geometries, with clear implications for numerical analysis and boundary-value problem theory.

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 $Ω$. Then there exists a constant $C > 0$ such that if $u$ is a harmonic function in $Ω$ and vanishes at $\textit{\textbf x}_0$, 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 $Ω$. Using Grisvard's work and interpolation theory for subspaces, we complete the solvability of the inhomogeneous Dirichlet problem: $$ (\mathscr{L}_D^0)\ \ \ \ -Δu = f\quad \ \mbox{in}\ Ω\quad \mbox{and } \quad u = 0 \ \ \mbox{on }Γ, $$ in a framework of fractional Sobolev spaces $H^s(Ω)$, when $Ω$ 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 $H^{1/2}_{00}(Ω)\hookrightarrow E(\nabla;\, Ω) \hookrightarrow H^{1/2}(Ω)$ and the trace operator $γ_0$ from $E(\nabla;\, Ω)$ into $L^2(Γ)$ is well-defined and continuous. This leads to an alternative to the functions $H^{1/2}(Ω)$, non necessarily harmonic, for having a trace in $L^2(Γ)$ and also to a new characterization of $H^{1/2}_{00}(Ω)$ as the kernel of this operator. However, we show that if the domain $Ω$ is of class $\mathscr{C}^{1, 1}$, then the above inequalities are valid.

Theorems & Definitions (15)

• Theorem 3.1: Uniqueness criterion in $H^{1/2}(\Omega)$
• Theorem 3.2: Ivanov-Kalton
• Theorem 3.3: Asekritova-Cobos-Kruglyak
• Theorem 3.4: Grisvard, $H^2$-Regularity
• Corollary 3.5
• Theorem 3.6: $H^s$- Regularity I
• Proposition 3.7
• Theorem 3.8: $H^s$- Regularity II
• Theorem 3.9: $H^s$- Regularity III
• Proposition 4.1: Counterexample for Inequalities \ref{['inegaltraceL2Gamma1']} and \ref{['ineg1']}