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

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

Problem Statement
The paper addresses the validity of specific norm equivalence estimates for harmonic functions in bounded Lipschitz domains, particularly concerning the relationship between the $L^2$-norm of the trace on the boundary $\Gamma$ and weighted Sobolev norms involving the gradient and Hessian in the interior $\Omega$. Specifically, the authors investigate the inequalities proposed by Dahlberg [7] and others, which suggest that for a harmonic function $u$ vanishing at a fixed point $x_0 \in \Omega$:
$$C^{-1}\|u\|_{L^2(\Gamma)} \leq \left( \int_\Omega \varrho |\nabla u|^2 \right)^{1/2} \leq C\|u\|_{L^2(\Gamma)}$$
where $\varrho$ is the distance to the boundary. The central question is whether these estimates hold generally for harmonic functions in $H^{1/2}(\Omega)$ (and by extension $H^{3/2}(\Omega)$) when the domain $\Omega$ is merely Lipschitz, or if they fail in the presence of non-convex corners (polygonal/polyhedral domains).

Methodology
The authors employ a combination of interpolation theory for subspaces, regularity theory for the Laplace equation in non-smooth domains, and explicit counterexample construction.

1. Regularity Analysis in Polygonal/Polyhedral Domains: Utilizing the work of Grisvard and interpolation theory (specifically the Ivanov-Kalton and Asekritova-Cobos-Kruglyak theorems), the authors analyze the solvability of the inhomogeneous Dirichlet problem $-\Delta u = f$ with $u=0$ on $\Gamma$ in fractional Sobolev spaces $H^s(\Omega)$ for $0 \leq s \leq 2$. They characterize the kernels of the Laplacian in these spaces, showing that for non-convex polygons, the kernel $N_{-s}(\Omega)$ is non-trivial for certain $s$, affecting the isomorphism properties of the Laplacian operator.
2. Counterexample Construction: To test the validity of the trace inequalities, the authors construct a specific family of Lipschitz domains $\Omega_\varepsilon$ and harmonic functions based on an explicit function provided by Nečas. This function is designed to belong to $H^{3/2}(\Omega_\varepsilon)$ with a bounded weighted Hessian norm, while its tangential derivative on the boundary grows unbounded as the domain parameter $\varepsilon \to 0$.
3. Functional Space Identification: Recognizing the failure of standard embeddings, the authors define a new functional space $E(\nabla; \Omega) = \{v \in H^{1/2}(\Omega); \nabla v \in [H^{1/2}(\Omega)]'\}$. They investigate the properties of the trace operator $\gamma_0$ restricted to this space.
4. Regularity Assumptions: The paper contrasts the Lipschitz case with domains of class $C^{1,1}$, where standard regularity results hold, to delineate the precise conditions under which the trace estimates are valid.

Key Contributions and Results

• Refutation of General Trace Inequalities: The paper demonstrates that the inequalities (1.1) and (1.2) cited from Dahlberg and others cannot be valid in their current form for general Lipschitz domains. Specifically, for a non-convex polygonal domain, there exist harmonic functions in $H^{3/2}(\Omega)$ (and $H^{1/2}(\Omega)$) such that the weighted interior norms are bounded, but the boundary trace (in $H^1(\Gamma)$ or $L^2(\Gamma)$) is unbounded. This invalidates the claim that any harmonic function in $H^{1/2}(\Omega)$ automatically possesses an $L^2(\Gamma)$ trace in a general Lipschitz setting.
• Optimal Regularity for the Dirichlet Problem: The authors provide a complete characterization of the solvability of the Dirichlet problem in fractional Sobolev spaces for polygonal and polyhedral domains. They identify critical exponents related to the largest interior angle $\omega_\star$ (specifically $1 - \pi/\omega_\star$) where the regularity of the solution drops below the expected $H^{3/2}$ or $H^2$ levels unless compatibility conditions on the source term $f$ are met.
• Introduction of Space $E(\nabla; \Omega)$: The paper identifies $E(\nabla; \Omega)$ as the correct functional framework for trace theory in the limit case $s=1/2$.
  • The trace operator $\gamma_0: E(\nabla; \Omega) \to L^2(\Gamma)$ is well-defined and continuous.
  • The kernel of this operator is exactly $H^{1/2}_{00}(\Omega)$.
  • This provides a new characterization of $H^{1/2}_{00}(\Omega)$ and offers an alternative to $H^{1/2}(\Omega)$ for ensuring the existence of an $L^2$ trace.
• Validity in $C^{1,1}$ Domains: The authors prove that if the domain $\Omega$ is of class $C^{1,1}$, the original inequalities do hold. In this regular setting, the trace operator is an isomorphism between the space of harmonic functions in $H^{1/2}(\Omega)$ and $L^2(\Gamma)$, and similarly for $H^{3/2}(\Omega)$ and $H^1(\Gamma)$ (for the normal derivative).

Significance
The paper clarifies a critical gap in the understanding of harmonic function traces in non-smooth domains. It corrects the assumption that weighted gradient norms in Lipschitz domains are sufficient to guarantee $L^2$ boundary traces for harmonic functions. By establishing that these estimates fail in the presence of re-entrant corners, the work necessitates a more nuanced approach to regularity theory, relying on the specific space $E(\nabla; \Omega)$ rather than standard Sobolev spaces $H^{1/2}(\Omega)$ for limit cases. The results refine the understanding of the Dirichlet problem's solvability in fractional spaces for polygonal domains, highlighting the dependence of regularity on the domain's geometric singularities.