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Variational solutions of the Dirichlet problem, Lebesgue's cusp and non-local properties

Wolfgang Arendt, Daniel Daners, Manfred Sauter

TL;DR

The paper develops variational solutions to the Dirichlet problem for general continuous boundary data and proves their equivalence with the Perron solution, unifying variational and potential-theoretic perspectives. It characterizes finite-energy solutions via the existence of an $H^1$-admissible extension, and demonstrates independence from the chosen extension alongside a weak maximum principle. By analyzing Lebesgue's cusp and domain, it reveals that boundary regularity of data does not determine classical solvability and that boundary behavior can be highly nonlocal. The results further establish that nonlocality is generic in the presence of singular boundary points, using capacity, Wiener criteria, and barrier arguments, with implications for the prevalence of classical solutions on irregular domains.

Abstract

A recent result from [AtES24] allows one to define variational solutions of the Dirichlet problem for general continuous boundary data. We establish basic properties of this notion of solution and show that it coincides with the Perron solution. Variational solutions can elegantly be characterised in terms of the given boundary function when the variational solution has finite energy. However, it is impossible to decide in terms of the regularity of the given boundary function when a classical solution exists. We demonstrate this by analysing Lebesgue's cusp, and more precisely Lebesgue's domain which is associated with the potential of a thin rod with mass density going to zero at one end. We also show that the non-continuity of the Perron solution at a singular point is a generic and non-local property.

Variational solutions of the Dirichlet problem, Lebesgue's cusp and non-local properties

TL;DR

The paper develops variational solutions to the Dirichlet problem for general continuous boundary data and proves their equivalence with the Perron solution, unifying variational and potential-theoretic perspectives. It characterizes finite-energy solutions via the existence of an -admissible extension, and demonstrates independence from the chosen extension alongside a weak maximum principle. By analyzing Lebesgue's cusp and domain, it reveals that boundary regularity of data does not determine classical solvability and that boundary behavior can be highly nonlocal. The results further establish that nonlocality is generic in the presence of singular boundary points, using capacity, Wiener criteria, and barrier arguments, with implications for the prevalence of classical solutions on irregular domains.

Abstract

A recent result from [AtES24] allows one to define variational solutions of the Dirichlet problem for general continuous boundary data. We establish basic properties of this notion of solution and show that it coincides with the Perron solution. Variational solutions can elegantly be characterised in terms of the given boundary function when the variational solution has finite energy. However, it is impossible to decide in terms of the regularity of the given boundary function when a classical solution exists. We demonstrate this by analysing Lebesgue's cusp, and more precisely Lebesgue's domain which is associated with the potential of a thin rod with mass density going to zero at one end. We also show that the non-continuity of the Perron solution at a singular point is a generic and non-local property.

Paper Structure

This paper contains 5 sections, 22 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Variational solutions
  3. Quasi-everywhere convergence and the Perron solution
  4. Lebesgue's cusp and domain
  5. Non-locality

Key Result

Theorem 1.1

Given $\varphi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\varphi$ on $\Omega$ and an extension $\Phi\in C(\overline{\Omega})$ of $\varphi$ such that $\Phi-u_\varphi\in H^1_0(\Omega)$.

Figures (4)

  • Figure 4.1: Lebesgue's paper Leb1913 (Source https://gallica.bnf.fr/ark:/12148/bpt6k9446739 / Bibliothèque nationale de France).
  • Figure 4.2: Graph of $V(r,z)$ between the level sets $c=\frac{1}{2}$ and $c=2$. One clearly sees the singularity at $(r,z)=(0,0)$.
  • Figure 4.3: Contour map in the $xz$-plane showing $\Gamma_{\frac{1}{2}}$ and $\Gamma_2$ bounding Lebesgue's domain.
  • Figure 4.4: Lebesgue's domain cut open along its outer boundary $\Gamma_{\frac{1}{2}}$ to reveal the cusp of the inner boundary $\Gamma_{2}$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 2.1: Extension result
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['lem:2.2']}
  • Proposition 2.3
  • proof
  • Theorem 2.4: Maximum principle
  • Proposition 2.5
  • proof : Proof of Proposition \ref{['prop:2.5']} (cf. AtE19:kato)
  • Remark 2.6
  • ...and 32 more