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Estimates for harmonic functions near pseudo-corners

Martin Costabel, Monique Dauge

TL;DR

The paper analyzes how harmonic functions solving the Dirichlet problem for the Laplacian exhibit Sobolev-norm blow-up near conical singularities when the domain is perturbed in a self-similar way. It develops inner and converging generalized power-series expansions to derive explicit, scale-dependent estimates for Sobolev norms and singularity coefficients, applicable to both smooth pseudo-corner perturbations and polygons with approaching corners. The results quantify how the first corner singularity controls the leading blow-up across integer and fractional Sobolev spaces, and they reveal how operator norms and singular coefficients depend on the geometry and on whether the perturbation stays below or crosses regularity thresholds. These findings illuminate the delicate interaction between domain geometry, interpolation norms, and solution regularity, with implications for potential theory and numerical approximations on Lipschitz domains.

Abstract

It is well known that derivatives of solutions to elliptic boundary value problems may become unbounded near the corner of a domain with a conical singularity, even if the data are smooth. When the corner domain is approximated by more regular domains, then higher order Sobolev norms of the solutions on these domains can blow up in the limit. We study this blow-up in the simple example of the Laplace operator with Dirichlet conditions in two situations: The rounding of a corner in any dimension, and the two-dimensional situation where a polygonal corner is replaced by two or more corners with smaller angles. We show how an inner expansion derived from a more general recent result on converging expansions into generalized power series can be employed to prove simple and explicit estimations for Sobolev norms and singularity coefficients by powers of the approximation scale.

Estimates for harmonic functions near pseudo-corners

TL;DR

The paper analyzes how harmonic functions solving the Dirichlet problem for the Laplacian exhibit Sobolev-norm blow-up near conical singularities when the domain is perturbed in a self-similar way. It develops inner and converging generalized power-series expansions to derive explicit, scale-dependent estimates for Sobolev norms and singularity coefficients, applicable to both smooth pseudo-corner perturbations and polygons with approaching corners. The results quantify how the first corner singularity controls the leading blow-up across integer and fractional Sobolev spaces, and they reveal how operator norms and singular coefficients depend on the geometry and on whether the perturbation stays below or crosses regularity thresholds. These findings illuminate the delicate interaction between domain geometry, interpolation norms, and solution regularity, with implications for potential theory and numerical approximations on Lipschitz domains.

Abstract

It is well known that derivatives of solutions to elliptic boundary value problems may become unbounded near the corner of a domain with a conical singularity, even if the data are smooth. When the corner domain is approximated by more regular domains, then higher order Sobolev norms of the solutions on these domains can blow up in the limit. We study this blow-up in the simple example of the Laplace operator with Dirichlet conditions in two situations: The rounding of a corner in any dimension, and the two-dimensional situation where a polygonal corner is replaced by two or more corners with smaller angles. We show how an inner expansion derived from a more general recent result on converging expansions into generalized power series can be employed to prove simple and explicit estimations for Sobolev norms and singularity coefficients by powers of the approximation scale.

Paper Structure

This paper contains 17 sections, 14 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational solutions and their regularity in smooth domains
  4. Harmonic functions in regular cones
  5. Dirichlet problem in corner domains
  6. Dirichlet problem in families of smooth pseudo-corner domains
  7. Families of solutions
  8. Representation of solutions as converging series
  9. Blow-up of Sobolev norms in smooth pseudo-corner domains
  10. Integer order Sobolev spaces
  11. Extension to fractional order Sobolev spaces
  12. Dirichlet problem in polygons with approaching corners
  13. Blow up of Sobolev norms in polygons with approaching corners
  14. Solution operators in polygons
  15. Solution operators in families of polygons with approaching corners
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $u = \varphi h^+_j$ i.e. $u(x)=\varphi(x)\, r^{\lambda^+_j}\psi_j(\vartheta)$. Denote by ${\mathbb P}$ the space of polynomials in the variable $x$. Then

Figures (2)

  • Figure 1: Rounded corner: (a) Domain $\Omega$, (b) Pattern ${\mathsf{P}}$ , (c) $\Omega_{\varepsilon}$ (for $\varepsilon=0.2$)
  • Figure 2: Cut corner: (a) Domain $\Omega$, (b) Pattern ${\mathsf{P}}$ , (c) $\Omega_{\varepsilon}$ (for $\varepsilon=0.4$)

Theorems & Definitions (23)

  • Lemma 2.1
  • Theorem 2.2: CoDa2Mu:2024
  • Theorem 2.3: Kond:1967 for $p=2$, MazPla:1978 for general $p$
  • Remark 2.4
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • ...and 13 more