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<p>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.</p>

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ESTIMATES FOR HARMONIC FUNCTIONS NEAR PSEUDO-CORNERS

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Martin Costabel,
  • Monique Dauge

摘要

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.