The Dirichlet Problem for L\'evy-stable operators with $L^2$-data

Florian Grube; Thorben Hensiek; Waldemar Schefer

The Dirichlet Problem for L\'evy-stable operators with $L^2$-data

Analysis of PDEs 2023-07-31 v1

Authors: Florian Grube , Thorben Hensiek , Waldemar Schefer

Abstract

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of $2s$ -stable processes and exterior data, inhomogeneity in weighted $L^2$ -spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit $s\to 1-$ which allows us to recover the local theory.

Keywords

sobolev regularity of partial differential equations dirichlet problem regularity for elliptic equations

Cite

@article{arxiv.2307.15235,
  title  = {The Dirichlet Problem for L\'evy-stable operators with $L^2$-data},
  author = {Florian Grube and Thorben Hensiek and Waldemar Schefer},
  journal= {arXiv preprint arXiv:2307.15235},
  year   = {2023}
}

Comments

21 pages, 1 figure

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