The Dirichlet Problem for L\'evy-stable operators with L2L^2-data

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.Comment: 21 pages, 1 figur

Maximum principle for stable operators

Grube F, Hensiek T. Maximum principle for stable operators. Mathematische Nachrichten. 2023.We prove a weak maximum principle for nonlocal symmetric stable operatorsincluding the fractional Laplacian. The main focus of this work is on minimalregularity assumptions of the functions under consideratio

The Dirichlet Problem for Lévy-stable operators with L2L^2-data

Grube F, Hensiek T, Schefer W. The Dirichlet Problem for Lévy-stable operators with $L^2$-data. arXiv:2307.15235. 2023.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