Abstract: We study the Dirichlet problem for the Poisson equation in bounded Lipschitz domains. We show that its well-posedness in the higher order Sobolev space implies a discrete Hardy type inequality that contains a positive harmonic function with vanishing trace and the approximative numbers of the boundary of the domain. This necessary condition is also expected to be su cient for the well-posedness. A simpler condition occurring in the author's straightenability theory of Lipschitz domains is shown to be equivalent to the existence of a homeomorphism that straightens the boundary and preserves with respect to composition the subspace of zero trace functions in the considered Sobolev space.

Cite: Парфёнов А.И.
Критерий соболевской корректности задачи Дирихле для уравнения Пуассона в липшицевых областях. II
Сибирские электронные математические известия (Siberian Electronic Mathematical Reports). 2023. Т.20. №1. С.211-244. DOI: 10.33048/semi.2023.20.018 WOS Scopus РИНЦ

Dates:

Submitted:	May 3, 2022
Published print:	Mar 13, 2023
Published online:	Mar 13, 2023

Identifiers:

≡ Web of science:	WOS:000959070400012
≡ Scopus:	2-s2.0-85150778955
≡ Elibrary:	54768290

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