Criterion for the Sobolev well-posedness of the Dirichlet problem for the Poisson equation in Lipschitz domains. II

A. I. Parfenov

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 1, pp. 211-244.

Résumé

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 sufficient 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.

Détail

Keywords: approximative numbers, Dirichlet problem for the Poisson equation, Hardy type inequality, Lipschitz domain, straightening.

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