Solvability of the Poisson-Dirichlet problem in domains with boundaries of mixed dimension

2026/08/04 by Léo Mandô
Mathematics · #math.AP

paper · pdf

Travail effectué sous la direction de Joseph Feneuil, Maître de conférences à l'Université Paris-Saclay

arxiv created 2026/08/06 · arxiv updated 2026/08/07

Abstract

We extend several characterizations of the L^p-solvability of the homogeneous Dirichlet problem to (possibly degenerate) elliptic operators L=-\textrmdiv(wA∇) defined on a large class of open sets Ω in ℝⁿ. This framework encompasses not only uniformly elliptic operators in Lipschitz domains, but also Caffarelli-Sylvestre-type operators, and boundaries ∂Ω that are not (n-1)-dimensional, for instance. We prove that, for p∈ (1,∞), the L^p-solvability of the homogeneous Dirichlet problem is equivalent to the solvability of the Poisson-Dirichlet problem Lu=wf-\textrmdiv(wF), i.e. to the existence of a solution u satisfying a L^p non-tangential estimate whenever f and F belong to suitable weighted L^p tent spaces. Furthermore, it is also equivalent to the solvability of the Poisson-Regularity problem for data in appropriate weighted L^p' tent spaces, where estimates are obtained for ∇ u rather than u.

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