Solvability of the Poisson-Dirichlet problem with interior data in Lp' - Carleson spaces and its applications to the Lp' -regularity problem
Mourgoglou, Mihalis (Universidad del País Vasco)
Poggi, Bruno (Universitat Autònoma de Barcelona)
Tolsa Domènech, Xavier (Centre de Recerca Matemàtica)

Data:	2025
Resum:	We prove that the Lp' -solvability of the homogeneous Dirichlet problem for an elliptic operator L=-divA∇ with real and merely bounded coefficients is equivalent to the Lp' -solvability of the Poisson-Dirichlet problem Lw=H-divF, which is defined in terms of an Lp' -estimate on the non-tangential maximal function, assuming that dist(⋅,∂Ω)H and F lie in certain Lp' -Carleson-type spaces, and that the domain Ω⊂Rn+1, n≥2, satisfies the corkscrew condition and has n-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly n-rectifiable boundary that satisfies the corkscrew condition,Lp' -solvability of the homogeneous Dirichlet problem for an operator L=-divA∇ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the Lp -regularity problem for the adjoint operator L∗ =-divA T∇, where 1/p+ 1/p'=1 and AT is the transpose matrix of A. This result for Dahlberg-Kenig-Pipher operators is new even if Ω is the unit ball, despite the fact that the Lp' -solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001. Further novel applications include (i) new local estimates for the Green's function and its gradient in rough domains, (ii) a local T1-type theorem for the Lp -solvability of the "Poisson-regularity problem", itself equivalent to the Lp' -solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and (iii) new Lp -estimates for the eigenfunctions (and their gradients) of symmetric operators L on bounded rough domains.
Ajuts:	Agencia Estatal de Investigación PID2020-118986GB-I00 European Commission 101018680 Generalitat de Catalunya 2021/SGR-00071 Agencia Estatal de Investigación PID2020-114167GB-I00 Ministerio de Ciencia, Innovación y Universidades CEX2020-001084-M
Nota:	Altres ajuts: IT1615-22 (Basque Government)
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Dirichlet problem ; Regularity problem ; Boundary value problems ; Elliptic PDEs ; PDEs on domains with rough boundaries ; Poisson problem
Publicat a:	Journal of the European Mathematical Society, July 2025, ISSN 1435-9863

DOI: 10.4171/jems/1660

Postprint 48 p, 637.5 KB

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