The Poisson-Dirichlet problem in domains with Ahlfors regular boundary

2025/06/17 by Ariel Barton, Svitlana Mayboroda, Barton, Ariel +3 · 1 citation
Computer Science · Engineering · Mathematics · #35A02 (Secondary) #35J25 (Primary) 35A01 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Elasticity and Wave Propagation #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.14639

openalex publication_date 2025/06/17 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

We present an announcement of some recent results concerning well-posedness of the Poisson-Dirichlet problem with boundary data in Besov spaces with fractional smoothness. This is a far-reaching generalization as previously known theorems concerning well-posedness of the Poisson problem in such intermediate smoothness classes were mostly restricted to the context of Lipschitz domains and coefficients satisfying strong regularity assumptions.

Citations

• Elliptic operators in rough sets, and the Dirichlet problem with boundary data in Hölder spaces
• Solvability of the Poisson-Dirichlet problem with interior data in L^p'-Carleson spaces and its applications to the L^p-regularity problem
• Global Gradient Estimates for Dirichlet Problems of Elliptic Operators with a BMO Anti-Symmetric Part
• Harmonic measure is absolutely continuous with respect to the Hausdorff measure on all low-dimensional uniformly rectifiable sets
• Absolute continuity of the harmonic measure on low dimensional rectifiable sets
• Elliptic theory in domains with boundaries of mixed dimension
• Dirichlet problem in domains with lower dimensional boundaries
• Gradient weighted norm inequalities for linear elliptic equations with\n discontinuous coefficients
• Dahlberg's theorem in higher co-dimension
• Elliptic theory for sets with higher co-dimensional boundaries
• An end-point global gradient weighted estimate for quasilinear equations\n in non-smooth domains
• Layer potentials and boundary value problems for elliptic equations with complex L^∞ coefficients satisfying the small Carleson measure norm condition
• The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients

Cited by

• The parabolic Dirichlet problem with continuous and Hölder boundary data, and rough coefficients

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• Sobolev meets Besov: Regularity for the Poisson equation with Dirichlet, Neumann and mixed boundary values
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• Regularity of Singular Solutions to p-Poisson Equations