The regularity problem for the Laplace equation in rough domains
type: Item
instance of: scholarly article
title: The regularity problem for the Laplace equation in rough domains
zbMATH Open document ID: 1548.3102
DOI: 10.1215/00127094-2023-0044
author: Mihalis Mourgoglou
author: Xavier Tolsa
published in: Duke Mathematical Journal
publication date: 2024-07-03T00:00:00Z
review text: The present paper studies the equivalence between solvability of the \(L^{p'}\)-Dirichlet and solvability of the \(L^p\)-regularity problem for the Laplace operator in corkscrew domains \(\Omega\subset \mathbb{R}^{n+1}\), \(n\ge 2\) with uniformly rectifiable boundaries. The main result shows that the Dirichlet problem for the Laplacian with boundary data in \(L^{p'}(\partial \Omega)\) is solvable if and only if the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space \(W^{1,p}(\partial \Omega)\) is solvable, where \(p\in(1,2+\varepsilon)\) and \(1/p +1/p'=1\).\N\NLet us mention that the obtained result also solves the problem posed in [\textit{C. E. Kenig}, Harmonic analysis techniques for second order elliptic boundary value problems: dedicated to the memory of Professor Antoni Zygmund. Providence, RI: American Mathematical Society (1994; Zbl 0812.35001)] and reintroduced in [\textit{T. Toro}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. III: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 1485--1497 (2011; Zbl 1251.28002)] which can be stated as follows:\N\N\textit{In a bounded chord-arc domain \(\Omega \subset \mathbb{R}^{n+1}, n\ge 2\), does there exist \(p > 1\) such that the regularity problem for the Laplacian with boundary data in the Sobolev space \(W^{1,p}(\partial \Omega)\) is solvable? If so, are the layer potentials invertible in appropriate \(L^p\) spaces for such \(p > 1\)?}
reviewed by: D. Kapanadze
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B37
zbMATH DE Number: 7875323
zbMATH Keywords: Laplace equation
zbMATH Keywords: regularity problem
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: chord-arc domain
zbMATH Keywords: corkscrew domain
MaRDI profile type: Publication
cites work: Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems
cites work: Poincaré inequalities and uniform rectifiability
cites work: Uniform Rectifiability, Elliptic Measure, Square Functions, and ε-Approximability Via an ACF Monotonicity Formula
cites work: Rectifiability of harmonic measure
cites work: Harmonic measure and quantitative connectivity: geometric characterization of the \(L^p\)-solvability of the Dirichlet problem
cites work: Carleson measure estimates for caloric functions and parabolic uniformly rectifiable sets
cites work: Estimates of harmonic measure
cites work: On the Poisson integral for Lipschitz and $C^{1}$-domains
cites work: Hardy spaces and the Neumann problem in \(L^ p\) for Laplace's equation in Lipschitz domains
cites work: Opérateurs d'intégrale singulière sur les surfaces régulières
cites work: Q4713875
cites work: Q4001787
cites work: Q4278678
cites work: The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space \(HS^1\)
cites work: Q5178046
cites work: Q4137515
cites work: Potential techniques for boundary value problems on \(C^1\)-domains
cites work: Measure and area
cites work: \(H^p\) spaces of several variables
cites work: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs
cites work: Sobolev spaces on an arbitrary metric space
cites work: Traces of Sobolev functions on fractal type sets and characterization of extension domains
cites work: Q5698658
cites work: Q5294569
cites work: Quantitative absolute continuity of harmonic measure and the Dirichlet problem: a survey of recent progress
cites work: The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients
cites work: BMO solvability and absolute continuity of harmonic measure
cites work: The weak-\(A_\infty\) property of harmonic and \(p\)-harmonic measures implies uniform rectifiability
cites work: Uniform rectifiability, Carleson measure estimates, and approximation of harmonic functions
cites work: Q3586767
cites work: Bounded variation approximation of \(L_p\) dyadic martingales and solutions to elliptic equations
cites work: The failure of \(L^ p\) estimates for harmonic measure in chord-arc domains
cites work: The Dirichlet problem in non-smooth domains
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Hardy spaces, $A_\infty$, and singular integrals on chord-arc domains
cites work: The Poincaré inequality is an open ended condition
cites work: Q3762712
cites work: Q4305610
cites work: The Neumann problem for elliptic equations with non-smooth coefficients
cites work: Layer potential methods for elliptic homogenization problems
cites work: The 𝐿^{𝑝} regularity problem on Lipschitz domains
cites work: Lebesgue points and capacities via boxing inequality in metric spaces
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cites work: Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory
cites work: Capacity and measure densities
cites work: Harmonic measure and Riesz transform in uniform and general domains
cites work: Q4297491
cites work: A relationship between the Dirichlet and regularity problems for elliptic equations
cites work: Jump formulas for singular integrals and layer potentials on rectifiable sets
cites work: Q3096657
cites work: Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains
description: scientific article; zbMATH DE number 7875323