Peer-reviewedCC BY 4.0Archive for Rational Mechanics and Analysis1 Nov 2026

Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains

Mihalis Mourgoglou^1,2
Xavier Tolsa³

¹Departamento de Matemáticas, Universidad del País Vasco, UPV/EHU, Barrio Sarriena s/n, 48940 Leioa, Spain
²Ikerbasque, Basque Foundation for Science, Bilbao, Spain
³ICREA, Barcelona, Dept. de Matemàtiques, Universitat Autònoma de Barcelona, and Centre de Recerca Matemàtica, Barcelona, Catalonia Spain

Abstract

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Ω ⊂ ℝⁿ⁺¹\end{document}Ω⊂Rn+1 be a bounded chord-arc domain, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ=−div A∇\end{document}L=-divA∇ be an elliptic operator in Ω Ω associated with a matrix A having Dini mean oscillation coefficients, and let 1<p≤ 21<p≤2. In this paper we show that if the regularity problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^q\end{document}Lq for some q>pq>p in Ω Ω, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}∂ Ω\end{document}∂Ω supports a weak p-Poincaré inequality, and Ω Ω has very big pieces of superdomains for which the Neumann problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable uniformly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^q\end{document}Lq, then the Neumann problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^p\end{document}Lp in Ω Ω.

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BibTeX RIS

BibTeX

@article{mourgoglou2026solvability,
  title = {{Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains}},
  author = {Mihalis Mourgoglou and Xavier Tolsa},
  journal = {Archive for Rational Mechanics and Analysis},
  year = {2026},
  month = nov,
  doi = {10.1007/s00205-026-02236-x},
  url = {https://doi.org/10.1007/s00205-026-02236-x},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Solvability of the Neumann Problem for Elliptic Equations in Chord-Arc Domains with Very Big Pieces of Good Superdomains
AU  - Mihalis Mourgoglou
AU  - Xavier Tolsa
PY  - 2026
DA  - 2026/11/01
JO  - Archive for Rational Mechanics and Analysis
DO  - 10.1007/s00205-026-02236-x
UR  - https://doi.org/10.1007/s00205-026-02236-x
AB  - Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}Ω ⊂ ℝ^n+1\end{document}Ω⊂Rn+1 be a bounded chord-arc domain, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ=−div A∇\end{document}L=-divA∇ be an elliptic operator in Ω Ω associated with a matrix A having Dini mean oscillation coefficients, and let 1<p≤ 21<p≤2. In this paper we show that if the regularity problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^q\end{document}Lq for some q>pq>p in Ω Ω, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}∂ Ω\end{document}∂Ω supports a weak p-Poincaré inequality, and Ω Ω has very big pieces of superdomains for which the Neumann problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable uniformly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^q\end{document}Lq, then the Neumann problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}ℒ\end{document}L is solvable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}L^p\end{document}Lp in Ω Ω.
SN  - 0003-9527
SN  - 1432-0673
ER  -

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