Optimal boundary regularity for mixed local and nonlocal equations

Nicola Abatangelo; Elisa Affili; Matteo Cozzi; Nicola Abatangelo; Elisa Affili; Matteo Cozzi

doi:10.3934/mine.2026001

Mathematics in Engineering

2026, Volume 8, Issue 1: 1-42. doi: 10.3934/mine.2026001

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Optimal boundary regularity for mixed local and nonlocal equations

1.
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, P.zza di Porta S. Donato 5, 40126 Bologna, Italy
2.
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen Normandie, Avenue de l'Université BP.12, 76801 Saint-Étienne-du-Rouvray, France
3.
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, via Saldini 50, 20133 Milan, Italy

Received: 21 July 2025 Revised: 09 December 2025 Accepted: 23 December 2025 Published: 19 January 2026

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted Hölder spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.
- elliptic regularity,
- mixed operators,
- weighted Hölder spaces,
- boundary value problems,
- nonlocal operators,
- fractional Laplacian
Citation: Nicola Abatangelo, Elisa Affili, Matteo Cozzi. Optimal boundary regularity for mixed local and nonlocal equations[J]. Mathematics in Engineering, 2026, 8(1): 1-42. doi: 10.3934/mine.2026001

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Abstract

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted Hölder spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.

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