Mixed local and nonlocal elliptic operators: regularity and maximum principles

Stefano Biagi; Serena Dipierro; Enrico Valdinoci; Eugenio Vecchi

doi:10.1080/03605302.2021.1998908

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Stefano Biagi, Serena Dipierro, Enrico Valdinoci, Eugenio Vecchi

Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

130 Citations (Scopus)

Abstract

We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

Original language	English
Pages (from-to)	585-629
Number of pages	45
Journal	Communications in Partial Differential Equations
Volume	47
Issue number	3
Early online date	2021
DOIs	https://doi.org/10.1080/03605302.2021.1998908
Publication status	Published - 2022

Access to Document

10.1080/03605302.2021.1998908

https://arxiv.org/pdf/2005.06907.pdf

Cite this

@article{b120a76b49f14a3093dbd1375fd1731c,

title = "Mixed local and nonlocal elliptic operators: regularity and maximum principles",

abstract = "We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.",

keywords = "Existence, maximum principle, operators of mixed order, qualitative properties of solutions, regularity",

author = "Stefano Biagi and Serena Dipierro and Enrico Valdinoci and Eugenio Vecchi",

note = "Funding Information: The authors are members of INdAM. S. Biagi is partially supported by the INdAM-GNAMPA project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali. S. Dipierro and E. Valdinoci are members of AustMS and are supported by the Australian Research Council Discovery Project DP170104880 NEW “Nonlocal Equations at Work.” S. Dipierro is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications.” E. Valdinoci is supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations.” E. Vecchi is partially supported by the INdAM-GNAMPA project Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali. We thank the Referees for their very valuable comments. Publisher Copyright: {\textcopyright} 2021 Taylor \& Francis Group, LLC.",

year = "2022",

doi = "10.1080/03605302.2021.1998908",

language = "English",

volume = "47",

pages = "585--629",

journal = "Communications in Partial Differential Equations",

issn = "0360-5302",

publisher = "Taylor \& Francis",

number = "3",

}

TY - JOUR

T1 - Mixed local and nonlocal elliptic operators

T2 - regularity and maximum principles

AU - Biagi, Stefano

AU - Dipierro, Serena

AU - Valdinoci, Enrico

AU - Vecchi, Eugenio

N1 - Funding Information: The authors are members of INdAM. S. Biagi is partially supported by the INdAM-GNAMPA project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali. S. Dipierro and E. Valdinoci are members of AustMS and are supported by the Australian Research Council Discovery Project DP170104880 NEW “Nonlocal Equations at Work.” S. Dipierro is supported by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications.” E. Valdinoci is supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations.” E. Vecchi is partially supported by the INdAM-GNAMPA project Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali. We thank the Referees for their very valuable comments. Publisher Copyright: © 2021 Taylor & Francis Group, LLC.

PY - 2022

Y1 - 2022

N2 - We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

AB - We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

KW - Existence

KW - maximum principle

KW - operators of mixed order

KW - qualitative properties of solutions

KW - regularity

UR - https://www.scopus.com/pages/publications/85119373482

U2 - 10.1080/03605302.2021.1998908

DO - 10.1080/03605302.2021.1998908

M3 - Article

AN - SCOPUS:85119373482

SN - 0360-5302

VL - 47

SP - 585

EP - 629

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 3

ER -

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Abstract

Access to Document

Other files and links

Fingerprint

Minimal surfaces, free boundaries and partial differential equations

Partial Differential Equations, free boundaries and applications

Cite this

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Abstract

Access to Document

Other files and links

Fingerprint

Projects

Minimal surfaces, free boundaries and partial differential equations

Partial Differential Equations, free boundaries and applications

Cite this