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Cauchy problem and periodic homogenization for nonlocal Hamilton–Jacobi equations with coercive gradient terms

Part of: Miscellaneous topics - Partial differential equations Representations of solutions Partial differential equations General first-order equations and systems

Published online by Cambridge University Press:  17 September 2019

Martino Bardi ,
Annalisa Cesaroni  and
Erwin Topp
Martino Bardi
Affiliation:
Dipartimento di Matematica Tullio Levi Civita, Università di Padova, Via Trieste 63, Padova, Italy (bardi@math.unipd.it)
Annalisa Cesaroni
Affiliation:
Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 141, Padova, Italy (annalisa.cesaroni@unipd.it)
Erwin Topp
Affiliation:
Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, Chile (erwin.topp@usach.cl)
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Abstract

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

Keywords

HomogenizationHamilton–Jacobi equationsintegro-differential equationsfractional Laplacianscomparison principleviscosity solutions

MSC classification

Primary: 35R09: Integro-partial differential equations
Secondary: 35B27: Homogenization; equations in media with periodic structure 35F25: Initial value problems for nonlinear first-order equations 35D40: Viscosity solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3028 - 3059
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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