Heteroclinic connections for nonlocal equations

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Abstract

We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.

Original languageEnglish
Article number1950055
JournalMathematical Models and Methods in Applied Sciences
DOIs
Publication statusE-pub ahead of print - 26 Nov 2019

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Heteroclinic Connection
Nonlocal Equations
Integrodifferential equations
Dislocations (crystals)
Orbits
Viscosity
Penalization Method
Atoms
Heteroclinic Orbit
Crystals
Obstacle Problem
Energy Functional
Perturbation Method
Dislocation
Free Boundary
Variational Methods
Integro-differential Equation
Crystal
Energy
Model

Cite this

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title = "Heteroclinic connections for nonlocal equations",
abstract = "We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.",
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author = "Serena Dipierro and Stefania Patrizi and Enrico Valdinoci",
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doi = "10.1142/S0218202519500556",
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Heteroclinic connections for nonlocal equations. / Dipierro, Serena; Patrizi, Stefania; Valdinoci, Enrico.

In: Mathematical Models and Methods in Applied Sciences, 26.11.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Heteroclinic connections for nonlocal equations

AU - Dipierro, Serena

AU - Patrizi, Stefania

AU - Valdinoci, Enrico

PY - 2019/11/26

Y1 - 2019/11/26

N2 - We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.

AB - We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type. The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.

KW - crystal dislocation

KW - fractional Laplacian

KW - Heteroclinic solutions

KW - nonlocal equations

KW - Peierls-Nabarro model

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