Gradient estimates for a class of anisotropic nonlocal operators

Alberto Farina, Enrico Valdinoci

Research output: Contribution to journalArticle

Abstract

Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain Hölder estimates for the solution.

Original languageEnglish
Article number25
JournalNonlinear Differential Equations and Applications
Volume26
Issue number4
DOIs
Publication statusPublished - 1 Aug 2019

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Gradient Estimate
Nonlocal Equations
Operator
Elliptic Equations
Increment
Superposition
Logarithmic
Fractional
Linearly
Derivatives
Derivative
Estimate
Class

Cite this

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title = "Gradient estimates for a class of anisotropic nonlocal operators",
abstract = "Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain H{\"o}lder estimates for the solution.",
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Gradient estimates for a class of anisotropic nonlocal operators. / Farina, Alberto; Valdinoci, Enrico.

In: Nonlinear Differential Equations and Applications, Vol. 26, No. 4, 25, 01.08.2019.

Research output: Contribution to journalArticle

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AU - Valdinoci, Enrico

PY - 2019/8/1

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N2 - Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain Hölder estimates for the solution.

AB - Using a classical technique introduced by Achi E. Brandt for elliptic equations, we study a general class of nonlocal equations obtained as a superposition of classical and fractional operators in different variables. We obtain that the increments of the derivative of the solution in the direction of a variable experiencing classical diffusion are controlled linearly, with a logarithmic correction. From this, we obtain Hölder estimates for the solution.

KW - Anisotropic nonlocal elliptic equations

KW - Modulus of continuity of the solutions

KW - Regularity theory

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