Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

Nicola Garofalo; Arshak Petrosyan; Camelia A. Pop; Mariana Smit Vega Garcia

doi:10.1016/j.anihpc.2016.03.001

Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

Nicola Garofalo
, Arshak Petrosyan
, Camelia A. Pop
, Mariana Smit Vega Garcia

School of Mathematics

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

23 Scopus citations

Abstract

We establish the C^1+γ-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

Original language	English (US)
Pages (from-to)	533-570
Number of pages	38
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	34
Issue number	3
DOIs	https://doi.org/10.1016/j.anihpc.2016.03.001
State	Published - May 1 2017

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Masson SAS

Keywords

Epiperimetric inequality
Fractional Laplacian with drift
Free boundary regularity
Monotonicity formulas
Obstacle problem
Symmetric stable process

Publisher link

10.1016/j.anihpc.2016.03.001

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title = "Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift",

abstract = "We establish the C1+γ-H{\"o}lder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.",

keywords = "Epiperimetric inequality, Fractional Laplacian with drift, Free boundary regularity, Monotonicity formulas, Obstacle problem, Symmetric stable process",

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T1 - Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

AU - Garofalo, Nicola

AU - Petrosyan, Arshak

AU - Pop, Camelia A.

AU - Smit Vega Garcia, Mariana

PY - 2017/5/1

Y1 - 2017/5/1

N2 - We establish the C1+γ-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

AB - We establish the C1+γ-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss in [15] for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research [12], where two of us established the optimal interior regularity of solutions.

KW - Epiperimetric inequality

KW - Fractional Laplacian with drift

KW - Free boundary regularity

KW - Monotonicity formulas

KW - Obstacle problem

KW - Symmetric stable process

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DO - 10.1016/j.anihpc.2016.03.001

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VL - 34

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JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

IS - 3

ER -

Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

Abstract

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