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# Analysis

### International mathematical journal of analysis and its applications

Editor-in-Chief: Schulz, Friedmar

CiteScore 2017: 0.66

SCImago Journal Rank (SJR) 2017: 0.564
Source Normalized Impact per Paper (SNIP) 2017: 0.674

Mathematical Citation Quotient (MCQ) 2017: 0.38

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2196-6753
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Volume 38, Issue 4

# An overdetermined problem for the infinity-Laplacian around a set of positive reach

Antonio Greco
Published Online: 2018-09-20 | DOI: https://doi.org/10.1515/anly-2017-0030

## Abstract

We consider an overdetermined problem associated to an inhomogeneous infinity-Laplace equation. More precisely, the domain of the problem is required to contain a given compact set K of positive reach, and the boundary of the domain must lie within the reach of K. We look for a solution vanishing at the boundary and such that the outer derivative depends only on the distance from K. We prove that if the boundary gradient grows fast enough with respect to such distance (faster than the distance raised to $\frac{1}{3}$), then the problem is solvable if and only if the domain is a tubular neighborhood of K, thus extending a previous result valid in the case when K is made up of a single point.

MSC 2010: 35N25; 35R35

Dedicated to Gérard A. Philippin, a source of inspiration and encouragement

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Revised: 2018-06-04

Accepted: 2018-07-12

Published Online: 2018-09-20

Published in Print: 2019-01-01

The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was partially supported by the research project “Integro-differential Equations and Non-Local Problems”, funded by Fondazione di Sardegna (2017).

Citation Information: Analysis, Volume 38, Issue 4, Pages 155–165, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747,

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