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On global solutions to semilinear elliptic equations related to the one-phase free boundary problem


Fernández-Real, Xavier; Ros-Oton, Xavier (2019). On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems. Series A, 39(12):6945-6959.

Abstract

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of Δu=f(u) in Rn, where f is smooth, non-negative, with support in the interval [0,1]. In such setting, any "blow-down" of the solution u will converge to a global solution to the classical one-phase free boundary problem of Alt–Caffarelli.
In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions u that are energy minimizers. Our main result establishes that, in dimensions n<6, if u is axially symmetric and stable then it is 1D.

Abstract

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of Δu=f(u) in Rn, where f is smooth, non-negative, with support in the interval [0,1]. In such setting, any "blow-down" of the solution u will converge to a global solution to the classical one-phase free boundary problem of Alt–Caffarelli.
In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D symmetry of solutions u that are energy minimizers. Our main result establishes that, in dimensions n<6, if u is axially symmetric and stable then it is 1D.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Discrete Mathematics and Combinatorics
Physical Sciences > Applied Mathematics
Language:English
Date:1 January 2019
Deposited On:16 Dec 2019 09:21
Last Modified:22 Apr 2020 21:38
Publisher:American Institute of Mathematical Sciences (A I M S Press)
ISSN:1078-0947
OA Status:Green
Publisher DOI:https://doi.org/10.3934/dcds.2019238

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