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

Xavier Fernández-Real; Xavier Ros-Oton

doi:10.3934/dcds.2019238

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2019, Volume 39, Issue 12: 6945-6959. Doi: 10.3934/dcds.2019238

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

Xavier Fernández-Real^1, , and
Xavier Ros-Oton^2,

1.
Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
2.
Institut für Mathematik, Universität Zürich, Winterthurerstrasse, 8057 Zürich, Switzerland

^* Corresponding author
^* Corresponding author

Received: August 31, 2018

Revised: January 31, 2019

Published: September 2019

This work has received funding from the European Research Council (ERC) under the Grant Agreements No 721675 and No 801867. In addition, the second author was supported by the Swiss National Science Foundation and by MINECO grant MTM2017-84214-C2-1-P

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Abstract

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of $ \Delta u = f(u) $ in $ \mathbb{R}^n $, 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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Mathematics Subject Classification: Primary: 35R35, 35J91; Secondary: 35B07.

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Figure 1. Representation of $ \Phi_\varepsilon(t) = \int_0^t \beta_\varepsilon(s)\, ds $

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Figure 2. Representation of the cases (ⅰ) $ a > 1 $, (ⅱ) $ a = 1 $, and (ⅲ) $ a < 1 $

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