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Title: | On global solutions to semilinear elliptic equations related to the one-phase free boundary problem |
Author: | Fernandez-Real, Xavier Ros, Xavier |
Keywords: | Laplacià Equacions diferencials el·líptiques Equacions en derivades parcials Distribució (Teoria de la probabilitat) Laplacian operator Elliptic differential equations Partial differential equations Distribution (Probability theory) |
Issue Date: | Sep-2019 |
Publisher: | American Institute of Mathematical Sciences (AIMS) |
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, nonnegative, 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 onephase free boundary problem of Alt-Caffarelli. In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study here the $1 \mathrm{D}$ 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 $1 \mathrm{D}$. |
Note: | Versió postprint del document publicat a: https://doi.org/10.3934/dcds.2019238 |
It is part of: | Discrete and Continuous Dynamical Systems-Series A, 2019, vol. 39, num. 12, p. 6945-6959 |
URI: | http://hdl.handle.net/2445/194026 |
Related resource: | https://doi.org/10.3934/dcds.2019238 |
ISSN: | 1078-0947 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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