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

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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 $1 \mathrm{D}$.

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American Institute of Mathematical Sciences (AIMS)

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Discrete and Continuous Dynamical Systems-Series A, 2019, vol. 39, num. 12, p. 6945-6959

# metadata.dc.rights[].value
(c) American Institute of Mathematical Sciences (AIMS), 2019

# metadata.dc.source[].value
Articles publicats en revistes (Matemàtiques i Informàtica)

# metadata.dc.title[].value
On global solutions to semilinear elliptic equations related to the one-phase free boundary problem