Fernandez-Real, XavierRos, Xavier2023-02-232023-02-232019-091078-0947https://hdl.handle.net/2445/194026Motivated 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 $1 \mathrm{D}$.15 p.application/pdfeng(c) American Institute of Mathematical Sciences (AIMS), 2019LaplaciàEquacions diferencials el·líptiquesEquacions en derivades parcialsDistribució (Teoria de la probabilitat)Laplacian operatorElliptic differential equationsPartial differential equationsDistribution (Probability theory)info:eu-repo/semantics/article7085702023-02-23info:eu-repo/semantics/openAccess