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2019-09

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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/194026

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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https://doi.org/10.3934/dcds.2019238

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Laplacià, Equacions diferencials el·líptiques, Equacions en derivades parcials, Distribució (Teoria de la probabilitat)

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Laplacian operator, Elliptic differential equations, Partial differential equations, Distribution (Probability theory)

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FERNANDEZ-REAL, Xavier and ROS, Xavier. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and Continuous Dynamical Systems-Series A. 2019. Vol. 39, num. 12, pags. 6945-6959. ISSN 1078-0947. [consulted: 6 of October of 2026]. Available at: https://hdl.handle.net/2445/194026

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