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

2018/11/07 by Xavier Fernández‐Real, Xavier Ros‐Oton, Fernández-Real, Xavier +1 · 1 citation
Computer Science · Mathematics · #35B07 #35J91 #35R35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1811.02980

openalex publication_date 2018/11/07 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28

Abstract

Motivated by its relation to models of flame propagation, we study globally\nLipschitz solutions of \Δ u=f(u) in \ℝⁿ, where f is smooth,\nnon-negative, with support in the interval [0,1]. In such setting, any\n"blow-down" of the solution u will converge to a global solution to the\nclassical one-phase free boundary problem of Alt-Caffarelli.\n In analogy to a famous theorem of Savin for the Allen-Cahn equation, we study\nhere the 1D symmetry of solutions u that are energy minimizers. Our main\nresult establishes that, in dimensions n<6, if u is axially symmetric and\nstable then it is 1D.\n

Cited by

• Smoothness and stability in the Alt-Phillips problem

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