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

Abstract

Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of Δu=f(u) in Rn, where f is smooth, non-negative, 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 one-phase free boundary problem of Alt–Caffarelli.
In analogy to a famous theorem of Savin for the Allen–Cahn equation, we study here the 1D 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 1D.