Dedicado con afecto a Luis Caffarelli, cuyos trabajos han influenciado a toda una nueva generación de matemáticos. Abstract. Motivated by its relation to models of flame propagation, we study globally Lipschitz solutions of ∆u = f (u) in 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 one-phase free boundary problem of Alt-Caffarelli. In analogy to a famous theorem

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... 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. 2010 Mathematics Subject Classification. Primary: 35R35, 35J91; Secondary: 35B07.