Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family functionals arising in combustion models. The problems we consider converge, for infinitesimal values the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. prove C1,α estimates “interfaces” (level sets separating burnt and unburnt regions). As byproduct, obtain one-dimensional symmetry minimizers whole RN, N≤4, answering positively conjecture Fernández-Real Ros-Oton. Our results are problem what Savin's Allen-Cahn equation minimal surfaces.