Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one phase problem. We prove a C-1,C-alpha estimates for the "interfaces " (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole R-N, for N <= 4, answering positively a conjecture of Fernandez-Real and Ros-Oton. Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.(c) 2022 The Author(s). Published by Elsevier Inc.