We study minimizers of a family of functionals arising in combustion theory, which converge, for infinitesimal values of the parameter, to minimizers of the one-phase free boundary problem. We prove a $C^{1,\alpha}$ estimate for the "interfaces'' of critical points (i.e. the level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\mathbb{R}^N$ for $N \le 4$, answering positively a conjecture of Fernández-Real and Ros-Oton. Our results are to the one-phase free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces. This is a joint work with J. Serra (ETHZ).