A Local Monotonicity Formula for an Inhomogenous Singular Perturbation Problem and Applications

In this paper we prove a local monotonicity formula for solutions to an inhomogeneous singularly perturbed diffusion problem of interest in combustion. This type of monotonicity formula has proved to be very useful for the study of the regularity of limits u of solutions of the singular perturbation problem and of ∂{u > 0}, in the global homogeneous case. As a consequence of this formula we prove that u has an asymptotic development at every point in ∂{u > 0} where there is a nonhorizontal tangent ball. These type of developments have been essential for the proof of the regularity of ∂{u > 0} for Bernoulli and Stefan free boundary problems. We also present applications of our results to the study of ∂{u > 0} in the stationary case including, in particular, its regularity in the case of energy minimizers. We present as well a regularity result for travelling waves of a combustion model that relies on our monotonicity formula and its consequences. The fact that our results hold for the inhomogeneous problem allows a very wide applicability. In fact, they may be applied to problems with nonlocal diffusion and/or transport.