A local monotonicity formula for an inhomogeneous singular perturbation problem and applications: Part II

Abstract

In this paper we continue with our work in Lederman and Wolanski (Ann Math Pura Appl 187(2):197–220, 2008) where we developed a local monotonicity formula for solutions to an inhomogeneous singular perturbation problem of interest in combustion theory. There we proved local monotonicity formulae for solutions uε to the singular perturbation problem and for u = lim uε, assuming that both uε and u were defined in an arbitrary domain D in RN+1. In the present work we obtain global monotonicity formulae for limit functions u that are globally defined, while uε are not. We derive such global formulae from a local one that we prove here. In particular, we obtain a global monotonicity formula for blow up limits u0 of limit functions u that are not globally defined. As a consequence of this formula, we characterize blow up limits u0 in terms of the value of a density at the blow up point. We also present applications of the results in this paper to the study of the regularity of ∂{u > 0} (the flame front in combustion models). The fact that our results hold for the inhomogeneous singular perturbation problem allows a very wide applicability, for instance to problems with nonlocal diffusion and/or transport.