A two phase elliptic singular perturbation problem with a forcing term

Abstract

We study the following two phase elliptic singular perturbation problem: Δ u ε = β ε ( u ε ) + f ε , in Ω ⊂ R N , where ε > 0 , β ε ( s ) = 1 ε β ( s ε ) , with β a Lipschitz function satisfying β > 0 in ( 0 , 1 ) , β ≡ 0 outside ( 0 , 1 ) and ∫ β ( s ) d s = M . The functions u ε and f ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit ( ε → 0 ) and we show that limit functions are solutions to the two phase free boundary problem: Δ u = f χ { u ≢ 0 } in Ω ∖ ∂ { u > 0 } , | ∇ u + | 2 − | ∇ u − | 2 = 2 M on Ω ∩ ∂ { u > 0 } , where f = lim f ε , in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f ε ≡ 0 . The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary.