A two phase elliptic singular perturbation problem with a forcing term
We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(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 uε and fε are uniform...
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | JOUR |
| Materias: | |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman |
| Aporte de: |
| Sumario: | We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(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 uε and fε 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 {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on Ω ∩ ∂ {u > 0}, where f = lim fε, 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 fε ≡ 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. © 2006 Elsevier SAS. All rights reserved. |
|---|