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Abstract:

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.

Registro:

Documento: Artículo
Título:A two phase elliptic singular perturbation problem with a forcing term
Autor:Lederman, C.; Wolanski, N.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Combustion; Free boundary problem; Regularity; Two phase; Viscosity solutions
Año:2006
Volumen:86
Número:6
Página de inicio:552
Página de fin:589
DOI: http://dx.doi.org/10.1016/j.matpur.2006.10.008
Título revista:Journal des Mathematiques Pures et Appliquees
Título revista abreviado:J. Math. Pures Appl.
ISSN:00217824
CODEN:JMPAA
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00217824_v86_n6_p552_Lederman.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v86_n6_p552_Lederman

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Citas:

---------- APA ----------
Lederman, C. & Wolanski, N. (2006) . A two phase elliptic singular perturbation problem with a forcing term. Journal des Mathematiques Pures et Appliquees, 86(6), 552-589.
http://dx.doi.org/10.1016/j.matpur.2006.10.008
---------- CHICAGO ----------
Lederman, C., Wolanski, N. "A two phase elliptic singular perturbation problem with a forcing term" . Journal des Mathematiques Pures et Appliquees 86, no. 6 (2006) : 552-589.
http://dx.doi.org/10.1016/j.matpur.2006.10.008
---------- MLA ----------
Lederman, C., Wolanski, N. "A two phase elliptic singular perturbation problem with a forcing term" . Journal des Mathematiques Pures et Appliquees, vol. 86, no. 6, 2006, pp. 552-589.
http://dx.doi.org/10.1016/j.matpur.2006.10.008
---------- VANCOUVER ----------
Lederman, C., Wolanski, N. A two phase elliptic singular perturbation problem with a forcing term. J. Math. Pures Appl. 2006;86(6):552-589.
http://dx.doi.org/10.1016/j.matpur.2006.10.008