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...
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v86_n6_p552_Lederman_oai |
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I28-R145-paper_00217824_v86_n6_p552_Lederman_oai2024-08-16 Lederman, C. Wolanski, N. 2006 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. Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. application/pdf http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Math. Pures Appl. 2006;86(6):552-589 Combustion Free boundary problem Regularity Two phase Viscosity solutions A two phase elliptic singular perturbation problem with a forcing term info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v86_n6_p552_Lederman_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (UBA) |
| topic |
Combustion Free boundary problem Regularity Two phase Viscosity solutions |
| spellingShingle |
Combustion Free boundary problem Regularity Two phase Viscosity solutions Lederman, C. Wolanski, N. A two phase elliptic singular perturbation problem with a forcing term |
| topic_facet |
Combustion Free boundary problem Regularity Two phase Viscosity solutions |
| description |
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. |
| format |
Artículo Artículo publishedVersion |
| author |
Lederman, C. Wolanski, N. |
| author_facet |
Lederman, C. Wolanski, N. |
| author_sort |
Lederman, C. |
| title |
A two phase elliptic singular perturbation problem with a forcing term |
| title_short |
A two phase elliptic singular perturbation problem with a forcing term |
| title_full |
A two phase elliptic singular perturbation problem with a forcing term |
| title_fullStr |
A two phase elliptic singular perturbation problem with a forcing term |
| title_full_unstemmed |
A two phase elliptic singular perturbation problem with a forcing term |
| title_sort |
two phase elliptic singular perturbation problem with a forcing term |
| publishDate |
2006 |
| url |
http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00217824_v86_n6_p552_Lederman_oai |
| work_keys_str_mv |
AT ledermanc atwophaseellipticsingularperturbationproblemwithaforcingterm AT wolanskin atwophaseellipticsingularperturbationproblemwithaforcingterm AT ledermanc twophaseellipticsingularperturbationproblemwithaforcingterm AT wolanskin twophaseellipticsingularperturbationproblemwithaforcingterm |
| _version_ |
1809356775859683328 |