Uniform estimates and limits for a two phase parabolic singular perturbation problem
This is the first in a series of two papers, where we study the uniform properties and the limit, as ε → 0, of solutions uε (x,t) of the equation (Pε] Δuε-uεt=β ε(uε), where ε > 0, βε ≥ 0, βε(s) = (1/ε)β(s/ε), support β= [0,1] and ∫β(s)ds = M. In this paper we prove uniform estimates for unif...
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todo:paper_00222518_v46_n2_p453_Caffarelli2023-10-03T14:29:53Z Uniform estimates and limits for a two phase parabolic singular perturbation problem Caffarelli, L.A. Lederman, C. Wolanski, N. This is the first in a series of two papers, where we study the uniform properties and the limit, as ε → 0, of solutions uε (x,t) of the equation (Pε] Δuε-uεt=β ε(uε), where ε > 0, βε ≥ 0, βε(s) = (1/ε)β(s/ε), support β= [0,1] and ∫β(s)ds = M. In this paper we prove uniform estimates for uniformly bounded solutions to (Pε), we pass to the limit, and we analyze the limit function u in general situations. We show that u satisfies Δu - ut = μ, where μ is a measure supported on the free boundary ∂{u > 0}. In order to determine the free boundary condition, we study the case in which u = αcursive Greek chi+1 - γcursive Greek chi-1 with α ≥ 0, γ > 0. We find that (u+v)2-(u-v)2 = 2M on ∂{u > 0}, where v is the inward unit spatial normal to the free boundary ∂{u > 0}, u+ = max(u,0) and u- = max(-u, 0). In addition, we prove that for any limit function u and free boundary point (cursive Greek chi0, t0) there holds that if limsup(x,t)→(xC,t0)|▽u-| ≤ γ, then limsup(x,t)→(x0,t0)l▽u+| ≤ √/2M + γ2. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00222518_v46_n2_p453_Caffarelli |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
This is the first in a series of two papers, where we study the uniform properties and the limit, as ε → 0, of solutions uε (x,t) of the equation (Pε] Δuε-uεt=β ε(uε), where ε > 0, βε ≥ 0, βε(s) = (1/ε)β(s/ε), support β= [0,1] and ∫β(s)ds = M. In this paper we prove uniform estimates for uniformly bounded solutions to (Pε), we pass to the limit, and we analyze the limit function u in general situations. We show that u satisfies Δu - ut = μ, where μ is a measure supported on the free boundary ∂{u > 0}. In order to determine the free boundary condition, we study the case in which u = αcursive Greek chi+1 - γcursive Greek chi-1 with α ≥ 0, γ > 0. We find that (u+v)2-(u-v)2 = 2M on ∂{u > 0}, where v is the inward unit spatial normal to the free boundary ∂{u > 0}, u+ = max(u,0) and u- = max(-u, 0). In addition, we prove that for any limit function u and free boundary point (cursive Greek chi0, t0) there holds that if limsup(x,t)→(xC,t0)|▽u-| ≤ γ, then limsup(x,t)→(x0,t0)l▽u+| ≤ √/2M + γ2. |
| format |
JOUR |
| author |
Caffarelli, L.A. Lederman, C. Wolanski, N. |
| spellingShingle |
Caffarelli, L.A. Lederman, C. Wolanski, N. Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| author_facet |
Caffarelli, L.A. Lederman, C. Wolanski, N. |
| author_sort |
Caffarelli, L.A. |
| title |
Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| title_short |
Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| title_full |
Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| title_fullStr |
Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| title_full_unstemmed |
Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| title_sort |
uniform estimates and limits for a two phase parabolic singular perturbation problem |
| url |
http://hdl.handle.net/20.500.12110/paper_00222518_v46_n2_p453_Caffarelli |
| work_keys_str_mv |
AT caffarellila uniformestimatesandlimitsforatwophaseparabolicsingularperturbationproblem AT ledermanc uniformestimatesandlimitsforatwophaseparabolicsingularperturbationproblem AT wolanskin uniformestimatesandlimitsforatwophaseparabolicsingularperturbationproblem |
| _version_ |
1807321340634464256 |