Artículo
Abstract:
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.
Registro:
| Documento: | Artículo |
| Título: | Uniform estimates and limits for a two phase parabolic singular perturbation problem |
| Autor: | Caffarelli, L.A.; Lederman, C.; Wolanski, N. |
| Filiación: | Department of Mathematics, RLM 8.100, University of Texas at Austin, Austin, TX 78712, United States Departamento de Matemática, Facultad de Ciencias Exactas, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina |
| Año: | 1997 |
| Volumen: | 46 |
| Número: | 2 |
| Página de inicio: | 453 |
| Página de fin: | 489 |
| Título revista: | Indiana University Mathematics Journal |
| Título revista abreviado: | Indiana Univ. Math. J. |
| ISSN: | 00222518 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v46_n2_p453_Caffarelli |
Referencias:
- Berestycki, H., Caffarelli, L.A., Nirenberg, L., Uniform estimates for regularization of free boundary problems (1988) Lecture Notes in Pure and Applied Mathematics, 122. , Analysis and Partial Differential Equations (Cora Sadosky, eds.), Marcel Dekker, New York
- Buckmaster, J.D., Ludford, G.S.S., (1982) Theory of Laminar Flames, , Cambridge University Press, Cambridge
- Caffarelli, L.A., A monotonicity formula for heat functions in disjoint domains (1993) Boundary Value Problems for P.D.E.'s and Applications, pp. 53-60. , dedicated to E. Magenes (J. L. Lions, C. Baiocchi, eds.), Masson, Paris
- Uniform Lipschitz regularity of a singular perturbation problem (1995) Diff. and Int. Eqs., 8, pp. 1585-1590
- A Harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz (1989) Comm. Pure and Appl. Math, 42, pp. 55-78
- Caffarelli, L.A., Lederman, C., Wolanski, N., Pointwise and Viscosity Solutions for the Limit of a Two Phase Parabolic Singular Perturbation Problem, , preprint
- Caffarelli, L.A., Vazquez, J.L., A free boundary problem for the heat equation arising in flame propagation (1995) Trans. Amer. Math. Soc., 347, pp. 411-441
- Vázquez, J.L., The free boundary problem for the heat equation with fixed gradient condition (1995) Proceedings International Conference on Free Boundary Problems and Applications, , Zakopane, Poland
Citas:
---------- APA ----------
Caffarelli, L.A., Lederman, C. & Wolanski, N. (1997) . Uniform estimates and limits for a two phase parabolic singular perturbation problem. Indiana University Mathematics Journal, 46(2), 453-489.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v46_n2_p453_Caffarelli [ ]
---------- CHICAGO ----------
Caffarelli, L.A., Lederman, C., Wolanski, N. "Uniform estimates and limits for a two phase parabolic singular perturbation problem" . Indiana University Mathematics Journal 46, no. 2 (1997) : 453-489.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v46_n2_p453_Caffarelli [ ]
---------- MLA ----------
Caffarelli, L.A., Lederman, C., Wolanski, N. "Uniform estimates and limits for a two phase parabolic singular perturbation problem" . Indiana University Mathematics Journal, vol. 46, no. 2, 1997, pp. 453-489.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v46_n2_p453_Caffarelli [ ]
---------- VANCOUVER ----------
Caffarelli, L.A., Lederman, C., Wolanski, N. Uniform estimates and limits for a two phase parabolic singular perturbation problem. Indiana Univ. Math. J. 1997;46(2):453-489.Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v46_n2_p453_Caffarelli [ ]
