Abstract:
In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). Here βε (s) = 1/εβ(s/ε), with β Ε Lip(rdbl), β > 0in(0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C 1, α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [ Comm. Partial Differential Equations, 16 (1991), pp. 311-361]. © 2009 Society for Industrial and Applied Mathematics.
Registro:
Documento: |
Artículo
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Título: | A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
Autor: | Sandra, M.; Noemi, W. |
Filiación: | Departamento de Matemática, FCEyN, UBA, (1428) Buenos Aires, Argentina
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Palabras clave: | Free boundaries; Orlicz spaces; Singular perturbation; Following problem; Free boundary; Free-boundary problems; Growth conditions; Limiting problem; Orlicz spaces; Quasi-linear; Singular perturbation problems; Singular perturbations; Weak solution; Differential equations; Mathematical operators |
Año: | 2009
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Volumen: | 41
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Número: | 1
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Página de inicio: | 318
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Página de fin: | 359
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DOI: |
http://dx.doi.org/10.1137/070703740 |
Título revista: | SIAM Journal on Mathematical Analysis
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Título revista abreviado: | SIAM J. Math. Anal.
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ISSN: | 00361410
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Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00361410_v41_n1_p318_Sandra |
Referencias:
- Alt, H.W., Caffarelli, L.A., Existence and regularity for a minimum problem with free boundary (1981) J. Reine Angew. Math., 325, pp. 105-144
- Alt, H.W., Caffarelli, L.A., Friedman, A., A free boundary problem for quasilinear elliptic equations (1984) Ann. Sc. Norm. Super. Pisa Cl. Sci., 11, pp. 1-44
- Berestycki, H., Caffarelli, L.A., Nirenberg, L., Uniform estimates for regularization of free boundary problems (1990) Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., 122, pp. 567-619. , Dekker, New York
- Caffarelli, L.A., A harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1, α (1987) Rev. Mat. Iberoamericana, 3, pp. 139-162
- Caffarelli, L.A., A harnack inequality approach to the regularity of free boundaries. Part II: Flat free boundaries are Lipschitz (1989) Comm. Pure Appl. Math., 42, pp. 55-78
- Caffarelli, L.A., Jerison, D., Kenig, C.E., Regularity for Inhomogeneous Two-phase Free Boundary Problems. Part I: Flat Free Boundaries Are C1,α, , preprint
- Caffarelli, L.A., Lederman, C., Wolanski, N., Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem (1997) Indiana University Mathematics Journal, 46 (3), pp. 719-740
- Caffarelli, L.A., Lederman, C., Wolanski, N., Uniform estimates and limits for a two phase parabolic singular perturbation problem (1997) Indiana University Mathematics Journal, 46 (2), pp. 453-489
- Caffarelli, L.A., Salsa, S., A geometric approach to free boundary problems (2005) Grad. Stud. Math., 68. , American Mathematical Society, Providence, RI
- Caffarelli, L.A., Vázquez, J.L., A free-boundary problem for the heat equation arising in flame propagation (1995) Trans. Amer. Math. Soc., 347, pp. 411-441
- Cerutti, M.C., Ferrari, F., Salsa, S., Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,y (2004) Arch. Ration. Mech. Anal., 171, pp. 329-348
- Danielli, D., Petrosyan, A., Shahgholian, H., A singular perturbation problem for the p-Laplace operator (2003) Indiana Univ. Math. J., 52, pp. 457-476
- Ferrari, F., Salsa, S., Regularity of the free boundary in two-phase problems for linear elliptic operators (2007) Adv. Math., 214, pp. 288-322
- Ladyzhenskaya, O., Ural'Tseva, N., (1968) Linear and Quasilinear Elliptic Equations, , Academic Press, New York
- Le, V.K., Schmitt, K., On boundary value problems for degenerate quasilinear elliptic equations and inequalities (1998) J. Differential Equations, 144, pp. 170-218
- Lederman, C., Wolanski, N., Viscosity solutions and regularity of the free boundary for the limit of an elliptic two phase singular perturbation problem (1998) Ann. Sc. Norm. Super. Pisa Cl. Sci., 27, pp. 253-288
- Lederman, C., Wolanski, N., A two phase elliptic singular perturbation problem with a forcing term (2006) J. Math. Pures Appl., 86, pp. 552-589
- Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations (1988) Nonlinear Anal, 12, pp. 1203-1219
- Lieberman, G.M., The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations (1991) Comm. Partial Differential Equations, 16, pp. 311-361
- Martínez, S., An optimization problem with volume constraint in Orlicz spaces (2008) J. Math. Anal. Appl., 340, pp. 1407-1421
- Martínez, S., Wolanski, N., A minimum problem with free boundary in Orlicz spaces (2008) Adv. Math., 218, pp. 1914-1971
- Moreira, D.R., Teixeira, E.V., A singular perturbation free boundary problem for elliptic equations in divergence form (2007) Calc. Var. Partial Differential Equations, 29, pp. 161-190
- Schmitt, K., Revisiting the method of sub- and supersolutions for nonlinear elliptic problems (2007) Proceedings of the Sixth Mississippi State-UAB Conference on Differential Equations and Computational Simulations, pp. 377-385. , Starkville, MS, Electron. J. Differ. Equ. Conf. 15, Southwest Texas State Univ., San Marcos, TX
- Teixeira, E.V., A variational treatment for general elliptic equations of the flame propagation type: Regularity of the free boundary (2008) Ann. Inst. H. Poincare Anal. Non Lineaire, 25, pp. 633-658
- Weiss, G.S., A singular limit arising in combustion theory: Fine properties of the free boundary (2003) Calc. Var. Partial Differential Equations, 17, pp. 311-340
- Zeldovich, Y.B., Frank-Kamenetskii, D.A., The theory of thermal propagation of flames (1938) Zh. Fiz. Khim, 12, pp. 100-105. , (in Russian); selected Works of Yakov Borisovich Zeldovich, 1, Princeton Univ. Press, Princeton, NJ (1992 in English)
Citas:
---------- APA ----------
Sandra, M. & Noemi, W.
(2009)
. A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman. SIAM Journal on Mathematical Analysis, 41(1), 318-359.
http://dx.doi.org/10.1137/070703740---------- CHICAGO ----------
Sandra, M., Noemi, W.
"A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman"
. SIAM Journal on Mathematical Analysis 41, no. 1
(2009) : 318-359.
http://dx.doi.org/10.1137/070703740---------- MLA ----------
Sandra, M., Noemi, W.
"A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman"
. SIAM Journal on Mathematical Analysis, vol. 41, no. 1, 2009, pp. 318-359.
http://dx.doi.org/10.1137/070703740---------- VANCOUVER ----------
Sandra, M., Noemi, W. A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman. SIAM J. Math. Anal. 2009;41(1):318-359.
http://dx.doi.org/10.1137/070703740