An inhomogeneous singular perturbation problem for the p(x)-Laplacian Dedicated to our dear friend and colleague Juan Luis Vázquez on the occasion of his 70th birthday
In this paper we study the following singular perturbation problem for the pϵ(x)-Laplacian: Δpϵ (x)uϵ:=div(|∇uϵ(x)|pϵ (x)-2∇ uϵ)=βϵ(uϵ)+fϵ,uϵ≥0, (Pϵ(fϵ, pϵ)) where ϵ>0, βϵ(s)=1/ϵβ(s/ϵ), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions...
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| Formato: | Capítulo de libro |
| Lenguaje: | Inglés |
| Publicado: |
Elsevier Ltd
2016
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| Acceso en línea: | Registro en Scopus DOI Handle Registro en la Biblioteca Digital |
| Aporte de: | Registro referencial: Solicitar el recurso aquí |
| Sumario: | In this paper we study the following singular perturbation problem for the pϵ(x)-Laplacian: Δpϵ (x)uϵ:=div(|∇uϵ(x)|pϵ (x)-2∇ uϵ)=βϵ(uϵ)+fϵ,uϵ≥0, (Pϵ(fϵ, pϵ)) where ϵ>0, βϵ(s)=1/ϵβ(s/ϵ), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1) and ∫β(s)ds=M. The functions uϵ, fϵ and pϵ are uniformly bounded. We prove uniform Lipschitz regularity, we pass to the limit (ϵ→0) and we show that, under suitable assumptions, limit functions are weak solutions to the free boundary problem: u≥0 and {Δp(x)u = f in {u>0}u=0,|∇u|=λ ∗(x)on ∂{u>0} (P(f, p, λ∗)) with λ∗ (x)=(p(x)/p(x)-1 M)1/p(x), p = lim pϵ and f = lim fϵ. In Lederman and Wolanski (submitted) we prove that the free boundary of a weak solution is a C1,α surface near flat free boundary points. This result applies, in particular, to the limit functions studied in this paper. © 2015 Elsevier Ltd. All rights reserved. |
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| Bibliografía: | Aboulaich, R., Meskine, D., Souissi, A., New diffusion models in image processing (2008) Comput. Math. Appl., 56 (4), pp. 874-882 Andersson, J., Weiss, G.S., A parabolic free boundary problem with Bernoulli type condition on the free boundary (2009) J. Reine Angew. Math., 627, pp. 213-235 Berestycki, H., Caffarelli, L.A., Nirenberg, L., Uniform estimates for regularization of free boundary problems (1990) Analysis and Partial Differential Equations, 122, pp. 567-619. , Cora Sadosky, Lecture Notes in Pure and Applied Mathematics Marcel Dekker New York Berestycki, H., Larrouturou, B., Quelques aspects mathématiques de la propagation des flammes prémélangées (1991) Nonlinear Partial Differential Equations and Their Applications, 10, pp. 65-129. , H. Brezis, J.L. Lions, Collège de France Seminar Pitman London Caffarelli, L.A., Lederman, C., Wolanski, N., Uniform estimates and limits for a two phase parabolic singular perturbation problem (1997) Indiana Univ. Math. J., 46 (2), pp. 453-490 Caffarelli, L.A., Lederman, C., Wolanski, N., Pointwise and viscosity solutions for the limit of a two phase parabolic singular perturbation problem (1997) Indiana Univ. Math. J., 46 (3), pp. 719-740 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 Challal, S., Lyaghfouri, A., Second order regularity for the p(x)-Laplace operator (2011) Math. Nachr., 284 (10), pp. 1270-1279 Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration (2006) SIAM J. Appl. Math., 66 (4), pp. 1383-1406 Danielli, D., Petrosyan, A., Shahgholian, H., A singular perturbation problem for the p-Laplace operator (2003) Indiana Univ. Math. J., 52 (2), pp. 457-476 Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M., (2011) Lebesque and Sobolev Spaces with Variable Exponents, 2017. , Lecture Notes in Mathematics Springer Fan, X., Global C1,α regularity for variable exponent elliptic equations in divergence form (2007) J. Differential Equations, 235, pp. 397-417 Fernandez Bonder, J., Martínez, S., Wolanski, N., A free boundary problem for the p(x)-Laplacian (2010) Nonlinear Anal., 72, pp. 1078-1103 Kováčik, O., Rákosník, J., On spaces Lp(x) and Wk,p(x) (1991) Czechoslovak Math. J., 41, pp. 592-618 Lederman, C., Oelz, D., A quasilinear parabolic singular perturbation problem (2008) Interfaces Free Bound., 10 (4), pp. 447-482 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. (4), 27 (2), pp. 253-288 Lederman, C., Wolanski, N., A two phase elliptic singular perturbation problem with a forcing term (2006) J. Math. Pures Appl., 86 (6), pp. 552-589 Lederman, C., Wolanski, N., Singular perturbation in a nonlocal diffusion problem (2006) Comm. Partial Differential Equations, 31 (2), pp. 195-241 Lederman, C., Wolanski, N., Weak Solutions and Regularity of the Interface in An Inhomogeneous Free Boundary Problem for the P(x)-Laplacian, , submitted Lederman, C., Wolanski, N., On Inhomogeneous Minimization Problems for the P(x)-Laplacian, , in preparation Martínez, S., Wolanski, N., A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman (2009) SIAM J. Math. Anal., 40 (1), pp. 318-359 Moreira, D., Wang, L., Singular perturbation method for inhomogeneous nonlinear free boundary problems (2014) Calc. Var. Partial Differential Equations, 49 (3-4), pp. 1237-1261 Ricarte, G., Teixeira, E., Fully nonlinear singularly perturbed equations and asymptotic free boundaries (2011) J. Funct. Anal., 261, pp. 1624-1673 Ruzicka, M., (2000) Electrorheological Fluids: Modeling and Mathematical Theory, , Springer-Verlag Berlin Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations (1984) J. Differential Equations, 51, pp. 126-150 Vazquez, J.L., The free boundary problem for the heat equation with fixed gradient condition (1996) Free Boundary Problems, Theory and Applications (Zakopane, 1995), 363, pp. 277-302. , M. Niezgódka, P. Strzelecki, Pitman Res. Notes Math. Ser. Longman Harlow Weiss, G.S., A singular limit arising in combustion theory: Fine properties of the free boundary (2003) Calc. Var. Partial Differential Equations, 17 (3), pp. 311-340 Wolanski, N., Local bounds, Harnack inequality and Hölder continuity for divergence type elliptic equations with non-standard growth (2015) Rev. Un. Mat. Argentina, 56 (1), pp. 73-105 Zeldovich, Ya.B., Frank-Kamenetski, D.A., The theory of thermal propagation of flames (1938) Zh. Fiz. Khim., 12, pp. 100-105. , (in Russian); English translation in "Collected Works of Ya. B. Zeldovich", vol. 1, Princeton Univ. Press, 1992 |
| ISSN: | 0362546X |
| DOI: | 10.1016/j.na.2015.09.026 |