Artículo

Lederman, C.; Wolanski, N. "Inhomogeneous minimization problems for the p(x)-Laplacian" (2019) Journal of Mathematical Analysis and Applications. 475(1):423-463
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Abstract:

This paper is devoted to the study of inhomogeneous minimization problems associated to the p(x)-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the one hand, we study the problem of minimizing the functional J(v)=∫ Ω ([Formula presented]+λ(x)χ {v>0} +fv)dx. We show that nonnegative local minimizers u are solutions to the free boundary problem: u≥0 and (P(f,p,λ ⁎ )){Δ p(x) u:=div(|∇u(x)| p(x)−2 ∇u)=fin {u>0}u=0,|∇u|=λ ⁎ (x)on ∂{u>0} with λ ⁎ (x)=([Formula presented]λ(x)) 1/p(x) and that the free boundary is a C 1,α surface with the exception of a subset of H N−1 -measure zero. On the other hand, we study the problem of minimizing the functional J ε (v)=∫Ω([Formula presented]+B ε (v)+f ε v)dx, where B ε (s)=∫ 0 s β ε (τ)dτ ε>0, β ε (s)=[Formula presented]β([Formula presented]), with β a Lipschitz function satisfying β>0 in (0,1), β≡0 outside (0,1). We prove that if u ε are nonnegative local minimizers, then u ε are solutions to (P ε (f ε ,p ε ))Δ p ε (x) u ε =β ε (u ε )+f ε ,u ε ≥0. Moreover, if the functions u ε , f ε and p ε are uniformly bounded, we show that limit functions u (ε→0) are solutions to the free boundary problem P(f,p,λ ⁎ ) with λ ⁎ (x)=([Formula presented]M) 1/p(x) , M=∫β(s)ds, p=lim⁡p ε , f=lim⁡f ε , and that the free boundary is a C 1,α surface with the exception of a subset of H N−1 -measure zero. In order to obtain our results we need to overcome deep technical difficulties and develop new strategies, not present in the previous literature for this type of problems. © 2019 Elsevier Inc.

Registro:

 Documento: Artículo Título: Inhomogeneous minimization problems for the p(x)-Laplacian Autor: Lederman, C.; Wolanski, N. Filiación: IMAS, CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina Palabras clave: Free boundary problem; Inhomogeneous problem; Minimization problem; Regularity of the free boundary; Singular perturbation; Variable exponent spaces Año: 2019 Volumen: 475 Número: 1 Página de inicio: 423 Página de fin: 463 DOI: http://dx.doi.org/10.1016/j.jmaa.2019.02.049 Título revista: Journal of Mathematical Analysis and Applications Título revista abreviado: J. Math. Anal. Appl. ISSN: 0022247X Registro: http://digital.bl.fcen.uba.ar/collection/paper/document/paper_0022247X_v475_n1_p423_Lederman

Referencias:

• Aboulaich, R., Meskine, D., Souissi, A., New diffusion models in image processing (2008) Comput. Math. Appl., 56, pp. 874-882
• 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. (4), 11 (1), pp. 1-44
• Berestycki, H., Caffarelli, L.A., Nirenberg, L., Uniform estimates for regularization of free boundary problems (1990) Analysis and Partial Differential Equations, Lect. Notes Pure Appl. Math., 122, pp. 567-619. , Cora Sadosky Marcel Dekker New York
• Bonder, J.F., Martínez, S., Wolanski, N., A free boundary problem for the p(x)-Laplacian (2010) Nonlinear Anal., 72, pp. 1078-1103
• 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., Salsa, S., A Geometric Approach to Free Boundary Problems (2005), Amer. Math. Soc. Providence, RI; 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
• Chen, Y., Levine, S., Rao, M., Variable exponent, linear growth functionals in image restoration (2006) SIAM J. Appl. Math., 66, pp. 1383-1406
• Danielli, D., Petrosyan, A., A minimum problem with free boundary for a degenerate quasilinear operator (2005) Calc. Var. Partial Differential Equations, 23 (1), pp. 97-124
• 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., Lebesque and Sobolev Spaces with Variable Exponents (2011) Lecture Notes in Math., 2017. , Springer
• Fan, X., Global C 1,α regularity for variable exponent elliptic equations in divergence form (2007) J. Differential Equations, 235, pp. 397-417
• Federer, H., Geometric Measure Theory (1969) Grundlehren Math. Wiss., 153. , Springer-Verlag New York Inc. New York
• Gustafsson, B., Shahgholian, H., Existence and geometric properties of solutions of a free boundary problem in potential theory (1996) J. Reine Angew. Math., 473, pp. 137-179
• Harjulehto, P., Hästö, P., Orlicz spaces and generalized Orlicz spaces (2018), http://cc.oulu.fi/~phasto/pp/orliczBook.pdf, Manuscript; Kováčik, O., Rákosník, J., On spaces L p(x) and W k,p(x) (1991) Czechoslovak Math. J., 41, pp. 592-618
• Lederman, C., A free boundary problem with a volume penalization (1996) Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 23 (2), pp. 249-300
• 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., Singular perturbation in a nonlocal diffusion problem (2006) Comm. Partial Differential Equations, 31 (2), pp. 195-241
• 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., An inhomogeneous singular perturbation problem for the p(x)-Laplacian (2016) Nonlinear Anal., 138, pp. 300-325
• Lederman, C., Wolanski, N., Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the p(x)-Laplacian (2017) Interfaces Free Bound., 19 (2), pp. 201-241
• Maly, J., Ziemer, W.P., Fine Regularity of Solutions of Elliptic Partial Differential Equations (1997) Math. Surveys Monogr., 51. , American Mathematical Society Providence, RI
• Martínez, S., Wolanski, N., A minimum problem with free boundary in Orlicz spaces (2008) Adv. Math., 218 (6), pp. 1914-1971
• 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
• Moreira, D., Wang, L., Hausdorff measure estimates and Lipschitz regularity in inhomogeneous nonlinear free boundary problems (2014) Arch. Ration. Mech. Anal., 213, pp. 527-559
• Radulescu, V.D., Repovs, D.D., Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis (2015) Monogr. Res. Notes in Math., 9. , Chapman & Hall/CRC Press Boca Raton, FL
• Ricarte, G., Teixeira, E., Fully nonlinear singularly perturbed equations and asymptotic free boundaries (2011) J. Funct. Anal., 261, pp. 1624-1673
• Ruzicka, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Springer-Verlag Berlin; 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

Citas:

---------- APA ----------
Lederman, C. & Wolanski, N. (2019) . Inhomogeneous minimization problems for the p(x)-Laplacian. Journal of Mathematical Analysis and Applications, 475(1), 423-463.
http://dx.doi.org/10.1016/j.jmaa.2019.02.049
---------- CHICAGO ----------
Lederman, C., Wolanski, N. "Inhomogeneous minimization problems for the p(x)-Laplacian" . Journal of Mathematical Analysis and Applications 475, no. 1 (2019) : 423-463.
http://dx.doi.org/10.1016/j.jmaa.2019.02.049
---------- MLA ----------
Lederman, C., Wolanski, N. "Inhomogeneous minimization problems for the p(x)-Laplacian" . Journal of Mathematical Analysis and Applications, vol. 475, no. 1, 2019, pp. 423-463.
http://dx.doi.org/10.1016/j.jmaa.2019.02.049
---------- VANCOUVER ----------
Lederman, C., Wolanski, N. Inhomogeneous minimization problems for the p(x)-Laplacian. J. Math. Anal. Appl. 2019;475(1):423-463.
http://dx.doi.org/10.1016/j.jmaa.2019.02.049