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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:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v475_n1_p423_Lederman

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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