Inhomogeneous minimization problems for the p(x)-Laplacian

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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 mini...

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Detalles Bibliográficos
Autor principal: Lederman, C.
Otros Autores: Wolanski, N.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Academic Press Inc. 2019
Acceso en línea:Registro en Scopus
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040 |a Scopus  |b spa  |c AR-BaUEN  |d AR-BaUEN 
100 1 |a Lederman, C. 
245 1 0 |a Inhomogeneous minimization problems for the p(x)-Laplacian 
260 |b Academic Press Inc.  |c 2019 
270 1 0 |m Wolanski, N.; IMAS, CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos AiresArgentina; email: wolanski@dm.uba.ar 
506 |2 openaire  |e Política editorial 
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520 3 |a 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.  |l eng 
593 |a IMAS, CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina 
690 1 0 |a FREE BOUNDARY PROBLEM 
690 1 0 |a INHOMOGENEOUS PROBLEM 
690 1 0 |a MINIMIZATION PROBLEM 
690 1 0 |a REGULARITY OF THE FREE BOUNDARY 
690 1 0 |a SINGULAR PERTURBATION 
690 1 0 |a VARIABLE EXPONENT SPACES 
700 1 |a Wolanski, N. 
773 0 |d Academic Press Inc., 2019  |g v. 475  |h pp. 423-463  |k n. 1  |p J. Math. Anal. Appl.  |x 0022247X  |w (AR-BaUEN)CENRE-271  |t Journal of Mathematical Analysis and Applications 
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