Inhomogeneous minimization problems for the p(x)-Laplacian
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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todo:paper_0022247X_v475_n1_p423_Lederman2023-10-03T14:29:26Z Inhomogeneous minimization problems for the p(x)-Laplacian Lederman, C. Wolanski, N. Free boundary problem Inhomogeneous problem Minimization problem Regularity of the free boundary Singular perturbation Variable exponent spaces 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=limp ε , f=limf 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_0022247X_v475_n1_p423_Lederman |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Free boundary problem Inhomogeneous problem Minimization problem Regularity of the free boundary Singular perturbation Variable exponent spaces |
| spellingShingle |
Free boundary problem Inhomogeneous problem Minimization problem Regularity of the free boundary Singular perturbation Variable exponent spaces Lederman, C. Wolanski, N. Inhomogeneous minimization problems for the p(x)-Laplacian |
| topic_facet |
Free boundary problem Inhomogeneous problem Minimization problem Regularity of the free boundary Singular perturbation Variable exponent spaces |
| description |
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=limp ε , f=limf 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. |
| format |
JOUR |
| author |
Lederman, C. Wolanski, N. |
| author_facet |
Lederman, C. Wolanski, N. |
| author_sort |
Lederman, C. |
| title |
Inhomogeneous minimization problems for the p(x)-Laplacian |
| title_short |
Inhomogeneous minimization problems for the p(x)-Laplacian |
| title_full |
Inhomogeneous minimization problems for the p(x)-Laplacian |
| title_fullStr |
Inhomogeneous minimization problems for the p(x)-Laplacian |
| title_full_unstemmed |
Inhomogeneous minimization problems for the p(x)-Laplacian |
| title_sort |
inhomogeneous minimization problems for the p(x)-laplacian |
| url |
http://hdl.handle.net/20.500.12110/paper_0022247X_v475_n1_p423_Lederman |
| work_keys_str_mv |
AT ledermanc inhomogeneousminimizationproblemsforthepxlaplacian AT wolanskin inhomogeneousminimizationproblemsforthepxlaplacian |
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1807319156684488704 |