The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞
In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in...
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Artículo publishedVersion |
| Publicado: |
2010
|
| Materias: | |
| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0022247X_v363_n2_p502_PerezLlanos https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0022247X_v363_n2_p502_PerezLlanos_oai |
| Aporte de: |
| id |
I28-R145-paper_0022247X_v363_n2_p502_PerezLlanos_oai |
|---|---|
| record_format |
dspace |
| spelling |
I28-R145-paper_0022247X_v363_n2_p502_PerezLlanos_oai2024-08-16 Pérez-Llanos, M. Rossi, J.D. 2010 In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in over(Ω, -). Under adequate hypotheses on the sequence pn, namely that the limits∇ ln pn (x) → ξ (x), and frac(pn, n) (x) → q (x) exist, we prove that the corresponding eigenvalues Λpn and eigenfunctions upn verify that(Λpn)1 / n → Λ∞, upn → u∞ uniformly in over(Ω, -), where Λ∞, u∞ is a nontrivial viscosity solution of the following problem{(min {- Δ∞ u∞ - | ∇ u∞ |2 log (| ∇ u∞ |) 〈 ξ, ∇ u∞ 〉, | ∇ u∞ |q - Λ∞ u∞ q} = 0, in Ω,; u∞ = 0, on ∂ Ω .). © 2009 Elsevier Inc. All rights reserved. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. application/pdf http://hdl.handle.net/20.500.12110/paper_0022247X_v363_n2_p502_PerezLlanos info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Math. Anal. Appl. 2010;363(2):502-511 Eigenvalue problems p (x)-Laplacian ∞-Laplacian The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0022247X_v363_n2_p502_PerezLlanos_oai |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-145 |
| collection |
Repositorio Digital de la Universidad de Buenos Aires (UBA) |
| topic |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian |
| spellingShingle |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian Pérez-Llanos, M. Rossi, J.D. The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| topic_facet |
Eigenvalue problems p (x)-Laplacian ∞-Laplacian |
| description |
In this paper we study the behaviour of the solutions to the eigenvalue problem corresponding to the p (x)-Laplacian operator{(- div (| ∇ u |p (x) - 2 ∇ u) = Λp (x) | u |p (x) - 2 u,, in Ω,; u = 0,, on ∂ Ω,) as p (x) → ∞. We consider a sequence of functions pn (x) that goes to infinity uniformly in over(Ω, -). Under adequate hypotheses on the sequence pn, namely that the limits∇ ln pn (x) → ξ (x), and frac(pn, n) (x) → q (x) exist, we prove that the corresponding eigenvalues Λpn and eigenfunctions upn verify that(Λpn)1 / n → Λ∞, upn → u∞ uniformly in over(Ω, -), where Λ∞, u∞ is a nontrivial viscosity solution of the following problem{(min {- Δ∞ u∞ - | ∇ u∞ |2 log (| ∇ u∞ |) 〈 ξ, ∇ u∞ 〉, | ∇ u∞ |q - Λ∞ u∞ q} = 0, in Ω,; u∞ = 0, on ∂ Ω .). © 2009 Elsevier Inc. All rights reserved. |
| format |
Artículo Artículo publishedVersion |
| author |
Pérez-Llanos, M. Rossi, J.D. |
| author_facet |
Pérez-Llanos, M. Rossi, J.D. |
| author_sort |
Pérez-Llanos, M. |
| title |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_short |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_full |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_fullStr |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_full_unstemmed |
The behaviour of the p (x)-Laplacian eigenvalue problem as p (x) → ∞ |
| title_sort |
behaviour of the p (x)-laplacian eigenvalue problem as p (x) → ∞ |
| publishDate |
2010 |
| url |
http://hdl.handle.net/20.500.12110/paper_0022247X_v363_n2_p502_PerezLlanos https://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_0022247X_v363_n2_p502_PerezLlanos_oai |
| work_keys_str_mv |
AT perezllanosm thebehaviourofthepxlaplacianeigenvalueproblemaspx AT rossijd thebehaviourofthepxlaplacianeigenvalueproblemaspx AT perezllanosm behaviourofthepxlaplacianeigenvalueproblemaspx AT rossijd behaviourofthepxlaplacianeigenvalueproblemaspx |
| _version_ |
1809357072442064896 |