The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p
In this paper we study the behavior as p→∞ of solutions up,q to −Δpu−Δqu=0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u=g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj→∞ such that upj,q→u∞ uniformly in Ω¯ and we prove tha...
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paper:paper_0362546X_v150_n_p104_Bonheure2023-06-08T15:35:14Z The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p Rossi, Julio Daniel p-Laplacian Viscosity solutions Nonlinear equations Viscosity Behavior of solutions Dirichlet boundary Elliptic equations Lipschitz constant P-Laplacian Smooth domains Variational problems Viscosity solutions Laplace transforms In this paper we study the behavior as p→∞ of solutions up,q to −Δpu−Δqu=0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u=g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj→∞ such that upj,q→u∞ uniformly in Ω¯ and we prove that this limit u∞ is a solution to a variational problem, that, when the Lipschitz constant of the boundary datum is less than or equal to one, is given by the minimization of the Lq-norm of the gradient with a pointwise constraint on the gradient. In addition we show that the limit is a viscosity solution to a limit PDE problem that involves the q-Laplacian and the ∞-Laplacian. © 2016 Elsevier Ltd Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v150_n_p104_Bonheure http://hdl.handle.net/20.500.12110/paper_0362546X_v150_n_p104_Bonheure |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
p-Laplacian Viscosity solutions Nonlinear equations Viscosity Behavior of solutions Dirichlet boundary Elliptic equations Lipschitz constant P-Laplacian Smooth domains Variational problems Viscosity solutions Laplace transforms |
| spellingShingle |
p-Laplacian Viscosity solutions Nonlinear equations Viscosity Behavior of solutions Dirichlet boundary Elliptic equations Lipschitz constant P-Laplacian Smooth domains Variational problems Viscosity solutions Laplace transforms Rossi, Julio Daniel The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| topic_facet |
p-Laplacian Viscosity solutions Nonlinear equations Viscosity Behavior of solutions Dirichlet boundary Elliptic equations Lipschitz constant P-Laplacian Smooth domains Variational problems Viscosity solutions Laplace transforms |
| description |
In this paper we study the behavior as p→∞ of solutions up,q to −Δpu−Δqu=0 in a bounded smooth domain Ω with a Lipschitz Dirichlet boundary datum u=g on ∂Ω. We find that there is a uniform limit of a subsequence of solutions, that is, there is pj→∞ such that upj,q→u∞ uniformly in Ω¯ and we prove that this limit u∞ is a solution to a variational problem, that, when the Lipschitz constant of the boundary datum is less than or equal to one, is given by the minimization of the Lq-norm of the gradient with a pointwise constraint on the gradient. In addition we show that the limit is a viscosity solution to a limit PDE problem that involves the q-Laplacian and the ∞-Laplacian. © 2016 Elsevier Ltd |
| author |
Rossi, Julio Daniel |
| author_facet |
Rossi, Julio Daniel |
| author_sort |
Rossi, Julio Daniel |
| title |
The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| title_short |
The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| title_full |
The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| title_fullStr |
The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| title_full_unstemmed |
The behavior of solutions to an elliptic equation involving a p-Laplacian and a q-Laplacian for large p |
| title_sort |
behavior of solutions to an elliptic equation involving a p-laplacian and a q-laplacian for large p |
| publishDate |
2017 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0362546X_v150_n_p104_Bonheure http://hdl.handle.net/20.500.12110/paper_0362546X_v150_n_p104_Bonheure |
| work_keys_str_mv |
AT rossijuliodaniel thebehaviorofsolutionstoanellipticequationinvolvingaplaplacianandaqlaplacianforlargep AT rossijuliodaniel behaviorofsolutionstoanellipticequationinvolvingaplaplacianandaqlaplacianforlargep |
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1768544597506523136 |