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Remarks on the p-Laplacian on thin domains

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Contributions to Nonlinear Elliptic Equations and Systems

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 86))

Abstract

The limiting behavior of solutions of quasilinear elliptic equations on thin domains is investigated. As we will see the boundary conditions play an important role. If one considers homogeneous Dirichlet boundary conditions, the sequence of solutions will converge to the null function, whereas, if one considers Neumann boundary conditions, there is a nontrivial equation which determines the limiting behavior.

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Notes

  1. 1.

    \(J_{\psi }: W_{0}^{1,p}(\varOmega ^{\epsilon }) \rightarrow 2^{W^{-1,p'}(\varOmega ^{\epsilon }) }\) is defined by \(J_{\psi }u:=\{ u^{{\ast}}\in W^{-1,p'}(\varOmega ^{\epsilon }):\| u^{{\ast}}\|_{W^{-1,p'}(\varOmega ^{\epsilon })} =\psi (\|u\|_{W_{0}^{1,p}(\varOmega ^{\epsilon })})\|u\|_{W_{0}^{1,p}(\varOmega ^{\epsilon })},\ \langle u^{{\ast}},u\rangle _{W_{0}^{-1,p'},W_{0}^{1,p}} =\psi (\|u\|_{W_{0}^{1,p}(\varOmega ^{\epsilon })})\}\)

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Acknowledgements

The authors would like to thank the anonymous referee whose comments have considerably improved the writing of the paper. Marcone C. Pereira was partially supported by FAPESP #2013/22275-1, CAPES DGU #127/07, CNPq #302847/2011-1, and #471210/2013-7, Brazil. Ricardo P. Silva was partially supported by FAPESP #2012/06753-8 and #2014/16165-1, CNPq #440371/2014-7, Brazil.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, Brazil

    Marcone C. Pereira

  2. Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Rio Claro, SP, Brazil

    Ricardo P. Silva

Authors
  1. Marcone C. Pereira
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  2. Ricardo P. Silva
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Corresponding author

Correspondence to Marcone C. Pereira .

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Editors and Affiliations

  1. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Alexandre Nolasco de Carvalho

  2. Dipartimento di Matematica, Università degli Studi di Milano, Lombardia, Milano, Italy

    Bernhard Ruf

  3. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Ederson Moreira dos Santos

  4. Département de Mathématique. CP 214, Université Libre de Bruxelles, Brussels, Belgium

    Jean-Pierre Gossez

  5. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Sergio Henrique Monari Soares

  6. Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, Paris, France

    Thierry Cazenave

Additional information

To Djairo G. de Figueiredo, on the occasion of his 80th birthday

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Pereira, M.C., Silva, R.P. (2015). Remarks on the p-Laplacian on thin domains. In: Nolasco de Carvalho, A., Ruf, B., Moreira dos Santos, E., Gossez, JP., Monari Soares, S., Cazenave, T. (eds) Contributions to Nonlinear Elliptic Equations and Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 86. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19902-3_23

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