Taiwanese Journal of Mathematics

MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION

Mao-Sheng Chang, Shu-Cheng Lee, and Chien-Chang Yen

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Abstract

This paper presents the existence of minimizers and $\Gamma$-convergence for the energey functionals \begin{eqnarray*} E_\epsilon(u) = \int_\Omega \left\{ W(u(x))+\epsilon{|\nabla u(x)|^p}\right\} dx, \mbox{ for all }\epsilon>0,\quad p>1 \end{eqnarray*} with Neumann boundary condition and the constraint \begin{eqnarray*} \int_\Omega u(x) dx = m|\Omega|, \mbox{ where }0 \lt m \lt 1. \end{eqnarray*} The energy functionals discussed in this paper are associated with the Euler-Lagrange $p$-Laplacian equation. We employ the direct method in the calculus of variations to show the existence of minimizers. The $\Gamma$-convergence is achieved with the help of coarea formula and Young's inequality.

Article information

Source
Taiwanese J. Math., Volume 13, Number 6B (2009), 2021-2036.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405655

Digital Object Identifier
doi:10.11650/twjm/1500405655

Mathematical Reviews number (MathSciNet)
MR2583535

Zentralblatt MATH identifier
1192.49017

Subjects
Primary: 49J45: Methods involving semicontinuity and convergence; relaxation

Keywords
minimizer gamma-convergence $p$-Laplacian equation functions of bounded variation

Citation

Chang, Mao-Sheng; Lee, Shu-Cheng; Yen, Chien-Chang. MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION. Taiwanese J. Math. 13 (2009), no. 6B, 2021--2036. doi:10.11650/twjm/1500405655. https://projecteuclid.org/euclid.twjm/1500405655


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