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.
Citation
Mao-Sheng Chang. Shu-Cheng Lee. Chien-Chang Yen. "MINIMIZERS AND GAMMA-CONVERGENCE OF ENERGY FUNCTIONALS DERIVED FROM $p$-LAPLACIAN EQUATION." Taiwanese J. Math. 13 (6B) 2021 - 2036, 2009. https://doi.org/10.11650/twjm/1500405655
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