Calculus of Variations and Geometric Measure Theory

G. Palatucci - Y. Sire

$\Gamma$-convergence of some super quadratic functionals with singular weights

created by palatucci on 05 Nov 2008
modified on 23 Dec 2011


Published Paper

Inserted: 5 nov 2008
Last Updated: 23 dec 2011

Journal: Math. Z.
Volume: 266
Number: 3
Pages: 533-560
Year: 2010


We study the $\Gamma$-convergence of the following functional ($p>2$) $$ F{\varepsilon}(u):=\varepsilon{p-2}\!\int{\Omega}\!
p d(x,\partial \Omega){a}dx+\frac{1}{\varepsilon{\frac{p-2}{p-1}}}\!\int{\Omega}\!W(u) d(x,\partial \Omega){-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}\!\int{\partial\Omega}\!V(Tu)d\mathcal{H}2, $$ where $\Omega$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $\alpha, \beta$ and $\alpha', \beta'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\partial\Omega$, $d(x,\partial \Omega)$ stands for the distance function to the boundary $\partial\Omega$. We show that the singular limit of the energies $F_{\varepsilon}$ leads to a coupled problem of bulk and surface phase transitions.

Keywords: phase transitions, G-convergence, Line tension, Nonlocal variational problems, weighted Sobolev spaces