Calculus of Variations and Geometric Measure Theory

G. Palatucci

Phase transitions with the line tension effect: the super-quadratic case

created by palatucci on 10 Jun 2008
modified on 23 Dec 2011


Published Paper

Inserted: 10 jun 2008
Last Updated: 23 dec 2011

Journal: Math. Models Methods Appl. Sciences (M3AS)
Volume: 19
Number: 10
Pages: 1765-1795
Year: 2009


Let $\Omega$ be an open bounded set of $\mathbb{R}^3$ and let $W$ and $V$ be two non-negative continuous functions vanishing at $\alpha, \beta$ and $\alpha', \beta'$, respectively. We analyze the asymptotic behavior as $\varepsilon \to 0$, in terms of $\Gamma$-convergence, of the following functional $$ F{\varepsilon}(u):=\varepsilon{p-2}\!\int{\Omega}\!
pdx+\frac{1}{\varepsilon{\frac{p-2}{p-1}}}\!\int{\Omega}\!W(u)dx+\frac{1}{\varepsilon}\!\int{\partial\Omega}\!V(Tu)d\mathcal{H}2 \ \ \ (p>2), $$ where $u$ is a scalar density function and $Tu$ denotes its trace on $\partial\Omega$. We show that the singular limit of the energies $F_{\varepsilon}$ leads to a coupled problem of bulk and surface phase transitions.

Keywords: functions of bounded variation, phase transitions, $\Gamma$-convergence, Line tension, Nonlocal variational problems