*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

**Links:**
http://www.worldscinet.com/m3as/19/preserved-docs/1910/S0218202509003991.pdf

**Abstract:**

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
$$
F_{{\varepsilon}}(u):=\varepsilon^{{p}-2}\!\int_{{\Omega}\!Du}^{pdx+\frac{1}{\varepsilon}^{{\frac{p}-2}{p-1}}}\!\int_{{\Omega}\!W}(u)dx+\frac{1}{\varepsilon}\!\int_{{\partial\Omega}\!V}(Tu)d\mathcal{H}^{2} \ \ \ (p>2),
$$
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

**Download:**