*Published Paper*

**Inserted:** 5 nov 2008

**Last Updated:** 23 dec 2011

**Journal:** Math. Z.

**Volume:** 266

**Number:** 3

**Pages:** 533-560

**Year:** 2010

**Links:**
http://www.springerlink.com/content/v437h222078x1276/

**Abstract:**

We study the $\Gamma$-convergence of the following functional ($p>2$)
$$
F_{{\varepsilon}}(u):=\varepsilon^{{p}-2}\!\int_{{\Omega}\!Du}^{p} d(x,\partial \Omega)^{{a}dx+\frac{1}{\varepsilon}^{{\frac{p}-2}{p-1}}}\!\int_{{\Omega}\!W}(u) d(x,\partial \Omega)^{{}-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}\!\int_{{\partial\Omega}\!V}(Tu)d\mathcal{H}^{2,
}
$$
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

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