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

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