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Publication 01-CNA-08

 

A $ \Gamma$-convergence result for the two-gradient theory of phase transitions

Sergio Conti
Max-Planck-Institut für Mathematik in den Naturwissenschaften
04103 Leipzig, Germany

Irene Fonseca
Department of Mathematical Sciences,
Carnegie Mellon University,
Pittsburgh, PA 15213, U.S.A.

Giovanni Leoni
Dipartimento di Scienze e Tecnologie Avanzate,
Università del Piemonte Orientale,
Alessandria, 15100 Italy

Abstract: The generalization to gradient vector fields of the classical double-well, singularly perturbed functionals,
$\displaystyle I_{\varepsilon}\left( u;\Omega\right) :=\int_{\Omega}{\frac{1}{\varepsilon}%% }W(\nabla u)+\varepsilon\vert\nabla^{2}u\vert^{2}\,\,dx, $

where $ W(\xi)=0$ if and only if $ \xi=A$ or $ \xi=B$, and $ A-B$ is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on $ W$ it is shown that $ I_{\varepsilon}$ $ \Gamma$-converge to

\begin{displaymath} I\left( u;\Omega\right) =\left\{ \begin{array}[c]{ll}%% K^{\... ...ight) $,}\\ +\infty & \text{otherwise,}%% \end{array}\right. \end{displaymath}

where $ K^{\ast}$ is the (constant) interfacial energy per unit area.

 


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