Coupled Singular Perturbations for Phase Transitions



Irene FONSECA
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
fonseca@andrew.cmu.edu

and

Cristina POPOVICI
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213 U.S.A.

The $\Gamma(L^{1}(\Omega;\mathbb{R}^{d}))$-limit of the sequence

\begin{displaymath}J_{\varepsilon}(u):=\frac{1}{\varepsilon}E_{\varepsilon}(u)\end{displaymath}

where $E_{\varepsilon}$ is the family of anisotropic singular perturbations

\begin{displaymath}E_{\varepsilon}(u):= \int\limits_{\Omega}f(x, u(x), \varepsilon \nabla u(x))\ dx\end{displaymath}

of a non-convex functional of vector-valued functions

\begin{displaymath}E(u):= \int\limits_{\Omega}f(x, u(x), \nabla u(x))\ dx\end{displaymath}

is obtained where $f$ is a nonnegative energy density satisfying $f(x, u, 0)=0$ if and only if $u \in \{ a, b\}$.

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