Second Order Singular Perturbation Models for Phase Transitions

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
USA
email: fonseca@andrew.cmu.edu

Carlo Mantegazza
Scuola Normale Superiore
Pisa 56126
Italy
email: mantegaz@sns.it

Abstract

Singular perturbation models involving a penalization of the first order derivatives have provided a new insight into the role played by surface energies in the study of phase transitions problems. It is known that if $W:{{\mathbb R}}^d \to [0,+\infty)$ grows at least linearly at infinity and it has exactly two potential wells of level zero at $a, b \in {{\mathbb R}}^d$ , then the $\Gamma(L^1)$ -limit of the family of functionals

$\begin{displaymath}\hspace{-.75in} {\mathcal F}_\varepsilon(u):= \begin{cases} ... ...^d)\setminus W^{1,2}(\Omega;{{\mathbb R}}^d)$ , } \end{cases}\end{displaymath}$

where $\Omega$ is a bounded, open set in ${{\mathbb R}}^N$ , is given by

$\begin{displaymath}{\mathcal F}(u):= \begin{cases} {\rm {\bf m }}\; {\rm Per}_... ...ga;\{a,b\})$ },\\ +\infty & \text{ otherwise, } \end{cases}\end{displaymath}$

for a suitable constant m depending on the energy density W. In this paper, and motivated by the study of phase transitions for nonlinear elastic materials, the $\Gamma(L^1)$ -limit is obtained in the case where in ${\mathcal F}_\varepsilon(u)$ the penalization term $\varepsilon \vert \nabla u\vert^2$ is replaced by $\varepsilon^3 \vert \nabla^2 u\vert^2$ , for $u \in W^{2,2}(\Omega;{{\mathbb R}}^d)$ . The resulting functional is of the same form as ${\mathcal F}(u)$ above.

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