Department of Mathematical Sciences

Center for Nonlinear Analysis

CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact

Publication 99-CNA-10

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.

Get the paper in its entirety as

Back to CNA Publications