Second Order Singular Perturbation Models for Phase Transitions

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 $$ {\mathcal F}_\varepsilon(u):= \begin{cases} \int_{\Omega} \left(\frac{W(u)}{\varepsilon}+\varepsilon \vert \nabla u\vert^2\right) \, dx & \hbox{ if $u\in W^{1,2}(\Omega;{\mathbb R}^d)$, }\\ \\ +\infty & \hbox{ if $u\in L^1(\Omega;{\mathbb R}^d)\setminus W^{1,2}(\Omega;{\mathbb R}^d)$, } \end{cases} $$ where $\Omega$ is a bounded, open set in ${\mathbb R}^N$, is given by $$ {\mathcal F}(u):= \begin{cases} {\rm {\bf m }}\; {\rm Per}_{\Omega} (\{u = a\}) & \hbox{ if $u\in BV(\Omega;\{a,b\})$, }\\ +\infty & \hbox{ otherwise, } $$ for a suitable constant {\bf 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.