Bulk and Contact Energies: Nucleation and Relaxation
Irene Fonseca, Giovanni Leoni · SIAM Journal on Mathematical Analysis · 1998
An integral representation formula in $BV(\Omega; \mbox{\smallBbb R}^p)$ for the relaxation ${\cal H}(u,\Omega)$ with respect to the L 1 topology of functionals of the general form $$ H(u,\Omega):=\int_\Omega h(x,u(x), abla u(x))\,dx+\int_{\partial \Omega} \theta(x, T\,u(x))\,d H_{N-1}(x),\quad u\in W^{1,1}(\Omega;\mbox{\smallBbb R}^p), $$ is obtained. Here $\Omega\subset\mbox{\smallBbb R}^N$ is an open, bounded set of class C 2 , T is the trace operator on $\partial\Omega$, and H N -1 is the N-1-dimensional Hausdorff measure. The main hypotheses on the functions h and $\theta$ are that h(x,u, /cdot) is quasiconvex and has linear growth, and that $\theta(x,\cdot)$ is Lipschitz. The understanding of nucleation phenomena for materials undergoing phase transitions leads to the study of constrained minimization problems of the type $$ \inf\left\{{\cal H}(u,\Omega)+\int_\Omega \tau(x,u(x))\,dx: \, u\in BV(\Omega;K)\right\}, $$ where K is a nonempty compact subset of $\mbox{\smallBbb R}^p$, and $\tau:\Omega\times K\to\mbox{\smallBbb R}$ is a continuous function. It is shown that if $\tau(x,\cdot)$ is a double well potential vanishing only at $\alpha$ and $\beta$, then minimizers u of the total energy are given by pure phases; that is, there exists $\Omega_u\subset\Omega$ such that u(x)=\alpha$ for ${\cal L}^N$ a.e. $x\in\Omega_u$ (liquid) and u(x=\beta$ for ${\cal L}^N$ a.e. $x\in\Omega\backslash\Omega_u$ (solid). This conclusion is closely related to results previously obtained by Visintin, and where the interfacial energy is assumed to satisfy a generalized co-area formula. Here this condition is replaced by a property which may be verified by energies for which the co-area formula might not hold.