BULK AND CONTACT ENERGIES - NUCLEATION AND RELAXATION

An integral representation formula in BV (Omega; R-p) for the relaxati on H(u, Omega) with respect to the L-1 topology of functionals of the general form [GRAPHICS] is obtained. Here Omega subset of R-N is an op en, bounded set of class C-2, T is the trace operator on partial deriv ative 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; .) is q uasiconvex and has linear growth, and that theta(x, .) is Lipschitz. T he understanding of nucleation phenomena for materials undergoing phas e transitions leads to the study of constrained minimization problems of the type [GRAPHICS] where K is a nonempty compact subset of R-p, an d tau : Omega x K --> R is a continuous function. It R is shown that i f tau(x; .) is a double well potential vanishing only at alpha and bet a, then minimizers u of the total energy are given by pure phases; tha t is, there exists Omega(u) subset of Omega such that u(x) = alpha for L-N a.e. x is an element of Omega(u) (liquid) and u(x) = beta for L-N a.e. x is an element of Omega/Omega(u) (solid). This conclusion is cl osely related to results previously obtained by Visintin, and where th e interfacial energy is assumed to satisfy a generalized co-area formu la. Here this condition is replaced by a property which may be verifie d by energies for which the co-area formula might not hold.