Abstract

Abstract. The equilibrium con gurations of a one-dimensional variational model that combines terms expressing the bulk energy of a deformable crystal and its surface energy are studied. After elimination of the displacement, the problem reduces to the minimization of a nonconvex and nonlocal functional of a single function, the thickness. Depending on a parameter which strengthens one of the terms comprising the energy at the expense of the other, it is shown that this functional may have a stable absolute minimum or only a minimizing sequence in which the term corresponding to the bulk energy is forced to zero by the production of a crack in the material. Key words. Equilibrium shape, non-convex energy functional, variational problem AMS(MOS) subject classi cations (1985 revision). Primary 49S, 73V25 1. Introduction. The morphological instabilities of interfaces is a topic of primary interest in physics (e.g., see [4]). Currently, many branches of the natural sciences, including low temperature physics, fracture, crystal growth, epitaxy of nano-scale lms, metallurgy, geology, and materials science show a rapidly growing interest in the so called stress driven rearrangement

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