On an equilibrium shape of a crystal (Q1358003)

The following minimization problem is considered \[ \inf_{\chi\in\text{BV}({\mathbb{R}}^k), \chi(x)\in \{0,1\}} \left( \int_{{\mathbb{R}}^k} f(\nu) |\nabla \chi|+ \int_{{\mathbb{R}}^k} \chi \theta \right), \tag{P} \] \noindent where \(f\colon {\mathbb{R}}^k \to {\mathbb{R}}\) is a convex, positively homogeneous function strictly positive on the unit sphere and \(\theta \in L^{\infty}({\mathbb{R}}^k)\). The quantity \(|\nabla \varphi|\) is a measure on \({\mathbb{R}}^k\) such that \(\int_{{\mathbb{R}}^k}|\nabla \varphi|\) coincides with the total variation of \(\varphi \in \text{BV} ({\mathbb{R}}^k)\). Interpreting the function \(f = f(\nu)\) as the surface free energy density and \(\theta\) as the temperature field, this problem can be viewed as that of finding an equilibrium shape of a crystal. It is known, by Wulff's method, that for \(f(\nu) = |\nu|\) and \(f(\nu) = |\nu_1|+ |\nu_2|\) the equilibrium shape is a sphere and a square, respectively. The main results are an existence theorem for (P) and a necessary condition for a minimum. They are proved by applying the direct method of the calculus of variations and tools from nonsmooth analysis.