We link covering spaces with the theory of functions of bounded variation, in order to study minimal networks in the plane and Plateau’s problem without fixing a priori the topology of solutions. We solve the minimization problem in the class of (possibly vector-valued) $\mathrm{BV}$ functions defined on a covering space of the complement of an ${(n-2)}$-dimensional compact embedded Lipschitz manifold S without boundary. This approach has several similarities with Brakke’s “soap films” covering construction. The main novelty of our method stands in the presence of a suitable constraint on the fibers, which couples together the covering sheets. In the case of networks, the constraint is defined using a suitable subset of transpositions of m elements, m being the number of points of S. The model avoids all issues concerning the presence of the boundary S, which is automatically attained. The constraint is lifted in a natural way to Sobolev spaces, allowing also an approach based on Γ-convergence.

Abstract:Given an anisotropy ϕ on R 3 , we discuss the relations between the ϕ-calibrability of a facet F ⊂ ∂E of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet︀ B F ϕ of the unit ball of ϕ, ϕ-calibrability is equivalent to show the existence of a ϕ-subunitary vector field in F, with suitable normal trace on ∂F, and with constant divergence equal to the ϕ-mean curvature of F. Assuming E convex at F,̃︀ B F ϕ a disk, and F (strictly) ϕ-calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict ϕ-calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C 1,1 , the solution of the total variation flow starting at 1 F .

We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m- equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies