Anisotropic energies in geometric measure theory

Abstract

In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] espectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane. In the paper [45], we apply the main theorem of [42] to the minimization of anisotropic energies in classes of rectifiable varifolds. We prove that the limit of a minimizing sequence of varifolds with density uniformly bounded from below is rectifiable. Moreover, with the further assumption that all the elements of the minimizing sequence are integral varifolds with uniformly locally bounded anisotropic first variation, we show that the limiting varifold is also integral. 2. Stability in branched transport: Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure \(μ^−\) onto a target measure \(μ^+\), along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power α ∈ (0,1) of the intensity of the flow. The transportation cost is called α-mass. In the paper [27] we address an open problem in the book [15] and we improve the stability for optimal traffic paths in the Euclidean space \(R^n\) with respect to variations of the given measures (\(μ^−\), \(μ^+\)), which was known up to now only for α > 1− \(\frac 1n\). We prove it for exponents α > 1− \(\frac{1}{n−1}\) (in particular, for every α ∈ (0,1) when n = 2), for a fairly large class of measures (\μ^+\) and (\μ^−\). The α- mass is a particular case of more general energies induced by even, subadditive, and lower semicontinuous functions H : R → [0,∞) satisfying H (0) = 0. In the paper [28], we prove that the lower semicontinuous envelope of these energy functionals defined on polyhedral chains coincides on rectifiable currents with the H -mass.