Rectifiability of Varifolds with Locally Bounded First Variation with Respect to Anisotropic Surface Energies

We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every $d$-dimensional varifold with locally bounded first variation and positive $d$-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane

Equivalence of the ellipticity conditions for geometric variational problems

We exploit the so called atomic condition, recently defined by De Philippis, De Rosa, and Ghiraldin in [Comm. Pure Appl. Math.] and proved to be necessary and sufficient for the validity of the anisotropic counterpart of the Allard rectifiability theorem. In particular, we address an open question of this seminal work, showing that the atomic condition implies the strict Almgren geometric ellipticity condition

Soap films with gravity and almost-minimal surfaces

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.Comment: 34 pages, 6 figures. Version 2: more detailed description of the proof of the estimates in Section 5 adde

Partial Differential Equations (hybrid meeting)

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge-Amp\`ere equations

Anisotropic energies in geometric measure theory

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

Construction of fillings with prescribed Gaussian image and applications

We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the prescribed measure is centered) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure, considered as a measure on $\bigwedge^d \mathbb{R}^n$, is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov [Geom. funct. anal., 2004]; our work gives an explicit construction. Furthermore, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove that for anisotropic surface energies, ellipticity for Lipschitz multivalued functions is equivalent to polyconvexity and to show that strict polyconvexity is necessary for the atomic condition to hold