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Antonio De Rosa

Publications and source records attributed to Antonio De Rosa.

34 records · Page 2Linked to original sources
arXiv2020

Regularity for graphs with bounded anisotropic mean curvature

We prove that $m$-dimensional Lipschitz graphs with anisotropic mean curvature bounded in $L^p$, $p>m$, are regular almost everywhere in every dimension and codimension. This provides partial or full answers to multiple open questions arising in the literature. The anisotropic energy is required to satisfy a novel ellipticity condition, which holds for instance in a $C^2$ neighborhood of the area functional. This condition is proved to imply the atomic condition. In particular we provide the first non-trivial class of examples of anisotropic energies in high codimension satisfying the atomic condition, addressing an open question in the field. As a byproduct, we deduce the rectifiability of varifolds (resp. of the mass of varifolds) with locally bounded anisotropic first variation for a $C^2$ (resp. $C^1$) neighborhood of the area functional. In addition to these examples, we also provide a class of anisotropic energies in high codimension, far from the area functional, for which the rectifiability of the mass of varifolds with locally bounded anisotropic first variation holds. To conclude, we show that the atomic condition excludes non-trivial Young measures in the case of anisotropic stationary graphs.

math.AP↗
arXiv2020

A nonlocal approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.

math.AP↗
arXiv2020

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.

math.AP↗
arXiv2019

On the well-posedness of branched transportation

We show in full generality the stability of optimal traffic paths in branched transport: namely we prove that any limit of optimal traffic paths is optimal as well. This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks, by Bernot, Caselles and Morel), which has been addressed up to now only under restrictive assumptions.

math.AP↗
arXiv2019

Existence results for minimizers of parametric elliptic functionals

We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem.

math.AP↗
arXiv2019

The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of (Invent. Math., 1987).

math.AP↗
arXiv2018

Stability for the mailing problem

We prove that optimal traffic plans for the mailing problem in $\mathbb{R}^d$ are stable with respect to variations of the given coupling, above the critical exponent $α=1-1/d$, thus solving an open problem stated in the book "Optimal transportation networks", by Bernot, Caselles and Morel. We apply our novel result to study some regularity properties of the minimizers of the mailing problem. In particular, we show that only finitely many connected components of an optimal traffic plan meet together at any branching point.

math.AP↗
arXiv2017

Quantitative stability for anisotropic nearly umbilical hypersurfaces

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that if the $L^p$ norm of the trace-free part of the anisotropic second fundamental form is small, then $Σ$ must be $W^{2, \, p}$-close to the Wulff shape, with a quantitative estimate.

math.DG↗
arXiv2017

A direct approach to the anisotropic Plateau problem

We prove a compactness principle for the anisotropic formulation of the Plateau problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding the Preiss' Rectifiability Theorem [Pre87].

math.AP↗
arXiv2017

On the lower semicontinuous envelope of functionals defined on polyhedral chains

In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by $H \colon \mathbb{R} \to \left[ 0,\infty \right)$ an even, subadditive, and lower semicontinuous function with $H(0)=0$, and by $Φ_H$ the functional induced by $H$ on polyhedral $m$-chains, namely \[ Φ_{H}(P) := \sum_{i=1}^{N} H(θ_{i}) \mathcal{H}^{m}(σ_{i}), \quad\mbox{for every }P=\sum_{i=1}^{N} θ_{i} [[ σ_{i} ]] \in\mathbf{P}_m(\mathbb{R}^n), \] we prove that the lower semicontinuous envelope of $Φ_H$ coincides on rectifiable $m$-currents with the $H$-mass \[ \mathbb{M}_{H}(R) := \int_E H(θ(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R= [[ E,τ,θ]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). \]

math.AP↗
arXiv2017

Improved stability of optimal traffic paths

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 $α\in (0,1)$ of the intensity of the flow. In this paper we address an open problem in the book "Optimal transportation networks" by Bernot, Caselles and Morel and we improve the stability for optimal traffic paths in the Euclidean space $\mathbb{R}^d$, with respect to variations of the given measures $(μ^-,μ^+)$, which was known up to now only for $α>1-\frac1d$. We prove it for exponents $α>1-\frac1{d-1}$ (in particular, for every $α\in (0,1)$ when $d=2$), for a fairly large class of measures $μ^+$ and $μ^-$.

math.AP↗
arXiv2016

Minimization of anisotropic energies in classes of rectifiable varifolds

We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence with density uniformly bounded from below converges (up to subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is integral, provided the minimizing sequence is made of integral varifolds with uniformly locally bounded anisotropic first variation.

math.AP↗
arXiv2016

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.

math.AP↗
arXiv2015

A direct approach to Plateau's problem in any codimension

This paper aims to propose a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of $d$-rectifiable closed subsets of $\mathbb R^n$: following the previous work \cite{DelGhiMag} the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than $(d-1)$.

math.AP↗