Searcharxiv⌕ Search

arXiv subjects

Andrea Marchese

Publications and source records attributed to Andrea Marchese.

At least 19 recordsLinked to original sources
arXiv2026

A model of anisotropic branched optimal transport

We propose a new anisotropic optimal transport model based on the theory of currents, where the anisotropic cost function splits as the product of a factor depending only on the spatial direction and a factor depending only on the multiplicity of the current. We prove that the planar transport problem admits a minimizer. In arbitrary dimension, we show that a minimizer exists provided that the ambient space endowed with the anisotropic norm is hypermetric.

math.OC↗
arXiv2026

Lipschitz solvability of prescribed Jacobian and divergence for singular measures

Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian equation with Lipschitz solutions. More precisely, for every $\varepsilon>0$ and every Borel datum $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C^1_c(Ω;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(Ω,μ)}$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $Φ-\operatorname{Id}\in C^1_c(Ω;\mathbb{R}^d)$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(Ω,μ)}$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.

math.AP↗
arXiv2025

A PDE perspective on the flat chain conjecture

This survey summarizes recent progress on the flat chain conjecture, which asserts the equivalence between metric currents and flat chains with finite mass in the Euclidean space. In particular, we focus on recent work showing that the conjecture is equivalent to a Lipschitz regularity estimate for a certain PDE.

math.AP↗
arXiv2025

Variational models of robust optimal transport

This paper introduces two variational formulations for a model of robust optimal transport, that is, the problem of designing optimal transport networks that are resilient to potential damages, balancing construction costs against the benefit of maintaining partial functionality when parts of the network are damaged. We propose a Eulerian formulation, where the network is modeled by a rectifiable measure and recovery plans are represented by 1-dimensional normal currents. This framework allows for changes in the direction of the transportation in response to damages but restricts damages to be characteristic functions of closed sets. We also propose a Lagrangian formulation, where the network is a traffic plan (that is, a measure on the space of Lipschitz curves) and recovery plans are sub-traffic plans. This approach prescribes the network's orientation but allows for a wider class of damages. We prove existence of minimizers in both settings. The two models are compared through examples that illustrate their main differences: the Eulerian formulation's necessity for an unoriented network to achieve existence, the Lagrangian formulation's ability to handle general damages and its requirement for a positive distance between the supports of the source and target measures.

math.AP↗
arXiv2025

Cryogenic pressure sensing with an ultrafast Meissner-levitated microrotor

Magnetically levitated spinning rotors are key elements in important technologies such as navigation by gyroscopes, energy storage by flywheels, ultra-high vacuum generation by turbomolecular pumps, and pressure sensing for process control. However, mechanical rotors are typically macroscopic and limited to room temperature and low rotation frequencies. In particular, sensing pressure at low temperatures remains a technological challenge, while emerging quantum technologies demand a precise evaluation of pressure conditions at low temperatures to cope with quantum-spoiling decoherence. To close this gap, we demonstrate wide range pressure sensing by a spinning rotor based on a micromagnet levitated by the Meissner effect at 4.2 Kelvin. We achieve rotational speeds of up to 138 million rotations per minute, resulting in very high effective quality factors, outperforming current platforms. Beside sensing applications, we envision the use of levitated rotors for probing fundamental science including quantum mechanics and gravity, enabled by ultralow torque noise.

quant-ph↗
arXiv2025

Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem

We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ in complete $C^2$ Riemannian manifolds $Σ$ of dimension $m+1$. For odd moduli we prove that, away from a closed rectifiable set of codimension $2$, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common $C^{1,α}$ boundary of dimension $m-1$, and the result is optimal. For even $p$ such structure holds in a neighborhood of any point where at least one tangent cone has $(m-1)$-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Leon Simon in "Cylindrical tangent cones and the singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of multiplicity one stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$.

math.AP↗
arXiv2025

On the closability of differential operators

We discuss the closability of directional derivative operators with respect to a general Radon measure $μ$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}(\mathbb{R}^d)$ to $L^p(μ)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(μ)$ to $L^p(μ)$, and give necessary and sufficient conditions for closability, but we do not have an exact characterization. As a corollary we obtain that classical differential operators such as gradient, divergence and Jacobian determinant are closable from $L^q(μ)$ to $L^p(μ)$ only if $μ$ is absolutely continuous with respect to the Lebesgue measure. We finally consider the closability of a certain class of multilinear differential operators; these results are then rephrased in terms of metric currents.

math.CA↗
arXiv2024

A refined Lusin type theorem for gradients

We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure $μ$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $μ(E)$, if the value of $f$ is $μ$-a.e. orthogonal to the decomposability bundle of $μ$. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^n$ and we state a suitable generalization for $k$-forms, which would imply the validity of the conjecture in full generality.

math.AP↗
arXiv2023

On the structure of flat chains with finite mass

We prove that every flat chain with finite mass in $\mathbb{R}^d$ with coefficients in a normed abelian group $G$ is the restriction of a normal $G$-current to a Borel set. We deduce a characterization of real flat chains with finite mass in terms of a pointwise relation between the associated measure and vector field. We also deduce that any codimension-one real flat chain with finite mass can be written as an integral of multiplicity-one rectifiable currents, without loss of mass. Given a Lipschitz homomorphism $ϕ:\tilde G\to G$ between two groups, we then study the associated map $π$ between flat chains in $\mathbb{R}^d$ with coefficients in $\tilde G$ and $G$ respectively. In the case $\tilde G=\mathbb{R}$ and $G=\mathbb{S}^1$, we prove that if $ϕ$ is surjective, so is the restriction of $π$ to the set of flat chains with finite mass of dimension $0$, $1$, $d-1$, $d$.

math.CA↗
arXiv2023

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,β}$ and an oriented, closed submanifold $Γ\subset \mathcal M$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{h,α}$-perturbations of $Γ$ inside $\mathcal{M}$, with $α<β$, enjoying the following property. For the typical element $b\in\mathcal B$, in the sense of Baire categories, there exists a unique $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem and it has multiplicity one.

math.AP↗
arXiv2023

Excess decay for minimizing hypercurrents mod $2Q$

We consider codimension $1$ area-minimizing $m$-dimensional currents $T$ mod an even integer $p=2Q$ in a $C^2$ Riemannian submanifold $Σ$ of the Euclidean space. We prove a suitable excess-decay estimate towards the unique tangent cone at every point $q\in \mathrm{spt} (T)\setminus \mathrm{spt}^p (\partial T)$ where at least one such tangent cone is $Q$ copies of a single plane. While an analogous decay statement was proved in arXiv:2111.11202 as a corollary of a more general theory for stable varifolds, in our statement we strive for the optimal dependence of the estimates upon the second fundamental form of $Σ$. This technical improvement is in fact needed in arXiv:2201.10204 to prove that the singular set of $T$ can be decomposed into a $C^{1,α}$ $(m-1)$-dimensional submanifold and an additional closed remaining set of Hausdorff dimension at most $m-2$.

math.AP↗
arXiv2022

On the converse of Pansu's Theorem

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures $μ$, then $μ$ must be absolutely continuous with respect to the Haar measure of the group.

math.MG↗
arXiv2022

Fine structure of the singular set of area minimizing hypersurfaces modulo $p$

Consider an area minimizing current modulo $p$ of dimension $m$ in a smooth Riemannian manifold of dimension $m+1$. We prove that its interior singular set is, up to a relatively closed set of dimension at most $m-2$, a $C^{1,α}$ submanifold of dimension $m-1$ at which, locally, $N\leq p$ regular sheets of the current join transversally, each sheet counted with a positive multiplicity $k_i$ so that $\sum_i k_i = p$. This completes the analysis of the structure of the singular set of area minimizing hypersurfaces modulo $p$, initiated by J. Taylor for $m=2$ and $p=3$ and extended by the authors to arbitrary $m$ and all odd $p$ in arXiv:2105.08135. We tackle the remaining case of even $p$ by showing that the set of singular points admitting a flat blow-up is of codimension at least two in the current. First, we prove a structural result for the singularities of minimizers in the linearized problem, by combining an epiperimetric inequality with an analysis of homogeneous minimizers to conclude that the corresponding degrees of homogeneity are always integers; second, we refine Almgren's blow-up procedure to prove that all flat singularities of the current persist as singularities of the $\mathrm{Dir}$-minimizing limit. An important ingredient of our analysis is the uniqueness of flat tangent cones at singular points, recently established by Minter and Wickramasekera in arXiv:2111.11202.

math.AP↗
arXiv2021

Characterization of rectifiability via Lusin type approximation

We prove that a Radon measure $μ$ on $\mathbb{R}^n$ can be written as $μ=\sum_{i=0}^nμ_i$, where each of the $μ_i$ is an $i$-dimensional rectifiable measure if and only if for every Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ and every $\varepsilon>0$ there exists a function $g$ of class $C^1$ such that $μ(\{x\in\mathbb{R}^n:g(x)\neq f(x)\})<\varepsilon$.

math.CA↗
arXiv2021

A multi-material transport problem with arbitrary marginals

In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures $μ^-$, $μ^+$ on $\mathbb{R}^n$ with values in $\mathbb{R}^m$. When the measures are finite atomic, a discrete transportation network is a measure $T$ on $\mathbb{R}^n$ with values in $\mathbb{R}^{n\times m}$ represented by an oriented graph $\mathcal{G}$ in $\mathbb{R}^n$ whose edges carry multiplicities in $\mathbb{R}^m$. The constraint is encoded in the relation ${\rm div}(T)=μ^--μ^+$. The cost of the discrete transportation $T$ is obtained integrating on $\mathcal{G}$ a general function $\mathcal{C}:\mathbb{R}^m\to\mathbb{R}$ of the multiplicity. When the initial data $\left(μ^-,μ^+\right)$ are arbitrary (possibly diffuse) measures, the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data $\left(μ^-,μ^+\right)$. Furthermore, under additional assumptions on the cost integrand $\mathcal{C}$, we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.

math.AP↗