arXiv subjects
Publications and source records attributed to Andrea Marchese.
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension.
We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure.
In this note we introduce a new model for the mailing problem in branched transportation in order to allow the cost functional to take into account the orientation of the moving particles. This gives an effective answer to [Problem 15.9] of the book "Optimal transportation networks" by Bernot, Caselles, and Morel. Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. With such approach we provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points.
We prove the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This is the Lagrangian counterpart of the recent Eulerian version proved in [CDM19a].
In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $μ$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $μ$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(μ,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point $x$ if it is null at the origin and it is the sum of a linear function on $V(μ,x)$ and a Lipschitz function on $V(μ,x)^{\perp}$.
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.
In this paper we study a variant of the branched transportation problem, that we call multi-material transport problem. This is a transportation problem, where distinct commodities are transported simultaneously along a network. The cost of the transportation depends on the network used to move the masses, as it is common in models studied in branched transportation. The main novelty is that in our model the cost per unit length of the network does not depend only on the total flow, but on the actual quantity of each commodity. This allows to take into account different interactions between the transported goods. We propose an Eulerian formulation of the discrete problem, describing the flow of each commodity through every point of the network. We provide minimal assumptions on the cost, under which existence of solutions can be proved. Moreover, we prove that, under mild additional assumptions, the problem can be rephrased as a mass minimization problem in a class of rectifiable currents with coefficients in a group, allowing to introduce a notion of calibration. The latter result is new even in the well studied framework of the "single-material" branched transportation.
Based on Smirnov's decomposition theorem we prove that every rectifiable $1$-current $T$ with finite mass $\mathbb{M}(T)$ and finite mass $\mathbb{M}(\partial T)$ of its boundary $\partial T$ can be approximated in mass by a sequence of rectifiable $1$-currents $T_n$ with polyhedral boundary $\partial T_n$ and $\mathbb{M}(\partial T_n)$ no larger than $\mathbb{M}(\partial T)$. Using this result we can compute the relaxation of the $h$-mass for polyhedral $1$-currents with respect to the joint weak-$\ast$ convergence of currents and their boundaries. We obtain that this relaxation coincides with the usual $h$-mass for normal currents. This shows that the concepts of so-called generalized branched transport and the $h$-mass are equivalent.
Let $Q=(0,1)^2$ be the unit square in $\mathbb{R}^2$. We prove that in a suitable complete metric space of $BV$ homeomorphisms $f:Q\rightarrow Q$ with $f_{|\partial Q}=Id$, the generical homeomorphism (in the sense of Baire categories) maps a null set in a set of full measure and vice versa. Moreover we observe that, for $1\leq p<2$, in the most reasonable complete metric space for such problem, the family of $W^{1,p}$ homemomorphisms satisfying the above property is of first category, instead.
In this paper we extend the results of "A strong minimax property of nondegenerate minimal submanifolds" by White, where it is proved that any smooth, compact submanifold, which is a strictly stable critical point for an elliptic parametric functional, is the unique minimizer in a certain geodesic tubular neighbourhood. We prove a similar result, replacing the tubular neighbourhood with one induced by the flat distance and we provide quantitative estimates. Our proof is based on the introduction of a penalized minimization problem, in the spirit of "A selection principle for the sharp quantitative isoperimetric inequality" by Cicalese and Leonardi, which allows us to exploit the regularity theory for almost minimizers of elliptic parametric integrands.
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.
We prove that for any singular measure $μ$ on $\mathbb{R}^n$ it is possible to cover $μ$-almost every point with $n$ families of Lipschitz slabs of arbitrarily small total width. More precisely, up to a rotation, for every $δ>0$ there are $n$ countable families of $1$-Lipschitz functions $\{f_i^1\}_{i\in\mathbb{N}},\ldots, \{f_i^n\}_{i\in\mathbb{N}},$ $f_i^j:\{x_j=0\}\subset\mathbb{R}^n\to\mathbb{R}$, and $n$ sequences of positive real numbers $\{\varepsilon_i^1\}_{i\in\mathbb{N}},\ldots, \{\varepsilon_i^n\}_{i\in\mathbb{N}}$ such that, denoting $\hat x_j$ the orthogonal projection of the point $x$ onto $\{x_j=0\}$ and $$I_i^j:=\{x=(x_1,\ldots,x_n)\in \mathbb{R}^n:f_i^j(\hat x_j)-\varepsilon_i^j< x_j< f_i^j(\hat x_j)+\varepsilon_i^j\},$$ it holds $\sum_{i,j}\varepsilon_i^j\leq δ$ and $μ(\mathbb{R}^n\setminus\bigcup_{i,j}I_i^j)=0.$ We apply this result to show that, if $μ$ is not absolutely continuous, it is possible to approximate the identity with a sequence $g_h$ of smooth equi-Lipschitz maps satisfying $$\limsup_{h\to\infty}\int_{\mathbb{R}^n}{\rm{det}}(\nabla g_h) dμ<μ(\mathbb{R}^n).$$ From this, we deduce a simple proof of the fact that every top-dimensional Ambrosio-Kirchheim metric current in $\mathbb{R}^n$ is a Federer-Fleming flat chain.
In this paper, we prove that every equivalence class in the quotient group of integral $1$-currents modulo $p$ in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for $m$-dimensional integral currents modulo $p$ implies that the family of $(m-1)$-dimensional flat chains of the form $pT$, with $T$ a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for $0$-dimensional flat chains, and, using a proposition from "The structure of minimizing hypersurfaces mod $4$" by Brian White, also for flat chains of codimension $1$.
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 \[ \mathbb{M}_{H}(R) := \int_E H(θ(x)) \, d\mathcal{H}^m(x) \quad \mbox{ for every } R= [[ E,τ,θ]] \in \mathbf{R}_{m}(\mathbb{R}^{n}). \]
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 $μ^-$.
This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.
We add to the literature the following observation. If $μ$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $μ$-a.e. then for every $C^1$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$μ\{x\in\mathbb{R}^n: f(x)=g(x)\}=0.$$ In other words the Lusin type approximation property of Lipschitz functions with $C^1$ functions does not hold with respect to a general Radon measure.
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $μ$. In particular we show that the differentiability properties of Lipschitz functions at $μ$-almost every point are related to the decompositions of $μ$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.