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Giovanni Alberti

Publications and source records attributed to Giovanni Alberti.

17 recordsLinked to original sources
arXiv2025

Frobenius theorem and fine structure of tangency sets to non-involutive distributions

In this paper we provide a complete answer to the question whether Frobenius' Theorem can be generalized to surfaces below the $C^{1,1}$ threshold. We study the fine structure of the tangency set in terms of involutivity of a given distribution and we highlight a tradeoff behavior between the regularity of a tangent surface and that of the tangency set. First of all, we prove a Frobenius-type result, that is, given a $k$-dimensional surface $S$ of class $C^1$ and a non-involutive $k$-distribution $V$, if $E$ is a Borel set contained in the tangency set $\tau(S,V)$ of $S$ to $V$ and $\mathbb1_E\in W^{s,1}(S)$ with $s>1/2$ then $E$ must be $\mathscr{H}^k$-null in $S$. In addition, if $S$ is locally a graph of a $C^1$ function with gradient in $W^{\alpha,q}$ and if a Borel set $E \subset \tau(S,V)$ satisfies $ \mathbb1_E\in W^{s,1}(S)$ with \[ s \in \bigl(0,\tfrac{1}{2}\bigr]\qquad\text{and}\qquad\alpha \;>\; 1 - \Bigl(2 - \tfrac{1}{q}\Bigr) \, s, \] then $\mathscr{H}^k(E) = 0$. We show this exponents' condition to be sharp by constructing, for any $\alpha < 1 - \bigl(2 - \tfrac{1}{q}\bigr) s$, a surface $S $ in the same class as above and a set $E \subset \tau(S,V)$ with $\mathbb1_E \in W^{s,1}(S)$ and $\mathscr{H}^k(E) > 0$. Our methods combine refined fractional Sobolev estimates on rectifiable sets, a Stokes-type theorem for rough forms on finite-perimeter sets, and a generalization of the Lusin's Theorem for gradients.

math.DG↗
arXiv2025

Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carath\'{e}odory spaces

In this paper, we establish refined versions of the Frobenius Theorem for non-involutive distributions and use these refinements to prove an unrectifiability result for Carnot-Carath\'{e}odory spaces. We also introduce a new class of metric spaces that extends the framework of Carnot-Carath\'{e}odory geometry and show that, within this class, Carnot-Carath\'{e}odory spaces are, in some sense, extremal. Our results provide new insights into the relationship between integrability, non-involutivity, and rectifiability in both classical and sub-Riemannian settings.

math.DG↗
arXiv2024

Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II

We consider a *continuous* solution $u$ of the balance law \[ \partial_{\mathit t} u + \partial_{\mathit x} (f(u)) = g\] in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term.

math.AP↗
arXiv2023

On the closability of differential operators

We discuss the closability of directional derivative operators with respect to a general Radon measure $\mu$ 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 $L^p(\mu)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(\mu)$ to $L^p(\mu)$, 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 $L^q(\mu)$ to $L^p(\mu)$ only if $\mu$ 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↗
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↗
arXiv2022

On the geometric structure of currents tangent to smooth distributions

It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces, namely integral and normal currents. We find out that integral currents behave to this regard exactly as smooth surfaces, while the behaviour of normal currents is rather multifaceted. This issue is strictly related to a geometric property of the boundary of currents, which is also discussed in details.

math.DG↗
arXiv2019

Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to Itô and Stratonovich

We show that geometric integrals of the type $\int_Ωf\, d g^1\wedge \, d g^2$ can be defined over a two-dimensional domain $Ω$ when the functions $f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just Hölder continuous with sufficiently large Hölder exponents and the boundary of $Ω$ has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R.~Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map $g=(g^1,g^2)$ when $f(x)=F(x,g(x))$ with $F$ sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.

math.FA↗
arXiv2019

Optimal reinforcing networks for elastic membranes

In this paper we study the optimal reinforcement of an elastic membrane, fixed at its boundary, by means of a network (connected onedimensional structure), that has to be found in a suitable admissible class. We show the existence of an optimal solution that may present multiplicities, that is regions where the optimal structure overlaps. Some numerical simulations are shown to confirm this issue and to illustrate the complexity of the optimal network when their total length becomes large.

math.AP↗
arXiv2019

Loss of regularity for the continuity equation with non-Lipschitz velocity field

We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper "Exponential self-similar mixing by incompressible flows" (J. Amer. Math. Soc. 32 (2019), no. 2), and have been announced in "Exponential self-similar mixing and loss of regularity for continuity equations" (C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11).

math.AP↗
arXiv2018

Exponential self-similar mixing by incompressible flows

We study the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field, which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the case of Lipschitz continuous velocities, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound, established in previous works, on the time decay of both scales. We also present several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields.

math.AP↗
arXiv2018

Lossless Analog Compression

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where K(x) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x. We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution -- with respect to the s-dimensional Hausdorff measure -- supported on countable unions of s-dimensional differentiable submanifolds of the m-dimensional real coordinate space. Countable unions of differentiable submanifolds include essentially all signal models used in the compressed sensing literature. Specifically, we prove that, for almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n>s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n<s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n greater than or equal to s being necessary for recovery with probability of error smaller than one. The central conceptual tools in the development of our theory are geometric measure theory and the theory of real analytic functions.

math.FA↗
arXiv2017

On some geometric properties of currents and Frobenius theorem

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1].

math.DG↗
arXiv2016

Lossless Linear Analog Compression

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ from the noiseless linear measurements ${\boldsymbol{\mathsf{y}}}=\boldsymbol{A}{\boldsymbol{\mathsf{x}}}$ with measurement matrix $\boldsymbol{A}\in{\mathbb R}^{n\times m}$. Specifically, for a random vector ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ of arbitrary distribution we show that ${\boldsymbol{\mathsf{x}}}$ can be recovered with zero error probability from $n>\inf\underline{\operatorname{dim}}_\mathrm{MB}(U)$ linear measurements, where $\underline{\operatorname{dim}}_\mathrm{MB}(\cdot)$ denotes the lower modified Minkowski dimension and the infimum is over all sets $U\subseteq{\mathbb R}^{m}$ with $\mathbb{P}[{\boldsymbol{\mathsf{x}}}\in U]=1$. This achievability statement holds for Lebesgue almost all measurement matrices $\boldsymbol{A}$. We then show that $s$-rectifiable random vectors---a stochastic generalization of $s$-sparse vectors---can be recovered with zero error probability from $n>s$ linear measurements. From classical compressed sensing theory we would expect $n\geq s$ to be necessary for successful recovery of ${\boldsymbol{\mathsf{x}}}$. Surprisingly, certain classes of $s$-rectifiable random vectors can be recovered from fewer than $s$ measurements. Imposing an additional regularity condition on the distribution of $s$-rectifiable random vectors ${\boldsymbol{\mathsf{x}}}$, we do get the expected converse result of $s$ measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as $s$-analytic random vectors.

cs.IT↗
arXiv2015

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

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.

math.CA↗
arXiv2014

Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.

math.AP↗