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Daniela Di Donato

Publications and source records attributed to Daniela Di Donato.

At least 19 recordsLinked to original sources
arXiv2026

Discretization and regularization for the reconstruction of inhomogeneities by scattering measurements

We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem with regularization. Such a problem depends on a variety of parameters, that is, the number of measurements, the regularization parameter and the discretization parameter, namely the size of the mesh on which we discretize the unknown coefficients of the Helmholtz type equation modelling our physical system. We show, through a convergence analysis, that one can carefully choose these parameters in such a way that the solution to this discrete regularized minimum problem is a good approximation of the looked-for solution to the inverse problem.

math.AP↗
arXiv2023

Intrinsically Lipschitz sections and applications to metric groups

We introduce a notion of intrinsically Lipschitz graphs in the context of metric spaces. This is a broad generalization of what in Carnot groups has been considered by Franchi, Serapioni, and Serra Cassano, and later by many others. We proceed by focusing our attention on the graphs as subsets of a metric space given by the image of a section of a quotient map and we require an intrinsically Lipschitz condition. We shall not have any function on a topological product, not we shall consider a metric on the base of the quotient map. Our results are: an Ascoli-Arzelà compactness theorem, an Ahlfors regularity theorem, and some extension theorems for partially defined intrinsically Lipschitz sections. Known results by Franchi, Serapioni, and Serra Cassano, and by Vittone will be our corollaries.

math.MG↗
arXiv2023

Normed spaces using intrinsically Lipschitz sections and Extension Theorem for the intrinsically Hölder sections

The purpose of this article is twofold: first of all, we want to define two norms using the space of intrinsically Lipschitz sections. On the other hand, we want to generalize an Extension Theorem proved by the author in the context of the intrinsically Hölder sections with target a topological space $Y.$ Here our target will be $Y\times \R^s$ with $s \geq 1$ instead of $Y.$

math.MG↗
arXiv2022

Non-symmetric intrinsic Hopf-Lax semigroup vs. intrinsic Lagrangian

In this paper, we analyze the 'symmetrized' of the intrinsic Hopf-Lax semigroup introduced by the author in the context of the intrinsically Lipschitz sections in the setting of metric spaces. Indeed, in the usual case, we have that $d(x,y) =d(y,x)$ for any point $x$ and $y$ belong to the metric space $X$; on the other hand, in our intrinsic context, we have that $d(f(x),π^{-1} (y)) \ne d(f(y),π^{-1} (x)),$ for every $x,y \in X$. Therefore, it is not trivial that we get the same result obtained for the "classical" intrinsic Hopf-Lax semigroup, i.e., the 'symmetrized' Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equation. Here, an important observation is that $f$ is just a continuous section of a quotient map $π$ and it can not intrinsic Lipschitz. However, following Evans, the main result of this note is to show that the "new" intrinsic Hopf-Lax semigroup satisfies a suitable variational problem where the functional contained an intrinsic Lagrangian. Hence, we also define and prove some basic properties of the intrinsic Fenchel-Legendre transform of this intrinsic Lagrangian that depends on a continuous section of $π$.

math.DG↗
arXiv2022

Intrinsic Cheeger energy for the intrinsically Lipschitz constants

Recently, in the metric spaces, Le Donne and the author introduced the so-called intrinsically Lipschitz sections. The main aim of this note is to adapt Cheeger theory for the classical Lipschitz constants in our new context. More precisely, we define the intrinsic Cheeger energy from $L^2(Y,\R^s)$ to $[0,+\infty],$ where $(Y,d_Y,\mm)$ is a metric measure space and we characterize it in terms of a suitable notion of relaxed slope. In order to get this result, in more general context, we establish some properties of the intrinsically Lipschitz constants like the Leibniz formula, the product formula and the upper semicontinuity of the asymptotic intrinsically Lipschitz constant.

math.DG↗
arXiv2022

Intrinsically Hölder sections in metric spaces

We introduce a notion of intrinsically Hölder graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over $\R$ or $\C,$ a convex set and an equivalence relation for intrinsically Hölder graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical tools.

math.MG↗
arXiv2022

Ahlfors-David regularity of intrinsically quasi-symmetric sections in metric spaces

We introduce a definition of intrinsically quasi-symmetric sections in metric spaces and we prove the Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano. We do this by focusing our attention on the graph property instead of the map one.

math.MG↗
arXiv2022

The intrinsic Hopf-Lax semigroup vs. The intrinsic slope

In this note, we introduce a natural notion of intrinsic Hopf-Lax semigroup in the context of the so-called intrinsically Lipschitz sections. The main aims are to prove the link between the intrinsic Hopf-Lax semigroup and the intrinsic slope and to show that the intrinsic Hopf-Lax semigroup is a subsolution of Hamilton-Jacobi type equality.

math.MG↗
arXiv2022

Intrinsic Lipschitz sections of no-linear quotient maps

Le Donne and the author introduced the so-called intrinsically Lipschitz sections of a fixed quotient map $π$ in the context of metric spaces. Moreover, the author introduced the concept of intrinsic Cheeger energy when the quotient map is also linear. In this note we investigate about the non linearity of $π$. In particular, we find a Leibniz formula for the intrinsic slope when $π$ satisfies a weaker condition. After that, we focus our attention on Carnot groups and using the properties of intrinsic dilations we show that the dilation of a Lipschitz section is so too. Finally, in Carnot groups of step 2, we give a suitable additional condition in order to get the sum of two intrinsically Lipschitz sections is so too.

math.DG↗
arXiv2022

Intrinsically quasi-isometric sections in metric spaces

This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.

math.MG↗
arXiv2022

Intrinsically Lipschitz graphs on semidirect products of groups

In the metric spaces, we give some equivalent condition of intrinsically Lipschitz maps introduce by Franchi, Serapioni and Serra Cassano in subRiemannian Carnot groups. Unlike what happens in the Carnot groups, in our context intrinsic dilation do not exist but we can prove the same results using the Lipschitz property of the projection maps.

math.MG↗
arXiv2022

A note about Intrinsically Lipschitz constants

Recently, Le Donne and the author introduce a notion of intrinsically Lipschitz graphs in metric spaces. The idea of this paper is to investigate about the properties of the intrinsically Lipschitz constants. More precisely, we give the Leibniz formula and the product formula for the intrinsic slope.

math.MG↗
arXiv2021

Intrinsic Lipschitz maps vs. Lagrangian type solutions in Carnot groups of step 2

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a subclass of Carnot groups of step 2 which includes a corank 1 Carnot groups (and so the Heisenberg groups), Free groups of step 2 and the complexified Heisenberg group. More precisely, we prove the equivalence between intrinsic Lipschitz map and a weak solution to a suitable non linear first order PDE system, which generalizes Lagrangian solution in the context of Heisenberg groups.

math.DG↗
arXiv2021

Metric rectifiability of $\mathbb{H}$-regular surfaces with Hölder continuous horizontal normal

Two definitions for the rectfiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on $\mathbb{H}$-regular surfaces, and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole-Pauls, Bigolin-Vittone, and Antonelli-Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of $\mathbb{H}$-regular surfaces. We prove that $\mathbb{H}$-regular surfaces in $\mathbb{H}^{n}$ with $α$-Hölder continuous horizontal normal, $α> 0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli-Le Donne, where the same conclusion was obtained for $C^{\infty}$-surfaces. In $\mathbb{H}^{1}$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $ε$ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between "big pieces" of metric spaces.

math.CA↗
arXiv2021

Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups

This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that were known previously only in the first Heisenberg group $\mathbb{H}^1$. The main difference to this case arises from the fact that for $1\leq k<n$, the complementary vertical subgroups of $k$-dimensional horizontal subgroups in $\mathbb{H}^n$ are not commutative.

math.CA↗
arXiv2020

Distributional solutions of Burgers' type equations for intrinsic graphs in Carnot groups of step 2

We prove that in arbitrary Carnot groups $\mathbb G$ of step 2, with a splitting $\mathbb G=\mathbb W\cdot\mathbb L$ with $\mathbb L$ one-dimensional, the graph of a continuous function $φ\colon U\subseteq \mathbb W\to \mathbb L$ is $C^1_{\mathrm{H}}$-regular precisely when $φ$ satisfies, in the distributional sense, a Burgers' type system $D^φφ=ω$, with a continuous $ω$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $φ$ to a Burgers' type system $D^φφ=ω$, with $ω$ continuous, is actually a broad solution to $D^φφ=ω$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^φφ=ω$, with $ω$ continuous, enjoy $1/2$-little Hölder regularity along vertical directions.

math.MG↗
arXiv2020

Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups

In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph. We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group. As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.

math.MG↗