cvgmt Papershttps://cvgmt.sns.it/papers/en-usMon, 06 Feb 2023 16:20:29 +0000Carnot rectifiability and Alberti representationshttps://cvgmt.sns.it/paper/5900/G. Antonelli, E. Le Donne, A. Merlo.<p>This paper introduces and studies the analogue of the notion of Lipschitz differentiability space by Cheeger, using Carnot groups and Pansu derivatives as models. We call such metric measure spaces Pansu differentiability spaces (PDS).</p><p>After fixing a Carnot group $\mathbb G$, we prove three main results.</p><p>1) Being a PDS with $\mathbb G$-valued charts is equivalent to having $\mathrm{rank}(\mathbb G)$ independent and horizontally universal Alberti representations with respect to complete $\mathbb G$-valued charts. This result leverages on a characterization by D. Bate, and extends it to our setting. For non-Abelian Carnot groups, the completeness assumption cannot be removed as in the Euclidean case. One direction of this equivalence can be seen as a metric analogue of Pansu--Rademacher theorem. </p><p>2) In every PDS the push-forward of the measure with respect to every $\mathbb G$-valued chart is absolutely continuous with respect to the Haar measure on $\mathbb G$. This extends the proof of Cheeger's conjecture by De Philippis--Marchese--Rindler to our setting.</p><p>3) For $Q$ being the homogeneous dimension of $\mathbb G$, being a PDS with $\mathbb G$-valued charts, with finite $Q$-upper density, and positive $Q$-lower density almost everywhere, is equivalent to being $\mathbb G$-biLipschitz rectifiable. This extends a result by D. Bate and S. Li to our setting. Moreover, the previous equivalence is false if we substitute $\mathbb G$-biLipschitz with $\mathbb G$-Lipschitz, contrarily to what happens in the Euclidean realm.</p>https://cvgmt.sns.it/paper/5900/On the regularity of optimal potentials in control problems governed by elliptic equationshttps://cvgmt.sns.it/paper/5899/G. Buttazzo, J. Casado-Diaz, F. Maestre.<p>In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of a Schr\"odinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the $BV$ one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the $BV$ regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.</p>https://cvgmt.sns.it/paper/5899/A general criterion for jump set slicing and applicationshttps://cvgmt.sns.it/paper/5898/S. Almi, E. Tasso.<p>In this paper a novel criterion for the slicing of the jump set of a function is provided, which bypasses the codimension-one and the parallelogram law techniques developed in BD-spaces. The approach builds upon a recent rectifiability result of integralgeometric measures and is further applied to the study of the structure of the jump set of functions with generalized bounded deformation in a Riemannian setting.</p>https://cvgmt.sns.it/paper/5898/Steiner and tube formulae in 3D contact sub-Riemannian geometryhttps://cvgmt.sns.it/paper/5897/D. Barilari, T. Bossio.<p>We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in 10.1016<i>j.na.2015.05.006 and arXiv:1703.01592v3 for the Heisenberg group.</i></p>https://cvgmt.sns.it/paper/5897/Surface measure on, and the local geometry of, sub-Riemannian manifoldshttps://cvgmt.sns.it/paper/5896/S. Don, V. Magnani.<p>We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds.Among various technical tools, we establish a general criterion for the uniform convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.</p>https://cvgmt.sns.it/paper/5896/Curved thin-film limits of chiral Dirichlet energieshttps://cvgmt.sns.it/paper/5895/G. Di Fratta, V. Slastikov.<p> We investigate the curved thin-film limit of a family of perturbed Dirichletenergies in the space of $H^1$ Sobolev maps defined in a tubular neighborhoodof an $(n - 1)$-dimensional submanifold $N$ of $\mathbb{R}^n$ and with valuesin an $(m - 1)$-dimensional submanifold $M$ of $\mathbb{R}^m$. The perturbation$\mathsf{K}$ that we consider is represented by a matrix-valued functiondefined on $M$ and with values in $\mathbb{R}^{m \times n}$. Under naturalregularity hypotheses on $N$, $M$, and $\mathsf{K}$, we show that the family ofthese energies converges, in the sense of $\Gamma$-convergence, to an energyfunctional on $N$ of an unexpected form, which is of particular interest in thetheory of magnetic skyrmions. As a byproduct of our results, we get that in thecurved thin-film limit, antisymmetric exchange interactions also manifest underan anisotropic term whose specific shape depends both on the curvature of thethin film and the curvature of the target manifold. Various types ofantisymmetric exchange interactions in the variational theory of micromagnetismare a source of inspiration and motivation for our work.</p>https://cvgmt.sns.it/paper/5895/On the numerical approximation of Blaschke-Santaló diagrams using Centroidal Voronoi Tessellationshttps://cvgmt.sns.it/paper/5894/B. Bogosel, G. Buttazzo, E. Oudet.<p>Identifying Blaschke-Santal\'o diagrams is an important topic that essentially consists in determining the image $Y=F(X)$ of a map $F:X\to{\mathbb{R}}^d$, where the dimension of the source space $X$ is much larger than the one of the target space. In some cases, that occur for instance in shape optimization problems, $X$ can even be a subset of an infinite-dimensional space. The usual Monte Carlo method, consisting in randomly choosing a number $N$ of points $x_1,\dots,x_N$ in $X$ and plotting them in the target space ${\mathbb{R}}^d$, produces in many cases areas in $Y$ of very high and very low concentration leading to a rather rough numerical identification of the image set. On the contrary, our goal is to choose the points $x_i$ in an appropriate way that produces a uniform distribution in the target space. In this way we may obtain a good representation of the image set $Y$ by a relatively small number $N$ of samples which is very useful when the dimension of the source space $X$ is large (or even infinite) and the evaluation of $F(x_i)$ is costly. Our method consists in a suitable use of {\it Centroidal Voronoi Tessellations} which provides efficient numerical results. Simulations for two and three dimensional examples are shown in the paper.</p>https://cvgmt.sns.it/paper/5894/Generic uniqueness for the Plateau problemhttps://cvgmt.sns.it/paper/5893/G. Caldini, A. Marchese, A. Merlo, S. Steinbrüchel.<p>Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{3,\beta}$, without boundary and an oriented, closed submanifold $\Gamma \subset \mathcal M$ of dimension $m-1$, of class $C^{3,\alpha}$ with $\alpha<\beta$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{3,\alpha}$-perturbations of $\Gamma$ inside $\mathcal{M}$ with the following property. For the typical element $b\in\mathcal B$, in the sense of Baire categories, every $m$-dimensional integral current in $\mathcal{M}$ which solves the corresponding Plateau problem has an open dense set of boundary points with density $1/2$. We deduce that the typical element $b\in\mathcal{B}$ admits a unique solution to the Plateau problem. Moreover we prove that, in a complete metric space of integral currents without boundary in $\mathbb{R}^{m+n}$, metrized by the flat norm, the typical boundary admits a unique solution to the Plateau problem.</p>https://cvgmt.sns.it/paper/5893/Scaling limits for fractional polyharmonic Gaussian fieldshttps://cvgmt.sns.it/paper/5892/N. De Nitti, F. Schweiger.<p>This work is concerned with fractional Gaussian fields, i.e. Gaussian fields whose covariance operator is given by the inverse fractional Laplacian $(-\Delta)^{-s}$ (where, in particular, we include the case $s >1$). We define a lattice discretization of these fields and show that their scaling limits -- with respect to the optimal Besov space topology -- are the original continuous fields. As a byproduct, in dimension $d<2s$, we prove the convergence in distribution of the maximum of the fields as well. A key tool in the proof is a sharp error estimate for the natural finite difference scheme for $(-\Delta)^s$ under minimal regularity assumptions, which is also of independent interest.</p>https://cvgmt.sns.it/paper/5892/Sobolev embeddings and distance functionshttps://cvgmt.sns.it/paper/5891/L. Brasco, F. Prinari, A. C. Zagati.<p>On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space $\mathcal{D}^{1,p}_0$ into $L^q$ and the summability properties of the distance function. We prove that in the superconformal case (i.e. when $p$ is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when $p$ is less than or equal to the dimension) we construct counterexamples to this equivalence.In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the $p-$Laplacian with sub-homogeneous right-hand side, as the exponent $p$ diverges to $\infty$. The case of first eigenfunctions of the $p-$Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.</p>https://cvgmt.sns.it/paper/5891/Sharp conditions for the validity of the Bourgain-Brezis-Mironescu formulahttps://cvgmt.sns.it/paper/5890/E. Davoli, G. Di Fratta, V. Pagliari.<p>Following the seminal paper by Bourgain, Brezis and Mironescu, we focus on the asymptotic behavior of some nonlocal functionals that, for each $u\in L^2(\mathbb{R}^N)$, are defined as the double integrals of weighted, squared difference quotients of $u$. Given a family of weights $\{\rho_\varepsilon\}$, $\varepsilon\in(0,1)$, we devise sufficient and necessary conditions on $\{\rho_\varepsilon\}$ for the associated nonlocal functionals to converge as $\varepsilon \to 0$ to a variant of the Dirichlet integral. Finally, some comparison between our result and the existing literature is provided.</p>https://cvgmt.sns.it/paper/5890/Sharp behavior of Dirichlet-Laplacian eigenvalues for a class of singularly perturbed problemshttps://cvgmt.sns.it/paper/5889/L. Abatangelo, R. Ognibene.<p>We deepen the study of Dirichlet eigenvalues in bounded domains where a thintube is attached to the boundary. As its section shrinks to a point, theproblem is spectrally stable and we quantitatively investigate the rate ofconvergence of the perturbed eigenvalues. We detect the proper quantity whichsharply measures the perturbation's magnitude. It is a sort of torsionalrigidity of the tube's section relative to the domain. This allows us tosharply describe the asymptotic behavior of the perturbed spectrum, even wheneigenvalues converge to a multiple one. The final asymptotics of eigenbranchesdepend on the local behavior near the junction of eigenfunctions chosen in aspecial way. The present techniques also apply when the perturbation of the Dirichleteigenvalue problem consists in prescribing homogeneous Neumann boundaryconditions on a small portion of the boundary of the domain.</p>https://cvgmt.sns.it/paper/5889/Lagrangian stability for a system of non-local continuity equations under Osgood conditionhttps://cvgmt.sns.it/paper/5888/M. Inversi, G. Stefani.<p>We extend known existence and uniqueness results of weak measure solutions for systems of non-local continuity equations beyond the usual Lipschitz regularity. Existence of weak measure solutions holds for uniformly continuous vector fields and convolution kernels, while uniqueness follows from a Lagrangian stability estimate under an additional Osgood condition.</p>https://cvgmt.sns.it/paper/5888/The double and triple bubble problem for stationary varifolds: the convex casehttps://cvgmt.sns.it/paper/5887/A. De Rosa, R. Tione.<p>We characterize the critical points of the double bubble problem in $\mathbb{R}^n$ and the triple bubble problem in $\mathbb{R}^3$, in the case the bubbles are convex.</p>https://cvgmt.sns.it/paper/5887/Mean-to-max ratio of the torsion function and honeycomb structureshttps://cvgmt.sns.it/paper/5885/L. Briani, D. Bucur.<p>In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.</p>https://cvgmt.sns.it/paper/5885/A Courant nodal domain theorem for linearized mean field type equationshttps://cvgmt.sns.it/paper/5883/D. Bartolucci, A. Jevnikar, R. Wu.<p>We are concerned with the analysis of a mean field type equation and its linearization, which is a nonlocal operator, for which we estimate the number of nodal domains for the radial eigenfunctions and the related uniqueness properties.</p>https://cvgmt.sns.it/paper/5883/An optimal lower bound in fractional spectral geometry for planar sets with topological constraintshttps://cvgmt.sns.it/paper/5882/F. Bianchi, L. Brasco.<p>We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical result proved independently by Croke, Osserman and Taylor, in the limit as $s$ goes to $1$. The limit as $s$ goes to $1/2$ is carefully analyzed, as well.</p>https://cvgmt.sns.it/paper/5882/Boundedness estimates for nonlinear nonlocal kinetic Kolmogorov-Fokker-Planck equationshttps://cvgmt.sns.it/paper/5881/F. Anceschi, M. Piccinini.<p> We investigate local properties of weak solutions to nonlocal and nonlinearkinetic equations whose prototype is given by $$ \partial<sub>t</sub> u +v\cdot\nabla<sub>x</sub> u +(-\Delta<sub>v)</sub><sup>s</sup><sub>p</sub> u = f(u). $$ We consider equations whose diffusion part is a (possibly degenerate)integro-differential operator of differentiability order $s \in (0,1)$ andsummability exponent $p\in (1,\infty)$. Amongst other results, we provide anexplicit local boundedness estimate by combining together a suitable Sobolevembedding theorem and a fractional Caccioppoli-type inequality with tail. Forthis, we introduce in the kinetic framework a new definition of nonlocal tailof a function and of its related tail spaces, also by establishing some usefulestimates for the tail of weak solutions. Armed with the aforementioned resultswe give a precise control of the long-range interactions arising from thenonlocal behaviour of the involved diffusion operator.</p>https://cvgmt.sns.it/paper/5881/Lower semicontinuity and relaxation for free discontinuity functionals with non-standard growthhttps://cvgmt.sns.it/paper/5880/S. Almi, D. Reggiani, F. Solombrino.<p>A lower semicontinuity result and a relaxation formula for free discontinuity functionals with non-standard growth in the bulk energy are provided. Our analysis is based on a non-trivial adaptation of the blow-up of Ambrosio (1994) and of the global method for relaxation of Bouchitté-Fonseca-Leoni-Mascarenhas (2002) to the setting of generalized special function of bounded variation with Orlicz growth. Key tools developed in this paper are an integral representation result and a Poincaré inequality under non-standard growth.</p>https://cvgmt.sns.it/paper/5880/Shape Optimization Problems and Regularity of the Free Boundarieshttps://cvgmt.sns.it/paper/5879/F. P. Maiale.https://cvgmt.sns.it/paper/5879/