cvgmt Papershttps://cvgmt.sns.it/papers/en-usTue, 18 Jan 2022 22:33:57 +0000Infinite multidimensional scaling for metric measure spaceshttps://cvgmt.sns.it/paper/5424/A. Kroshnin, E. Stepanov, D. Trevisan.<p>For a given metric measure space $(X,d,\mu)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithmwith this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure $\mu$. This limit can beviewed as ``infinite MDS'' embedding of the original space, now not anymore into a finite-dimensional spacebut rather into an infinite-dimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show thatin many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).</p>https://cvgmt.sns.it/paper/5424/Sharp isoperimetric comparison and asymptotic isoperimetry on non collapsed spaces with lower Ricci boundshttps://cvgmt.sns.it/paper/5423/G. Antonelli, E. Pasqualetto, M. Pozzetta, D. Semola.<p>This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called $N$-dimensional $\mathrm{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when $K=0$ under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.\\ The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination withan asymptotic mass decomposition result of perimeter-minimizing sequences.\\Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies.</p>https://cvgmt.sns.it/paper/5423/Distortion Coefficients of the $\alpha$-Grushin Planehttps://cvgmt.sns.it/paper/5420/S. Borza.<p>We compute the distortion coefficients of the $\alpha$-Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a measure contraction property condition for the generalised Grushin planes is suggested.</p>https://cvgmt.sns.it/paper/5420/Long Time Behaviour of the Discrete Volume Preserving Mean Curvature Flow in the Flat Torushttps://cvgmt.sns.it/paper/5418/D. De Gennaro, A. Kubin.<p> We show that the discrete approximate volume preserving mean curvature flowin the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set$E$ of the perimeter converges in the long time to a translate of $E$exponentially fast. As an intermediate result we establish a new quantitativeestimate of Alexandrov type for periodic strictly stable constant meancurvature hypersurfaces. Finally, in the two dimensional case a completecharacterization of the long time behaviour of the discrete flow with arbitraryinitial sets of finite perimeter is provided.</p>https://cvgmt.sns.it/paper/5418/A time-dependent switching mean-field game on networks motivated by optimal visiting problemshttps://cvgmt.sns.it/paper/5417/F. Bagagiolo, L. Marzufero.<p>Motivated by an optimal visiting problem, we study a switching mean-fieldgame on a network, where both a decisional and a switching time-variable is at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from $0$ to $1$ of one component of the string $p =(p_1,\ldots, p_n)$ which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by $i=1,\ldots,n$. The goal is to reach the final string $(1, \ldots, 1)$ in the final time $T$, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated$\varepsilon$-mean-field equilibrium and then address the passage to the limitwhen $\varepsilon$ goes to 0.</p>https://cvgmt.sns.it/paper/5417/Rigidity of the ball for an isoperimetric problem with strong capacitary repulsionhttps://cvgmt.sns.it/paper/5416/M. Goldman, M. Novaga, B. Ruffini.<p>We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range)interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity.Combining this with the stability of the ball under small $C^{1,\gamma}$ perturbations, this ultimately leads to the minimality of the ball for small charges.We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.</p>https://cvgmt.sns.it/paper/5416/A new space of generalised functions with bounded variation motivated by fracture mechanicshttps://cvgmt.sns.it/paper/5415/G. Dal Maso, R. Toader.<p>We introduce a new space of generalised functions with bound\-ed variation to prove the existence of a solution to a minimum problem that arises in the variational approach to fracture mechanics in elastoplastic materials. We study the fine properties of the functions belonging to this space and prove a compactness result. In order to use the Direct Method of the Calculus of Variations we prove a lower semicontinuity result for the functional occurring in this minimum problem. Moreover, we adapt a nontrivial argument introduced by Friedrich to show that every minimizing sequence can be modified to obtain a new minimizing sequence that satisfies the hypotheses of our compactness result.</p>https://cvgmt.sns.it/paper/5415/The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci boundshttps://cvgmt.sns.it/paper/5414/G. Antonelli, S. Nardulli, M. Pozzetta.<p>We establish a structure theorem for minimizing sequences for the isoperimetric problem on noncompact $\mathsf{RCD}(K,N)$ spaces $(X,\mathsf{d},\mathcal{H}^N)$. Under the sole (necessary) assumption that the measure of unit balls is uniformly bounded away from zero, we prove that the limit of such a sequence is identified by a finite collection of isoperimetric regions possibly contained in pointed Gromov--Hausdorff limits of the ambient space $X$ along diverging sequences of points. The number of such regions is bounded linearly in terms of the measure of the minimizing sequence.</p><p>The result follows from a new generalized compactness theorem, which identifies the limit of a sequence of sets $E_i\subset X_i$ with uniformly bounded measure and perimeter, where $(X_i,\mathsf{d}_i,\mathcal{H}^N)$ is an arbitrary sequence of $\mathsf{RCD}(K,N)$ spaces.</p><p>An abstract criterion for a minimizing sequence to converge without losing mass at infinity to an isoperimetric set is also discussed. The latter criterion is new also for smooth Riemannian spaces.</p>https://cvgmt.sns.it/paper/5414/Uniqueness of extremals for some sharp Poincaré-Sobolev constantshttps://cvgmt.sns.it/paper/5413/L. Brasco, E. Lindgren.<p>We study the sharp constant for the embedding of $W^{1,p}_0(\Omega)$ into $L^q(\Omega)$, in the case $2<p<q$.We prove that for smooth connected sets, when $q>p$ and $q$ is sufficiently close to $p$, extremals functions attaining the sharp constant are unique, up to a multiplicative constant. This in turn gives the uniqueness of solutions with minimal energy to the Lane-Emden equation, with super-homogeneous right-hand side.The result is achieved by suitably adapting a linearization argument due to C.-S. Lin. We rely on some fine estimates for solutions of $p-$Laplace--type equations by L. Damascelli and B. Sciunzi.</p>https://cvgmt.sns.it/paper/5413/Semilinear Li & Yau inequalitieshttps://cvgmt.sns.it/paper/5412/D. Castorina, G. Catino, C. Mantegazza.<p>We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.</p>https://cvgmt.sns.it/paper/5412/Origami and fractal solutions of differential systemshttps://cvgmt.sns.it/paper/5411/P. Marcellini, E. Paolini.<p>To appear in Imagine Math 8 by Springer Nature on April 7, 2022</p>https://cvgmt.sns.it/paper/5411/Geometric Flows on Planar Latticeshttps://cvgmt.sns.it/paper/5410/A. Braides, M. Solci.https://cvgmt.sns.it/paper/5410/Transport type metrics on the space of probability measures involving singular base measureshttps://cvgmt.sns.it/paper/5409/L. Nenna, B. Pass.<p>We develop the theory of a metric, which we call the $\nu$-based Wasserstein metric and denote by $W_\nu$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $\nu$ and is relevant in particular for the case when $\nu$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $\nu$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $\nu$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities when measures are disintegrated with respect to optimal transport to $\nu$, and through limits of certain multi-marginal optimal transport problems. We also introduce a class of metrics which are dual in a certain sense to $W_\nu$, defined relative to a fixed based measure $\mu$, on the set of measures which are absolutely continuous with respect to a second fixed based measure $\sigma$. As we vary the base measure $\nu$, the $\nu$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance (obtained when $\nu$ is a Dirac mass) and a metric associated with the uniquely defined generalized geodesics obtained when $\nu$ is sufficiently regular (eg, absolutely continuous with respect to Lebesgue). When $\nu$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $\nu$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $\mu$ such that optimal transport between $\mu$ and $\nu$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}. We also present two applications of the ideas introduced here. First, our dual metric (in fact, a slight variant of it) is used to prove convergence of an iterative scheme to solve a variational problem arising in game theory. We also use the multi-marginal formulation to characterize solutions to the multi-marginal problem by an ordinary differential equation, yielding a new numerical method for it.</p>https://cvgmt.sns.it/paper/5409/Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domainshttps://cvgmt.sns.it/paper/5408/L. Dello Schiavo, L. Portinale, F. Sau.<p>We consider the open symmetric exclusion (SEP) and inclusion (SIP) processeson a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. Forthe random walks on $\Omega$ dual to SEP$/$SIP we establish: afunctional-CLT-type convergence to the Brownian motion on $\Omega$ with eitherNeumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality)boundary conditions; the discrete-to-continuum convergence of the correspondingharmonic profiles. As a consequence, we rigorously derive the hydrodynamic andhydrostatic limits for SEP$/$SIP on $\Omega$, and analyze their stationarynon-equilibrium fluctuations.</p>https://cvgmt.sns.it/paper/5408/Characterization of rectifiability via Lusin type approximationhttps://cvgmt.sns.it/paper/5407/A. Marchese, A. Merlo.<p>We prove that a Radon measure $\mu$ on $\mathbb{R}^n$ can be written as $\mu=\sum_{i=0}^n\mu_i$, where each of the $\mu_i$ 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 $\mu(\{x\in\mathbb{R}^n:g(x)\neq f(x)\})<\varepsilon$.</p>https://cvgmt.sns.it/paper/5407/The relaxed energy of fractional Sobolev maps with values into the circlehttps://cvgmt.sns.it/paper/5406/D. Mucci.<p>We deal with the weak sequential density of smooth maps in the fractional Sobolev classes of $W^{s,p}$ maps in high dimension domains and with values into the circle.When $s$ is lower than one, using interpolation theory we introduce a natural energy in terms of optimal extensions on suitable weighted Sobolev spaces.The relaxation problem is then discussed in terms of Cartesian currents.When $sp=1$, the energy gap in the relaxed functional is always finite and is given by the minimal connection of the singularities times an energy weight, obtained through a minimum problem for one dimensional $W^{1/p,p}$ maps with degree one.When $sp>1$, instead, concentration on codimension one sets needs unbounded energy.We finally treat the case where $s$ is greater than one, obtaining an almost complete picture.</p>https://cvgmt.sns.it/paper/5406/Non-uniqueness of Leray solutions of the forced Navier-Stokes equationshttps://cvgmt.sns.it/paper/5405/D. Albritton, E. Bruè, M. Colombo.<p>In the seminal work <a href='39'>39</a>, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in <a href='43,44'>43,44</a>. The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák in <a href='32,33'>32,33</a>. Our solutions live precisely on the borderline of the known well-posedness theory.</p>https://cvgmt.sns.it/paper/5405/Instability and nonuniqueness for the 2d Euler equations in vorticity form, after M. Vishikhttps://cvgmt.sns.it/paper/5404/D. Albritton, E. Bruè, M. Colombo, C. De Lellis, V. Giri, M. Janisch, H. Kwon.<p>In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space,with initial vorticity in $L^p$ and body force in $L^1_tL^p_x$, $p<\infty$. His theorem demonstrates, in particular, the sharpness of the Yudovich class. An important intermediate step is the rigorous construction of an unstable vortex, which is of independent physical and mathematical interest. We follow the strategy of Vishik but allow ourselves certain deviations in the proof and substantial deviations in our presentation, which emphasizes the underlying dynamical point of view.</p>https://cvgmt.sns.it/paper/5404/Existence of minimizers for a generalized liquid drop model with fractional perimeterhttps://cvgmt.sns.it/paper/5403/M. Novaga, F. Onoue.<p>We consider the minimization problem of the functional given by the sum of the fractional perimeter and a general Riesz potential, which is one generalization of Gamow's liquid drop model. We first show the existence of minimizers for any volumes if the kernel of the Riesz potential decays faster than that of the fractional perimeter. Secondly, we show the existence of generalized minimizers for any volumes if the kernel of the Riesz potential just vanishes at infinity. Finally, we study the asymptotic behavior of minimizers when the volume goes to infinity and we prove that a sequence of minimizers converges to the Euclidean ball up to translations if the kernel of the Riesz potential decays sufficiently fast.</p>https://cvgmt.sns.it/paper/5403/Stability of ellipsoids as the energy minimisers of perturbed Coulomb energieshttps://cvgmt.sns.it/paper/5401/J. Mateu, M. G. Mora, L. Rondi, L. Scardia, J. Verdera.<p>In this paper we characterise the minimiser for a class of nonlocal perturbations of the Coulomb energy. We show that the minimiser is the normalised characteristic function of an ellipsoid, under the assumption that the perturbation kernel has the same homogeneity as the Coulomb potential, is even, smooth off the origin and sufficiently small. This result can be seen as the stability of ellipsoids as energy minimisers, since the minimiser of the Coulomb energy is the normalised characteristic function of a ball.</p>https://cvgmt.sns.it/paper/5401/