cvgmt Papershttps://cvgmt.sns.it/papers/en-usThu, 17 Jun 2021 05:51:26 +0000First order expansion in the semiclassical limit of the Levy-Lieb functionalhttps://cvgmt.sns.it/paper/5171/M. Colombo, S. Di Marino, F. Stra.<p> We prove the conjectured first order expansion of the Levy-Lieb functional inthe semiclassical limit, arising from Density Functional Theory (DFT). This isaccomplished by interpreting the problem as the singular perturbation of anOptimal Transport problem via a Dirichlet penalization.</p>https://cvgmt.sns.it/paper/5171/On the convex components of a set in $\mathbb{R}^n$https://cvgmt.sns.it/paper/5170/F. Giannetti, G. Stefani.<p>We prove a lower bound on the number of the convex components of a compact set with non-empty interior in $\mathbb{R}^n$ for all $n\ge2$. Our result generalizes and improves the inequalities previously obtained in <a href='https://doi.org/10.1142/S0219199718500360'>M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, "Convex components", in Communications in Contemporary Mathematics, Vol. 21, No. 06, 1850036 (2019)</a> and in M. La Civita and F. Leonetti, "Convex components of a set and the measure of its boundary", Atti. Sem. Mat. Fis. Univ. Modena Reggio Emilia 56 (2008–2009) 71–78.</p>https://cvgmt.sns.it/paper/5170/A Bourgain-Brezis-Mironescu representation for functions with bounded deformationhttps://cvgmt.sns.it/paper/5169/A. Arroyo-Rabasa, P. Bonicatto.<p> We establish a difference quotient integral representation for symmetricgradient semi-norms in $W^{1,p}(\Omega)$, $LD(\Omega)$ and $BD(\Omega)$. Therepresentation, which is inspired by the formulas for the $W^{1,p}(\Omega)$semi-norm introduced by Bourgain, Brezis and Mironescu and for the totalvariation semi-norm of $BV(\Omega)$ by Davila, provides a criterion for the$L^p$ and total-variation boundedness of symmetric gradients that does notrequire the understanding of distributional derivatives.</p>https://cvgmt.sns.it/paper/5169/Crystallinity of the homogenized energy density of periodic lattice systemshttps://cvgmt.sns.it/paper/5168/A. Chambolle, L. Kreutz.<p>We study the homogenized energy densities of periodic ferromagnetic Ising systems. We prove that, for finite range interactions, the homogenized energy density, identifying the effective limit, is crystalline, i.e. its Wulff crystal is a polytope, for which we can (exponentially) bound the number of vertices. This is achieved by deriving a dual representation of the energy density through a finite cell formula. This formula also allows easy numerical computations: we show a few experiments where we compute periodic patterns which minimize the anisotropy of the surface tension.</p>https://cvgmt.sns.it/paper/5168/Local density of Solutions to Fractional Equationshttps://cvgmt.sns.it/paper/5167/A. Carbotti, S. Dipierro, E. Valdinoci.<p>This book presents in a detailed and self-contained way a new and important density result in the analysis of fractional partial differential equations, while also covering several fundamental facts about space- and time-fractional equations.</p><p>De Gruyter studies in mathematics.</p>https://cvgmt.sns.it/paper/5167/Non-asymptotic convergence bounds for Wasserstein approximation using point cloudshttps://cvgmt.sns.it/paper/5166/Q. Mérigot, F. Santambrogio, C. Sarrazin.<p>Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm - in which Voronoi cells are replaced by Power cells - leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, as well as the existence of spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a modified Poliak-Lojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.</p>https://cvgmt.sns.it/paper/5166/On the Steiner property for planar minimizing clusters. The anisotropic case.https://cvgmt.sns.it/paper/5165/V. Franceschi, A. Pratelli, G. Stefani.<p>In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using two densities. In particular, the perimeter density may also depend on the direction of the normal vector. The classical ''Steiner property'' for the Euclidean case (which corresponds to both densities being equal to $1$) says that minimal clusters are made by finitely many ${\rm C}^{1,\gamma}$ arcs, meeting in finitely many ''triple points''. We can show that this property holds under very weak assumptions on the densities. In the parallel <a href='https://cvgmt.sns.it/paper/5164/'>paper</a> we consider the isotropic case, i.e., when the perimeter density does not depend on the direction, which makes most of the construction much simpler. In particular, in the present case the three arcs at triple points do not necessarily meet with three angles of $120^\circ$, which is instead what happens in the isotropic case.</p>https://cvgmt.sns.it/paper/5165/On the Steiner property for planar minimizing clusters. The isotropic case.https://cvgmt.sns.it/paper/5164/V. Franceschi, A. Pratelli, G. Stefani.<p>We consider the isoperimetric problem for clusters in the plane with a double density, that is, perimeter and volume depend on two weights. In this paper we consider the isotropic case, in the parallel <a href='https://cvgmt.sns.it/paper/5165/'>paper</a> the anisotropic case is studied. Here we prove that, in a wide generality, minimal clusters enjoy the ''Steiner property'', which means that the boundaries are made by ${\rm C}^{1,\gamma}$ regular arcs, meeting in finitely many triple points with the $120^\circ$ property.</p>https://cvgmt.sns.it/paper/5164/Interaction between oscillations and singular perturbations in a one-dimensional phase-field modelhttps://cvgmt.sns.it/paper/5163/A. Bach, T. Esposito, R. Marziani, C. I. Zeppieri.<p>In this note we study the relative impact of fine-scale heterogeneities and singular perturbations in aone-dimensional phase-field model of Ambrosio-Tortorelli type. We show that the limit functional isalways of Mumford-Shah type, with a surface term depending on the mutual converging rate of theoscillation and the perturbation parameter.</p>https://cvgmt.sns.it/paper/5163/Mean-field selective optimal control via transient leadershiphttps://cvgmt.sns.it/paper/5162/G. Albi, S. Almi, M. Morandotti, F. Solombrino.<p>A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process.This way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via $\Gamma$-convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion.Specific applications in the context of opinion dynamics are discussed with some numerical experiments.</p>https://cvgmt.sns.it/paper/5162/Two phase models for elastic membranes with soft inclusionshttps://cvgmt.sns.it/paper/5161/M. Santilli, B. Schmidt.<p>We derive an effective membrane theory in the thin film limit within atwo phase material model for a specimen consisting of an elastic matrix andsoft inclusions. The soft inclusions may lead to the formation of cracks withinthe elastic matrix and the corresponding limiting models are described byGriffith type fracture energy functionals. We also provide simplified proofs ofrelaxation results for bulk materials</p>https://cvgmt.sns.it/paper/5161/Functional inequalities for some generalised Mehler semigroupshttps://cvgmt.sns.it/paper/5160/L. Angiuli, S. Ferrari, D. Pallara.https://cvgmt.sns.it/paper/5160/Asymptotics of the $s$-fractional Gaussian perimeter as $s\to 0^+$https://cvgmt.sns.it/paper/5158/A. Carbotti, S. Cito, D. A. La Manna, D. Pallara.<p>We study the asymptotic behaviour of the renormalised $s$-fractional Gaussian perimeter of a set $E$ inside a domain $\Omega$ as $s\to 0^+$. Contrary to the Euclidean case, as the Gaussianmeasure is finite, the shape of the set at infinity does not matter, but, surprisingly, the limit set function is never additive.</p>https://cvgmt.sns.it/paper/5158/Large mass rigidity for a liquid drop model in 2D with kernels of finite momentshttps://cvgmt.sns.it/paper/5157/B. Merlet, M. Pegon.<p> Motivated by Gamow's liquid drop model in the large mass regime, we consideran isoperimetric problem in which the standard perimeter $P(E)$ is replaced by$P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ anonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes.We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, connected minimizers arenecessarily convex, provided that $\varepsilon$ is small enough. In turn, thisimplies that minimizers have nearly circular boundaries, that is, theirboundary is a small Lipschitz perturbation of the circle. Then, using aFuglede-type argument, we prove that (in arbitrary dimension $n\geq 2$) theunit ball in $\mathbb{R}^n$ is the unique unit-volume minimizer of the problemamong centered nearly spherical sets. As a consequence, up to translations, theunit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquiddrop model for the atomic nucleus introduced by Gamow, where the nonlocalrepulsive potential is given by a radial, sufficiently integrable kernel. Inthat formulation, our main result states that if the first moment of the kernelis smaller than an explicit threshold, there exists a critical mass $m_0$ suchthat for any $m>m_0$, the disk is the unique minimizer of area $m$ up totranslations. This is in sharp contrast with the usual case of Riesz kernels,where the problem does not admit minimizers above a critical mass.</p>https://cvgmt.sns.it/paper/5157/Infinitely many solutions for Schrödinger-Newton equationshttps://cvgmt.sns.it/paper/5156/Y. Hu, A. Jevnikar, W. Xie.<p>We prove the existence of infinitely many non-radial positive solutions for the Schrödinger-Newton system</p><p>$ \begin{cases} \Delta u- V(<br>x<br>)u + \Psi u=0,\quad &x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12 u^2=0, &x\in\mathbb{R}^3,\end{cases}$</p><p>provided that $V(r)$ has the following behavior at infinity:</p><p>$ V(r)=V_0+\frac{a}{r^m}+O\left(\frac{1}{r^{m+\theta}}\right) \quad\mbox{ as } r\rightarrow\infty,$</p><p>where $\frac12\le m<1$ and $a, V_0, \theta$ are some positive constants. In particular, for any $s$ large we use a reduction method to construct $s-$bump solutions lying on a circle of radius $r\sim (s\log s)^{\frac{1}{1-m}}$.</p>https://cvgmt.sns.it/paper/5156/Generic properties of free boundary problems in plasma physicshttps://cvgmt.sns.it/paper/5155/D. Bartolucci, Y. Hu, A. Jevnikar, W. Yang.<p>We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape of the branch of solutions resembles the monotone one of the model case of the two-dimensional disk, or it is a continuous simple curve without bifurcation points which ends up at a point where the boundary density vanishes. On the other hand, we deduce a general criterion ensuring the existence of a free boundary in the interior of the domain. Application to a classic nonlinear eigenvalue problem is also discussed.</p>https://cvgmt.sns.it/paper/5155/Transport and control problems with boundary costs: regularity and summability of optimal and equilibrium densitieshttps://cvgmt.sns.it/paper/5154/S. Dweik.https://cvgmt.sns.it/paper/5154/Quasistatic limit of a dynamic viscoelastic model with memoryhttps://cvgmt.sns.it/paper/5153/G. Dal Maso, F. Sapio.<p>We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.</p>https://cvgmt.sns.it/paper/5153/Maximal distance minimizers for a rectanglehttps://cvgmt.sns.it/paper/5152/D. D. Cherkashin, A. S. Gordeev, G. A. Strukov, Y. Teplitskaya.<p>\emph{A maximal distance minimizer} for a given compact set $M \subset\mathbb{R}^2$ and some given $r > 0$ is a set having the minimal length(one-dimensional Hausdorff measure) over the class of closed connected sets$\Sigma \subset \mathbb{R}^2$ satisfying the inequality \[ \max_{y\in M} dist(y, \Sigma) \leq r. \] This paper deals with the set of maximal distanceminimizers for a rectangle $M$ and small enough $r$.</p>https://cvgmt.sns.it/paper/5152/On minimizers of the maximal distance functional for a planar convex closed smooth curvehttps://cvgmt.sns.it/paper/5151/D. D. Cherkashin, A. S. Gordeev, G. A. Strukov, Y. Teplitskaya.<p>Fix a compact $M \subset \mathbb{R}^2$ and $r>0$. A minimizer of the maximaldistance functional is a connected set $\Sigma$ of the minimal length, suchthat \[ max_{y \in M} dist(y,\Sigma) \leq r. \] The problem of finding maximaldistance minimizers is connected to the Steiner tree problem. In this paper we consider the case of a convex closed curve $M$, with theminimal radius of curvature greater than $r$ (it implies that $M$ is smooth).The first part is devoted to statements on structure of $\Sigma$: we show thatthe closure of an arbitrary connected component of $B_r(M) \cap \Sigma$ is alocal Steiner tree which connects no more than five vertices. In the second part we "derive in the picture". Assume that the left and rightneighborhoods of $y \in M$ are contained in $r$-neighborhoods of differentpoints $x_1$, $x_2 \in \Sigma$. We write conditions on the behavior of $\Sigma$in the neighborhoods of $x_1$ and $x_2$ under the assumption by moving $y$along $M$.</p>https://cvgmt.sns.it/paper/5151/