cvgmt Papershttp://cvgmt.sns.it/papers/en-usSun, 15 Dec 2019 05:17:31 +0000Dissipative boundary conditions and entropic solutions in dynamical perfect plasticityhttp://cvgmt.sns.it/paper/4541/J. F. Babadjian, V. Crismale.<p> We prove the well--posedness of a dynamical perfect plasticity model undergeneral assumptions on the stress constraint set and on the referenceconfiguration. The problem is studied by combining both calculus of variationsand hyperbolic methods. The hyperbolic point of view enables one to derive aclass of dissipative boundary conditions, somehow intermediate betweenhomogeneous Dirichlet and Neumann ones. By using variational methods, we showthe existence and uniqueness of solutions. Then we establish the equivalencebetween the original variational solutions and generalizedentropic--dissipative ones, derived from a weak hyperbolic formulation forinitial--boundary value Friedrichs' systems with convex constraints.</p>http://cvgmt.sns.it/paper/4541/Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to Itô and Stratonovichhttp://cvgmt.sns.it/paper/4540/G. Alberti, E. Stepanov, D. Trevisan.<p>We show that geometric integrals of the type $\int_\Omega f\,d g^1\wedge \,d g^2$can be defined over a two-dimensional domain $\Omega$ when the functions$f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just Hölder continuouswith sufficiently large Hölder exponents and the boundary of $\partial \Omega$ 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 toalternating functions of two-dimensional oriented simplices, also proven in the paper.</p>http://cvgmt.sns.it/paper/4540/Brake orbits and heteroclinic connections for first order Mean Field Gameshttp://cvgmt.sns.it/paper/4539/A. Cesaroni, M. Cirant.<p>We consider first order variational MFG in the whole space, with aggregative interactions and density constraints, having stationary equilibria consisting of two disjoint compact sets of distributions with finite quadratic moments. Under general assumptions on the interaction potential, we provide a method for the construction of periodic in time solutions to the MFG, which oscillate between the two sets of static equilibria, for arbitrarily large periods. Moreover, as the period increases to infinity, we show that these periodic solutions converge, in a suitable sense, to heteroclinic connections. As a model example, we consider a MFG system where theinteractions are of (aggregative) Riesz-type.</p>http://cvgmt.sns.it/paper/4539/The $N$-link swimmer in three dimensions: controllability and optimality resultshttp://cvgmt.sns.it/paper/4538/R. Marchello, M. Morandotti, H. Shum, M. Zoppello.<p>The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory.The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimisation of the power expended are addressed and a qualitative description of the optimal strategies is provided.</p>http://cvgmt.sns.it/paper/4538/Asymptotic Mean Value Laplacian in Metric Measure Spaceshttp://cvgmt.sns.it/paper/4537/A. Minne, D. Tewodrose.<p>We use the mean value property in an asymptotic way to provide a notion of apointwise Laplacian, called AMV Laplacian, that we study in several contextsincluding the Heisenberg group and weighted Lebesgue measures. We focusespecially on a class of metric measure spaces including intersectingsubmanifolds of $\mathbb{R}^n$, a context in which our notion brings newinsights; the Kirchhoff law appears as a special case. In the general case, wealso prove a maximum and comparison principle, as well as a Green-type identityfor a related operator.</p>http://cvgmt.sns.it/paper/4537/The Bernstein problem in Heisenberg groupshttp://cvgmt.sns.it/paper/4536/F. Serra Cassano, M. Vedovato.<p>These notes aim to review results and open problems concerning the so-calledBernstein’s problem for entire area minimizing graphs of (topological) codi-mension 1, in the setting of sub-Riemannian Heisenberg groups.</p>http://cvgmt.sns.it/paper/4536/An existence theorem for Brakke flow with fixed boundary conditionshttp://cvgmt.sns.it/paper/4535/S. Stuvard, Y. Tonegawa.<p>Consider an arbitrary closed, countably $n$-rectifiable set in a strictly convex $(n+1)$-dimensional domain, and suppose that the set has finite $n$-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $t \uparrow \infty$, the flow sequentially converges to non-trivial solutions of Plateau's problem in the setting of stationary varifolds.</p>http://cvgmt.sns.it/paper/4535/Kinetic correlation functionals from the entropic regularisation of the strictly-correlated electrons problemhttp://cvgmt.sns.it/paper/4534/A. Gerolin, P. Gori-Giorgi, J. Grossi.<p> We investigate whether the entropic regularisation of thestrictly-correlated-electrons problem can be used to build approximations forthe kinetic correlation energy functional at large coupling strengths and, moregenerally, to gain new insight in the problem of describing and understandingstrong correlation within Density Functional Theory.</p>http://cvgmt.sns.it/paper/4534/How Well Do WGANs Estimate the Wasserstein Metric?http://cvgmt.sns.it/paper/4533/A. Gerolin, A. Mallasto, G. Montúfar.<p> Generative modelling is often cast as minimizing a similarity measure betweena data distribution and a model distribution. Recently, a popular choice forthe similarity measure has been the Wasserstein metric, which can be expressedin the Kantorovich duality formulation as the optimum difference of theexpected values of a potential function under the real data distribution andthe model hypothesis. In practice, the potential is approximated with a neuralnetwork and is called the discriminator. Duality constraints on the functionclass of the discriminator are enforced approximately, and the expectations areestimated from samples. This gives at least three sources of errors: theapproximated discriminator and constraints, the estimation of the expectationvalue, and the optimization required to find the optimal potential. In thiswork, we study how well the methods, that are used in generative adversarialnetworks to approximate the Wasserstein metric, perform. We consider, inparticular, the $c$-transform formulation, which eliminates the need to enforcethe constraints explicitly. We demonstrate that the $c$-transform allows for amore accurate estimation of the true Wasserstein metric from samples, butsurprisingly, does not perform the best in the generative setting.</p>http://cvgmt.sns.it/paper/4533/Numerical approximation of the Steiner problem in dimension 2 and 3http://cvgmt.sns.it/paper/4532/M. Bonnivard, E. Bretin, A. Lemenant.<p>The aim of this work is to present some numerical computations of solutions of the Steiner Prob- lem, based on the recent phase field approximations proposed in <a href='12'>12</a> and analyzed in <a href='5, 4'>5, 4</a>. Our strategy consists in improving the regularity of the associated phase field solution by use of higher- order derivatives in the Cahn-Hilliard functional as in <a href='6'>6</a>. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the nu- merical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.</p><p>preprint april 2018</p>http://cvgmt.sns.it/paper/4532/A rectifiability result for finite-perimeter sets in Carnot groupshttp://cvgmt.sns.it/paper/4531/S. Don, E. Le Donne, T. Moisala, D. Vittone.<p>In the setting of Carnot groups, we are concerned with the rectifiabilityproblem for subsets that have finite sub-Riemannian perimeter. We introduce anew notion of rectifiability that is possibly, weaker than the one introducedby Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $\Gamma$that, similarly to intrinsic Lipschitz graphs, have a cone property: thereexists an open dilation-invariant subset $C$ whose translations by elements in$\Gamma$ don't intersect $\Gamma$. However, a priori the cone $C$ may not haveany horizontal directions in its interior. In every Carnot group, we prove thatthe reduced boundary of every finite-perimeter subset can be covered bycountably many subsets that have such a cone property. The cones are related tothe semigroups generated by the horizontal half-spaces determined by the normaldirections. We further study the case when one can find horizontal directionsin the interior of the cones, in which case we infer that finite-perimetersubsets are countably rectifiable with respect to intrinsic Lipschitz graphs. Asufficient condition for this to hold is the existence of a horizontalone-parameter subgroup that is not an abnormal curve. As an application, weverify that this property holds in every filiform group, of either first orsecond type.</p>http://cvgmt.sns.it/paper/4531/Existence of varifold minimizers for the multiphase Canham-Helfrich functionalhttp://cvgmt.sns.it/paper/4530/K. Brazda, L. Lussardi, U. Stefanelli.<p>We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of single- and multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.</p>http://cvgmt.sns.it/paper/4530/Homogenisation of high-contrast brittle materialshttp://cvgmt.sns.it/paper/4529/C. I. Zeppieri.<p>This paper is an overview on some recent results concerning the variational analysis of static fracture in the so-called high-contrast brittle composite materials. The paper is divided into two main parts. The first part is devoted to establish a compactness result for a general class of free-discontinuity functionals with degenerate (or high-contrast) integrands. The second part is focussed on some specific examples which show that the degeneracy of the integrands may lead to non-standard limit effects, which are specific to this high-contrast setting.</p>http://cvgmt.sns.it/paper/4529/Local asymptotics for nonlocal convective Cahn-Hilliard equations with $W^{1,1}$ kernel and singular potentialhttp://cvgmt.sns.it/paper/4528/E. Davoli, L. Scarpa, L. Trussardi.<p>We prove existence of solutions and study the nonlocal-to-local asymptotics for nonlocal, convective, Cahn-Hilliard equations in the case of a $W<sup>{1,1}</sup> convolution kernel and under homogeneous Neumann conditions. Any type of potential, possibly also of double-obstacle or logarithmic type, is included. Additionally, we highlight variants and extensions to the setting of periodic boundary conditions and viscosity contributions, as well as connections with the general theory of evolutionary convergence of gradient flows.</p>http://cvgmt.sns.it/paper/4528/Recent results on non-convex functionals penalizing oblique oscillationshttp://cvgmt.sns.it/paper/4527/M. Goldman, B. Merlet.<p>The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some variational problem related to pattern formation and are somewhat reminiscent of those introduced by Bourgain, Brezis and Mironescu to characterize Sobolev functions. We obtain both qualitative and quantitative results for functions of finite energy. It turns out that this problem naturally leads to the study of various differential inclusions and has connections with branched transportation models.</p>http://cvgmt.sns.it/paper/4527/JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: $L^p$ and Sobolev boundshttp://cvgmt.sns.it/paper/4526/S. Di Marino, F. Santambrogio.<p>We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the $L^p$ and $L^\infty$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher order results, and in particulat to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called ``five gradients inequality'' allows to recover some inequalities that can be deduced from the Bakry-Emery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller-Segel chemiotaxis model.</p>http://cvgmt.sns.it/paper/4526/Optimal transport: entropic regularizations, geometry and diffusion PDEshttp://cvgmt.sns.it/paper/4525/N. De Ponti.<p>The thesis is divided in three main parts:</p><p>In the first part we introduce the class of optimal Entropy-Transport problems, a recent generalization of optimal transport problems where also creation and destruction of mass is taken into account. <br> We focus in particular on the metric properties of these problems, computed in terms of an entropy function $F$ and a cost function. Starting from the power-like entropy $F(s)=(s^p-p(s-1)-1)/(p(p-1))$ and a suitable cost depending on a metric $\mathsf{d}$ on a space $X$, our main result ensures that for every $p>1$ the related Entropy-Transport cost induces a distance on the space of finite measures over $X$. Inspired by previous work of Gromov and Sturm, we then use these Entropy-Transport metrics to construct new complete and separable distances on the family of metric measure spaces with finite mass.We also study in detail the pure entropic setting, that can be recovered as a particular case when the transport is forbidden. In this situation, corresponding to the classical theory of Csiszár $F$-divergences, we analyse some structural properties of these entropic functionals and we highlight the important role played by the class of Matusita's divergences. </p><p>The second part is devoted to the study of bounds involving Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian. <br> A celebrated lower bound of $\lambda_{1}$ in terms of $h$, $\lambda_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $\lambda_{1}$ in terms of $h$ was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below.<br> The goal of this part is two fold. First: by adapting the approach of Ledoux via heat semigroup techniques, we sharpen the inequalities obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-Émery weighted) Ricci curvature bounded below by $K\in \mathbb{R}$ (the inequality is sharp for $K>0$ as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called $ \mathsf{RCD}(K,\infty)$ spaces.</p><p>In the third part, given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance.We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg.<br> The strategy of the proof mainly relies on a quantitative $L^1$--$L^\infty$ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting.</p>http://cvgmt.sns.it/paper/4525/Shape Derivative for Obstacles in Crowd Motionhttp://cvgmt.sns.it/paper/4524/B. Fall, F. Santambrogio, D. Seck.<p>We consider different PDE modeling for crowd motion scenarios, or other sort of fluid flows, and we insert in the given domain $R$ an obstacle $O$. We then compute the shape derivatives of a cost functional, the average exit time, in order to be able to optimize the geometry of the obstacle $O$ and so to minimize the average exit time of particles in the domain $R$. This computation could be used to derive numerical simulations and understand whether the presence of an obstacle is or not profitable for the evacuation, or to optimize its shape and position, for instance when the presence of a structure (column,\dots) is already necessary in the building plan of a public space.</p>http://cvgmt.sns.it/paper/4524/Regularity for the planar optimal $p$-compliance problemhttp://cvgmt.sns.it/paper/4523/B. Bulanyi, A. Lemenant.<p>In this paper we prove a partial $C^{1,\alpha}$ regularity result in dimension $N=2$ for the optimal $p$-compliance problem, extending for $p\not = 2$ some of the results obtained by A. Chambolle, J. Lamboley, A. Lemenant, E. Stepanov (2017). Because of the lack of good monotonicity estimates for the $p$-energy when $p\not = 2$, we employ an alternative technique based on a compactness argument leading to a $p$-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and $C^{1,\alpha}$ at $\mathcal{H}^1$-a.e. point for every $p \in (1 ,+\infty)$.</p>http://cvgmt.sns.it/paper/4523/Continuity and canceling operators of order $n$ on $\mathbb{R}^n$http://cvgmt.sns.it/paper/4522/B. Raita, A. Skorobogatova.<p> We prove that for elliptic and canceling linear differential operators$\mathbb{B}$ of order $n$ on $\mathbb{R}^n$, continuity of a map $u$ can beinferred from the fact that $\mathbb{B} u$ is a measure. We also prove strictcontinuity of the embedding of the space$\mathrm{BV}^{\mathbb{B}}(\mathbb{R}^n)$ of functions of bounded$\mathbb{B}$-variation into the space of continuous functions vanishing atinfinity.</p>http://cvgmt.sns.it/paper/4522/