CVGMT Papershttp://cvgmt.sns.it/papers/en-usFri, 25 May 2018 03:27:12 +0000Brezis-Gallouet-Wainger type inequality with critical fractional Sobolev space and BMOhttp://cvgmt.sns.it/paper/3881/N. A. Dao, Q. H. Nguyen.<p>In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of $\mathcal{\dot{C}}^\eta$, for $\eta\in(0,1)$.</p>http://cvgmt.sns.it/paper/3881/$\Gamma$-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeterhttp://cvgmt.sns.it/paper/3880/L. De Luca, M. Novaga, M. Ponsiglione.<p>We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e., the so called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. </p><p>Whenever the limit configuration is a single crystal, i.e., it has constant orientation, we show that the $\Gamma$-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal.</p>http://cvgmt.sns.it/paper/3880/$BMO$-type norms and anisotropic surface measureshttp://cvgmt.sns.it/paper/3879/G. E. Comi.<p>The purpose of this note is to present an anisotropic variant of the $BMO$-type norm introduced by Bourgain, Brezis and Mironescu, and to show its relation with a surface measure, which is indeed a multiple of the perimeter in the isotropic case. This is done in the spirit of the new characterization of the perimeter of a measurable set in $\mathbb{R}^{n}$ recently studied by Ambrosio, Bourgain, Brezis and Figalli.</p>http://cvgmt.sns.it/paper/3879/On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$http://cvgmt.sns.it/paper/3878/M. Carioni, A. Pluda.<p>Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of Steiner problem in several variants. Our goal is to compare the different notions of calibrationsfor the Steiner Problem and for planar minimal partitions.The paper is then complemented with remarks on the convexification of the problem, onnon—existence of calibrations and on calibrations in families.</p>http://cvgmt.sns.it/paper/3878/Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagramhttp://cvgmt.sns.it/paper/3877/A. Gloria, M. Ruf.<p>Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, itis known that strong ellipticity is not necessarily conserved through periodichomogenization in linear elasticity. This phenomenon is related to microscopicbuckling of composite materials. Consider a mixture of two isotropic phaseswhich leads to loss of strong ellipticity when arranged in a laminate manner,as considered by Gutiérrez and by Briane and Francfort. In this contributionwe prove that the laminate structure is essentially the only microstructurewhich leads to such a loss of strong ellipticity. We perform a more generalanalysis in the stationary, ergodic setting.</p>http://cvgmt.sns.it/paper/3877/Nonstationary Navier–Stokes equations with singular time-dependent external forceshttp://cvgmt.sns.it/paper/3876/N. A. Dao, Q. H. Nguyen.<p>We establish a sufficient condition for the existence of solutions to the incompressible Navier–Stokes equations, with singular time-dependent external forces defined in terms of capacity $Cap_{\mathcal{H}_{1,2}}$.</p>http://cvgmt.sns.it/paper/3876/Benamou-Brenier and duality formulas for the entropic cost on $RCD^*(K,N)$ spaceshttp://cvgmt.sns.it/paper/3875/N. Gigli, L. Tamanini.<p>In this paper we prove that, within the framework of $RCD^*(K,N)$ spaces with $N < \infty$, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:</p><p>- a threefold dynamical variational representation, in the spirit of the Benamou-Brenier formula for the Wasserstein distance;</p><p>- a Hamilton-Jacobi-Bellman dual representation, in line with Bobkov-Gentil-Ledoux and Otto-Villani results on the duality between Hamilton-Jacobi and continuity equation for optimal transport;</p><p>- a Kantorovich-type duality formula, where the Hopf-Lax semigroup is replaced by a suitable `entropic' counterpart.</p><p>We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem (still missing even in the Riemannian setting) as well as a perfect parallelism with the analogous formulas for the Wasserstein distance.</p>http://cvgmt.sns.it/paper/3875/On the Continuity of Center-Outward Distribution and Quantile Functionshttp://cvgmt.sns.it/paper/3874/A. Figalli.<p>To generalize the notion of distribution function to dimension $d\geq 2$, in some recent papers it has been proposed a concept of center-outward distribution function based on optimal transportation ideas,and the inferential properties of thecorresponding center-outward quantile function have been studied. A crucial tool needed to derive the desired inferential properties is the continuity and invertibility for the center-outward quantile function outside the origin, as this ensures the existence of closed and nested quantile contours. The aim of this paper is to prove such a continuity and invertibility result.</p>http://cvgmt.sns.it/paper/3874/On the Monge-Amp\`ere equationhttp://cvgmt.sns.it/paper/3873/A. Figalli.<p>This text contains the material discussed by the author in the Bourbakiseminar of June 2018, on the recent developments in the theory of theMonge-Amp\`ere equation.</p>http://cvgmt.sns.it/paper/3873/Derivation of Linearised Polycrystals from a 2D system of edge dislocationshttp://cvgmt.sns.it/paper/3872/S. Fanzon, M. Palombaro, M. Ponsiglione.<p>In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimisation. For this purpose, we introduce a variational model for two-dimensional systems of edge dislocations, within the so-called core radius approach, and we derive the $\Gamma$-limit of the elastic energy functional as the lattice space tends to zero. </p><p> In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimisers under suitable boundary conditions are piece-wise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.</p>http://cvgmt.sns.it/paper/3872/Spatially Inhomogeneous Evolutionary Gameshttp://cvgmt.sns.it/paper/3871/L. Ambrosio, M. Fornasier, M. Morandotti, G. Savaré.<p>We introduce and study a mean-field model for a system of spatially distributed players interacting through an evolutionary game driven by a replicator dynamics. Strategies evolve by a replicator dynamics influenced by the position and the interaction between different players and return a feedback on the velocity field guiding their motion.</p><p>One of the main novelties of our approach concerns the description of the whole system, which can be represented by an evolving probability measure $\Sigma$ on an infinite dimensional state space (pairs $(x,\sigma)$ of position and distribution of strategies). We provide a Lagrangian and a Eulerian description of the evolution, and we prove their equivalence, together with existence, uniqueness, and stability of the solution.As a byproduct of the stability result, we also obtain convergence of the finite agents model to our mean-field formulation, when the number $N$ of the players goes to infinity, and the initial discrete distribution of positions and strategies converge. To this aim we develop some basic functional analytic tools to deal with interaction dynamics and continuity equations in Banach spaces, that could be of independent interest.</p>http://cvgmt.sns.it/paper/3871/Symmetry of minimizers of a Gaussian isoperimetric problemhttp://cvgmt.sns.it/paper/3870/M. Barchiesi, V. Julin.<p>We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. This second term has a repulsive effect, and it is in competition with the perimeter.Because of that, in general the solution is not the half-space.We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the repulsive term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problemamong symmetric sets when the volume is close to one.</p>http://cvgmt.sns.it/paper/3870/Onsager's conjecture on the energy convervation for solutions of Euler's equation in bounded domainshttp://cvgmt.sns.it/paper/3869/P. T. NGUYEN, Q. H. Nguyen.<p>We prove the Onsager's conjecture on the energy conservation for weak solutions of the Euler equations in a bounded domain.</p>http://cvgmt.sns.it/paper/3869/A minimization approach to the wave equation on time-dependent domainshttp://cvgmt.sns.it/paper/3868/G. Dal Maso, L. De Luca.<p>We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.</p>http://cvgmt.sns.it/paper/3868/Generalised Sadowsky theories for ribbons from three-dimensional nonlinear elasticityhttp://cvgmt.sns.it/paper/3867/L. Freddi, P. Hornung, M. G. Mora, R. Paroni.<p>In the 1930s Sadowsky derived an asymptotic theory for narrow ribbons. Generalised Sadowsky theories were recently obtained by the authors as a Gamma-limit from two-dimensional plate models. In the present article, we provide a rigorous derivation of these generalised Sadowsky theories starting from nonlinear three-dimensional elasticity. On a technical level, this involves capturing a contribution to the asymptotic energy functional generated by a nonlinear constraint which is satisfied only approximately. It also involves the construction of fine-scale "corrugations" capable of reaching a bending energy regime which is strictly below that of the original Sadowsky functional.</p>http://cvgmt.sns.it/paper/3867/The adiabatic strictly-correlated-electrons functional: kernel and exact propertieshttp://cvgmt.sns.it/paper/3866/S. Di Marino, A. Gerolin, P. Gori-Giorgi, G. Lani, R. van Leeuwen.<p> We investigate a number of formal properties of the adiabaticstrictly-correlated electrons (SCE) functional, relevant for time-dependentpotentials and for kernels in linear response time-dependent density functionaltheory. Among the former, we focus on the compliance to constraints of exactmany-body theories, such as the generalised translational invariance and thezero-force theorem. Within the latter, we derive an analytical expression forthe adiabatic SCE Hartree exchange-correlation kernel in one dimensionalsystems, and we compute it numerically for a variety of model densities. Weanalyse the non-local features of this kernel, particularly the ones that arerelevant in tackling problems where kernels derived from local or semi-localfunctionals are known to fail.</p>http://cvgmt.sns.it/paper/3866/Non-existence of optimal transport maps for the multi-marginal repulsive harmonic costhttp://cvgmt.sns.it/paper/3865/A. Gerolin, A. Kausamo, T. Rajala.<p> We give an example of an absolutely continuous measure $\mu$ on $\mathbbR^d$, for any $d \ge 1$, such that no minimizer of the $3$-marginal harmonicrepulsive cost with all marginals equal to $\mu$ is supported on a graph overthe first variable.</p>http://cvgmt.sns.it/paper/3865/Duality theory for multi-marginal optimal transport with repulsive costs in metric spaceshttp://cvgmt.sns.it/paper/3864/A. Gerolin, A. Kausamo, T. Rajala.<p> In this paper we extend the duality theory of the multi-marginal optimaltransport problem for cost functions depending on a decreasing function of thedistance (not necessarily bounded). This class of cost functions appears in thecontext of SCE Density Functional Theory introduced in "Strong-interactionlimit of density-functional theory" by M. Seidl.</p>http://cvgmt.sns.it/paper/3864/Quasilinear elliptic equations with a source reaction term involving the function and its gradient and measure datahttp://cvgmt.sns.it/paper/3863/M. F. Bidaut-Véron , Q. H. Nguyen, L. Véron.<p>We study the equation $-div(A(x,\nabla u))=g(x,u,\nabla u)+\mu$where $\mu$ is a measure and either $g(x,u,\nabla u)\sim <br>u<br>^{q_1}u<br>\nabla u<br>^{q_2}$ or $g(x,u,\nabla u)\sim <br>u<br>^{s_1}u+<br>\nabla u<br>^{s_2}$. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potentials or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.</p>http://cvgmt.sns.it/paper/3863/Porous medium equation with nonlocal pressure in a bounded domainhttp://cvgmt.sns.it/paper/3862/Q. H. Nguyen, J. L. Vazquez.<p>We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space domain. We prove existence of weak solutions and suitable a priori bounds and regularity estimates.</p>http://cvgmt.sns.it/paper/3862/