cvgmt Papershttps://cvgmt.sns.it/papers/en-usWed, 24 Feb 2021 21:05:07 +0000Ultralimits of pointed metric measure spaceshttps://cvgmt.sns.it/paper/5041/E. Pasqualetto, T. Schultz.<p>The aim of this paper is to study ultralimits of pointed metric measurespaces (possibly unbounded and having infinite mass). We prove that ultralimitsexist under mild assumptions and are consistent with the pointed measuredGromov-Hausdorff convergence. We also introduce a weaker variant of pointedmeasured Gromov-Hausdorff convergence, for which we can prove a version ofGromov's compactness theorem by using the ultralimit machinery. Thiscompactness result shows that, a posteriori, our newly introduced notion ofconvergence is equivalent to the pointed measured Gromov one. Another byproductof our ultralimit construction is the identification of direct and inverselimits in the category of pointed metric measure spaces.</p>https://cvgmt.sns.it/paper/5041/Shape optimization of light structures and the vanishing mass conjecturehttps://cvgmt.sns.it/paper/5040/J. F. Babadjian, F. Iurlano, F. Rindler.<p> This work proves rigorous results about the vanishing-mass limit of theclassical problem to find a shape with minimal elastic compliance. Contrary toall previous results in the mathematical literature, which utilize a soft massconstraint by introducing a Lagrange multiplier, we here consider the hard massconstraint. Our results are the first to establish the convergence ofapproximately optimal shapes of (exact) size $\varepsilon \downarrow 0$ to alimit generalized shape represented by a (possibly diffuse) probabilitymeasure. This limit generalized shape is a minimizer of the limit compliance,which involves a new integrand, namely the one conjectured by Bouchitt\'e in2001 and predicted heuristically before in works of Allaire & Kohn and Kohn &Strang from the 1980s and 1990s. This integrand gives the energy of the limitgeneralized shape understood as a fine oscillation of (optimal)lower-dimensional structures. Its appearance is surprising since the integrandin the original compliance is just a quadratic form and the non-convexity ofthe problem is not immediately obvious. In fact, it is the interaction of themass constraint with the requirement of attaining the loading (in the form of adivergence-constraint) that gives rise to this new integrand. We also presentconnections to the theory of Michell trusses, first formulated in 1904, andshow how our results can be interpreted as a rigorous justification of thattheory on the level of functionals in both two and three dimensions, finallysettling this long-standing open problem. Our proofs rest on compensatedcompactness arguments applied to an explicit family of (symmetric)div-quasiconvex quadratic forms, computations involving the Hashin-Shtrikmanbounds for the Kohn-Strang integrand, and the characterization of limitminimizers due to Bouchitt\'e & Buttazzo.</p>https://cvgmt.sns.it/paper/5040/On the Cauchy problem for the Muskat equation. II: Critical initial datahttps://cvgmt.sns.it/paper/5039/T. Alazard, Q. H. Nguyen.<p>We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in L2. Moreover, we prove that the solution exists globally in time under a smallness assumption.</p>https://cvgmt.sns.it/paper/5039/On symmetry of energy minimizing harmonic-type maps on cylindrical surfaceshttps://cvgmt.sns.it/paper/5038/G. Di Fratta, A. Fiorenza, V. Slastikov.<p>The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of S 2-valued maps defined in cylindrical surfaces, which naturally arises as curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. First, we show that minimal configurations are z-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequalities, which allow establishing a detailed picture of the energy landscape. Finally, we provide a complete characterization of in-plane minimizers.</p>https://cvgmt.sns.it/paper/5038/$C^{1,α}$-rectifiability in low codimension in Heisenberg groupshttps://cvgmt.sns.it/paper/5037/K. O. Idu, F. P. Maiale.<p> A natural notion of higher order rectifiability is introduced for subsets ofHeisenberg groups $\mathbb{H}^n$ in terms of covering a set almost everywhereby a countable union of $(\mathbf{C}_H^{1,\alpha},\mathbb{H})$-regularsurfaces, for some $0 < \alpha \leq 1$. We prove that a sufficient conditionfor $C^{1,\alpha}$-rectifiability of low-codimensional subsets in Heisenberggroups is the almost everywhere existence of suitable approximate tangentparaboloids.</p>https://cvgmt.sns.it/paper/5037/Towards a mathematical theory of trajectory inferencehttps://cvgmt.sns.it/paper/5036/Y. H. Kim, H. Lavenant, G. Schiebinger, S. Zhang.<p>We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which provide high dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real datasets.</p>https://cvgmt.sns.it/paper/5036/Crack occurrence in bodies with gradient polyconvex energieshttps://cvgmt.sns.it/paper/5035/M. Kruzik, P. M. Mariano, D. Mucci.<p>Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlet-type. Crack paths are described in terms of curvature varifolds so that we consider both "phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove existence of minimizers for such an energy. They are pairs of deformations and varifolds. The former ones are taken to be $SBV$ maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.</p>https://cvgmt.sns.it/paper/5035/On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernelshttps://cvgmt.sns.it/paper/5034/G. M. Coclite, N. De Nitti, A. Keimer, L. Pflug.<p>In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution $\gamma \ast q$, we weaken the standard assumption on the kernel $\gamma \in L^\infty\big((0,T); W^{1,\infty}(\mathbb R)\big)$ to the substantially more general condition $\gamma \in L^\infty((0,T); BV(\mathbb R))$, which allows for discontinuities in the kernel.</p>https://cvgmt.sns.it/paper/5034/Gamma-convergence of Cheeger energies with respect to increasing distanceshttps://cvgmt.sns.it/paper/5033/D. Lučić, E. Pasqualetto.<p> We prove a $\Gamma$-convergence result for Cheeger energies along sequencesof metric measure spaces, where the measure space is kept fixed, whiledistances are monotonically converging from below to the limit one. As aconsequence, we show that the infinitesimal Hilbertianity condition is stableunder this kind of convergence of metric measure spaces.</p>https://cvgmt.sns.it/paper/5033/Detection of chaos in the general relativistic Poynting-Robertson effect: Kerr equatorial planehttps://cvgmt.sns.it/paper/5032/W. Borrelli, V. De Falco.<p>The general relativistic Poynting-Robertson effect is a dissipative and non-linear dynamical system obtained by perturbing through radiation processes the geodesic motion of test particles orbiting around a spinning compact object, described by the Kerr metric. Using the Melnikov method we find that, in a suitable range of parameters, chaotic behavior is present in the motion of a test particle driven by the Poynting-Robertson effect in the Kerr equatorial plane.</p>https://cvgmt.sns.it/paper/5032/$\Gamma$-convergence and stochastic homogenisation of singularly-perturbed elliptic functionalshttps://cvgmt.sns.it/paper/5031/A. Bach, R. Marziani, C. I. Zeppieri.<p>We study the limit behaviour of singularly-perturbed elliptic functionals of the form\[F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla v)dx\,, \]where $u$ is a vector-valued Sobolev function, $v \in [0,1]$ a phase-field variable, and $\varepsilon_k>0$ a singular-perturbation parameter, i.e., $\varepsilon_k \to 0$, as $k\to +\infty$.</p><p>Under mild assumptions on the integrands $f_k$ and $g_k$, we show that if $f_k$ grows superlinearly in the gradient-variable, then the functionals $F_k$ $\Gamma$-converge (up to subsequences) to a <i>brittle</i> energy-functional, i.e., to a free-discontinuity functional whose surface integrand does <i>not</i> depend on the jump-amplitude of $u$. This result is achieved by providing explicit asymptotic formulas for bulk and surface integrands which show, in particular, that the volume and surface term in $F_k$ <i>decouple</i> in the limit.</p><p>The abstract $\Gamma$-convergence analysis is complemented by a stochastic homogenisation result for <i>stationary random</i> integrands.</p>https://cvgmt.sns.it/paper/5031/Embedded Delaunay tori and their Willmore energyhttps://cvgmt.sns.it/paper/5030/C. Scharrer.<p> A family of embedded rotationally symmetric tori in the Euclidean $3$-spaceconsisting of two opposite signed constant mean curvature surfaces thatconverge as varifolds to a round sphere of multiplicity $2$ is constructed.Using complete elliptic integrals, it is shown that their Willmore energy liesstrictly below $8\pi$. Combining such a strict inequality with previous worksby Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that forevery isoperimetric ratio there exists a smooth embedded torus minimising theWillmore functional under isoperimetric constraint, thus completing thesolution of the isoperimetric-constrained Willmore problem for tori. Moreover,because of their symmetry, the tori can be used to construct spheres of highisoperimetric ratio, leading to an alternative proof of the known result forthe genus zero case.</p>https://cvgmt.sns.it/paper/5030/Quantitative estimates for parabolic optimal control problems under L∞ and L1 constraints in the ball: Quantifying parabolic isoperimetric inequalitieshttps://cvgmt.sns.it/paper/5029/I. Mazari.https://cvgmt.sns.it/paper/5029/A Zero Sum Differential Game With Correlated Informations on the Initial Position. A case with a continuum of initial positions.https://cvgmt.sns.it/paper/5028/C. Jimenez.<p>We study a two player zero sum game where the initial position $z_0$ is not communicated to any player. The initial position is a function of a couple $(x_0,y_0)$ where $x_0$ is communicated to player I while $y_0$ is communicated to player II. The couple $(x_0,y_0)$ is chosen according a probability measure $dm(x,y)=h(x,y) d\mu(x) d\nu(y)$. We show that the game has a value and, under additional regularity assumptions, that the value is a solution of Hamilton Jacobi Isaacs equation in a dual sense.</p>https://cvgmt.sns.it/paper/5028/Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilingshttps://cvgmt.sns.it/paper/5027/G. Del Nin, M. Petrache.<p> We prove discrete-to-continuum convergence of interaction energies defined onlattices in the Euclidean space (with interactions beyond nearest neighbours)to a crystalline perimeter, and we discuss the possible Wulff shapes obtainablein this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is anextension of De Bruijn's "pentagrid" construction of Penrose tilings) we adaptthe same techniques to also find the macroscopical homogenized perimeter whenwe microscopically rescale a given quasiperiodic tiling.</p>https://cvgmt.sns.it/paper/5027/Phd Thesis: Asymptotic analysis of nonlinear models for line defects in materialshttps://cvgmt.sns.it/paper/5026/R. Marziani.<p>The thesis is devoted to the study, by means of a variational approach, the presence of dislocations in crystals, due to plastic slips of planes of atoms over each other.In the first part we consider a Geometrically nonlinear elastic model in the three-dimensional setting, that allows for large rotations. Adopting a core approach, which consists in regularizing the problem at scale epsilon around the dislocation lines, we perform the asymptotic analysis of the regularized energy as epsilon tends to zero. We focus in particular on the leading order regime and prove that the energy rescaled by $\epsilon^2\log(1/\epsilon)$ Gamma converges to the line-tension for a dislocation density derived by Conti, Garroni and Ortiz in a three-dimensional linear framework. The analysis is performed under the assumption that the dislocations are well separated at intermediate scale, this in fact will allow to treat individually each dislocation by means of a suitable cell formula. The nonlinear nature of the energy requires that in the characterization of the cell formula we take into account that the deformation gradient is close to a fixed rotation. In the second part we show that the same limit formulacan be obtained with a different type of regularization.Namely we consider an energy with mixed growth, that behaves quadratically far from the dislocations and sub-quadratically near the dislocation lines.</p>https://cvgmt.sns.it/paper/5026/Nonlinear diffusion in transparent mediahttps://cvgmt.sns.it/paper/5025/L. Giacomelli, S. Moll, F. Petitta.<p>We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions' support and in the bulk.</p>https://cvgmt.sns.it/paper/5025/De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensionshttps://cvgmt.sns.it/paper/5024/T. Laux, J. Lelmi.<p>We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoğlu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.</p>https://cvgmt.sns.it/paper/5024/Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicityhttps://cvgmt.sns.it/paper/5022/W. Gangbo, A. R. Mészáros, C. Mou, J. Zhang.<p>In this manuscript, we propose a structural condition on non-separable Hamiltonians, which we term displacement monotonicity condition, to study second order mean field games master equations. A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a--priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.</p>https://cvgmt.sns.it/paper/5022/The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysishttps://cvgmt.sns.it/paper/5021/G. Antonelli, M. Fogagnolo, M. Pozzetta.<p>In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.</p>https://cvgmt.sns.it/paper/5021/