cvgmt Papershttp://cvgmt.sns.it/papers/en-usThu, 21 Mar 2019 19:45:13 +0000Hardy-type inequalities for the Carnot-Carathéodory distance in the Heisenberg grouphttp://cvgmt.sns.it/paper/4263/V. Franceschi, D. Prandi.<p>In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carathéodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carathéodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma\subset \S^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.</p>http://cvgmt.sns.it/paper/4263/$L^{p,q}$ estimates on the transport densityhttp://cvgmt.sns.it/paper/4262/S. Dweik.<p>In this paper, we show a new regularity result on the transport density σ in theclassical Monge-Kantorovich optimal mass transport problem between two measures, μ and ν, having some summable densities, f<sup>+</sup> and f<sup>−.</sup> More precisely, we prove that the transport density σ belongs to L<sup>p,q</sup>(Ω) as soon as f<sup>+,</sup> f<sup>−</sup> ∈ L<sup>p,q</sup>(Ω).</p>http://cvgmt.sns.it/paper/4262/Dimensional reduction of the Kirchhoff-Plateau problemhttp://cvgmt.sns.it/paper/4261/G. Bevilacqua, L. Lussardi, A. Marzocchi.<p>We obtain the Plateau problem with elastic boundary as a variational limit of the Kirchhoff-Plateau problem when the cross section of the boundary rod vanishes. The boundary is a framed curve that can sustain bending and twisting.</p>http://cvgmt.sns.it/paper/4261/On the square distance function from a manifold with boundaryhttp://cvgmt.sns.it/paper/4260/G. Bellettini, A. Elshorbagy.<p>We characterize arbitrary codimensionalsmooth manifolds M with boundary embedded in Rn using the square distancefunction and the signed distance functionfrom M and from its boundary. The resultsare localized in an open set.</p>http://cvgmt.sns.it/paper/4260/Recent results on the essential self-adjointness of sub-Laplacians, with some remarks on the presence of characteristic pointshttp://cvgmt.sns.it/paper/4259/V. Franceschi, D. Prandi, L. Rizzi.<p>In this proceeding, we present some recent results obtained in <a href='4'>4</a> on the essential self-adjointness of sub-Laplacians on non-complete sub-Riemannian manifolds. A notable application is the proof of the essential self-adjointness of the Popp sub-Laplacian on the equiregular connected components of a sub-Riemannian manifold, when the singular region does not contain characteristic points. In their presence, the self-adjointness properties of (sub-)Laplacians are still unknown. We conclude the paper discussing the difficulties arising in this case.</p>http://cvgmt.sns.it/paper/4259/Weyl's law for singular Riemannian manifoldshttp://cvgmt.sns.it/paper/4258/Y. Chitour, D. Prandi, L. Rizzi.<p>In this paper, we study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics and the localization of the eigenfunctions for large frequencies. As a consequence of our results, we identify a class of singular structures such that the corresponding Laplace-Beltrami operator has the following non-classical Weyl's law:\[N(λ) \sim \frac{\omega_n}{(2\pi)^n} \lambda^{n/2} \upsilon(\lambda)\]where $\upsilon$ is slowly varying at infinity in the sense of Karamata. Finally, for any non-decreasing slowly varying function $\upsilon$, we construct singular Riemannian structures admitting the above Weyl's law. A key tool in our arguments is a universal estimate for the remainder of the heat trace on Riemannian manifolds, which is of independent interest.</p>http://cvgmt.sns.it/paper/4258/The local structure of the free boundary in the fractional obstacle problemhttp://cvgmt.sns.it/paper/4257/M. Focardi, E. Spadaro.<p>Building upon the recent results in \cite{FoSp17} we provide a thorough description of thefree boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $\varphi$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure.In particular, if $\varphi$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results:</p><p>(i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),</p><p>(ii) $\mathcal{H}^{n-1}$-rectifiability of the free boundary,</p><p>(iii) classification of the frequenciesup to a set of Hausdorff dimension at most $(n-2)$ and classification of the blow-ups at $\mathcal{H}^{n-1}$ almost every free boundary point.</p><p>Instead, if $\varphi\in C^{k+1}(\mathbb{R}^n)$, $k\geq 2$, similar results hold only for a distinguished subset of points in the free boundary where the order of contact of the solution and the obstacle is less than $k+1$.</p>http://cvgmt.sns.it/paper/4257/Quasi-continuous vector fields on $\sf RCD$ spaceshttp://cvgmt.sns.it/paper/4256/C. Debin, N. Gigli, E. Pasqualetto.<p>In the existing language for tensor calculus on $\sf RCD$ spaces, tensor fields are only defined $\mathfrak m$-a.e.. In this paper we introduce the concept of tensor field defined `2-capacity-a.e.' and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.</p>http://cvgmt.sns.it/paper/4256/None compactness versus critical points at infinity: an examplehttp://cvgmt.sns.it/paper/4255/M. Mayer.<p> We illustrate an example of a generic, positive function K on a Riemannianmanifold to be conformally prescribed as the scalar curvature, for which thecorresponding Yamabe type L2-gradient flow exhibits non compact flow lines,while a slight modification of it is compact.</p>http://cvgmt.sns.it/paper/4255/Characterization of BV functions on open domains: the Gaussian case and the general casehttp://cvgmt.sns.it/paper/4254/D. Addona, G. Menegatti, M. Miranda Jr.<p>We provide three different characterizations of the space $BV(O,\gamma)$ of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure $\gamma$ on open domains $O$ in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition for belonging to $BV(O,\gamma)$ by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our technique to Fomin differentiable probability measures $\nu$ on a Hilbert space $X$, inferring a characterization of the space $BV(O,\nu)$ of the functions of bounded variation with respect to $\nu$ on open domains $O\subseteq X$.</p>http://cvgmt.sns.it/paper/4254/Intrinsic Lipschitz graphs in Carnot groups of step 2http://cvgmt.sns.it/paper/4253/D. Di Donato.<p> We focus our attention on the notion of intrinsic Lipschitz graphs, inside aspecial class of metric spaces i.e. the Carnot groups. More precisely, weprovide a characterization of locally intrinsic Lipschitz functions in Carnotgroups of step 2 in terms of their intrinsic distributional gradients.</p>http://cvgmt.sns.it/paper/4253/A continuous dependence result for a dynamic debonding model in dimension onehttp://cvgmt.sns.it/paper/4252/F. Riva.<p>In this paper we address the problem of continuous dependence on initial and boundary data for a one-dimensional debonding model describing a thin film peeled away from a substrate. The system underlying the process couples the weakly damped wave equation with a Griffith's criterion which rules the evolution of the debonded region. We show that under general convergence assumptions on the data the corresponding solutions converge to the limit one with respect to different natural topologies.</p>http://cvgmt.sns.it/paper/4252/Optimal controlled transports with free end times subject to import/export tariffshttp://cvgmt.sns.it/paper/4251/S. Dweik, N. Ghoussoub, A. Z. Palmer.<p>We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle à la Kantorovich, that takes into consideration the additional tariffs. We then show that the primal optimal transport problem has an equivalent Eulerian formulation whose dual involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality with non-homogeneous boundary conditions. This allows us to prove existence and to describe the solutions for both the primal optimization problem and its Eulerian counterpart.</p>http://cvgmt.sns.it/paper/4251/A new variational approach to linearization of traction problems in elasticityhttp://cvgmt.sns.it/paper/4250/F. Maddalena, D. Percivale, F. Tomarelli.<p>A new energy functional for pure traction problems in elasticity has been deduced in a previous paper as the variational limit of nonlinear elastic energy functional for a material body subject to an equilibrated force field: a sort of Gamma limit with respect to the weak convergence of strains when a suitable small parameter tends to zero. This functional exhibits a gap that makes it different from the classical linear elasticity functional. Nevertheless a suitable compatibility condition on the force field ensures coincidence of related minima and minimizers. Here we show some relevant properties of the new functional and prove stronger convergence of minimizing sequences for suitable choices of nonlinear elastic energies.</p>http://cvgmt.sns.it/paper/4250/Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensionshttp://cvgmt.sns.it/paper/4249/S. Fanzon, M. Palombaro.<p>We study the higher gradient integrability of distributional solutions $u$ tothe equation $div(\sigma \nabla u) = 0$ in dimension two, in the case when theessential range of $\sigma$ consists of only two elliptic matrices, i.e.,$\sigma\in\{\sigma_1, \sigma_2\}$ a.e. in $\Omega$. In <a href='4'>4</a>, for every pair of elliptic matrices $\sigma_1$ and $\sigma_2$,exponents $p_{\sigma_1,\sigma_2}\in(2,+\infty)$ and $q_{\sigma_1,\sigma_2}\in(1,2)$ have been characterised so that if $u\inW^{1,q_{\sigma_1,\sigma_2}}(\Omega)$ is solution to the elliptic equation then$\nabla u\in L^{p_{\sigma_1,\sigma_2}}_{\rm weak}(\Omega)$ and the optimalityof the upper exponent $p_{\sigma_1,\sigma_2}$ has been proved. In this paper wecomplement the above result by proving the optimality of the lower exponent$q_{\sigma_1,\sigma_2}$. Precisely, we show that for every arbitrarily small$\delta$, one can find a particular microgeometry, i.e., an arrangement of thesets $\sigma^{-1}(\sigma_1)$ and $\sigma^{-1}(\sigma_2)$, for which thereexists a solution $u$ to the corresponding elliptic equation such that $\nablau \in L^{q_{\sigma_1,\sigma_2}-\delta}$, but $\nabla u \notinL^{q_{\sigma_1,\sigma_2}}.$ The existence of such optimal microgeometries isachieved by convex integration methods, adapting to the present setting thegeometric constructions provided in <a href='2'>2</a> for the isotropic case.</p>http://cvgmt.sns.it/paper/4249/An optimal transport approach for solving dynamic inverse problems in spaces of measureshttp://cvgmt.sns.it/paper/4248/K. Bredies, S. Fanzon.<p>In this paper we propose and study a novel optimal transport basedregularization of linear dynamic inverse problems. The considered inverse problems aim at recovering a measure valued curve and are dynamic in the sense that(i) the measured data takes values in a time dependent family of Hilbert spaces,and (ii) the forward operators are time dependent and map, for each time, Radonmeasures into the corresponding data space. The variational regularization wepropose bases on dynamic optimal transport which means that the measure valued curves to recover (i) satisfy the continuity equation, i.e., the Radon measureat time t is advected by a velocity field v and varies with a growth rate g, and (ii)are penalized with the kinetic energy induced by v and a growth energy inducedby g. We establish a functional-analytic framework for these regularized inverseproblems, prove that minimizers exist and are unique in some cases, and studyregularization properties. This framework is applied to dynamic image reconstruction in undersampled magnetic resonance imaging (MRI), modeling relevantexamples of time varying acquisition strategies, as well as patient motion andpresence of contrast agents.</p>http://cvgmt.sns.it/paper/4248/Inverse coefficient problems for a transport equation by local Carleman estimatehttp://cvgmt.sns.it/paper/4247/P. Cannarsa, G. Floridia, F. G\"olgeleyen, M. Yamamoto.<p>We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, anddiscuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$.Our results are conditional stability of H\"older type in a subdomain $D$ provided that the outward normal component of $H(x)$ is positive on $\ppp D \cap \ppp\OOO$.The proofs are based on a Carleman estimate where the weight function dependson $H$.</p>http://cvgmt.sns.it/paper/4247/Inverse coefficient problems for a transport equation by local Carleman estimatehttp://cvgmt.sns.it/paper/4246/P. Cannarsa, G. Floridia, F. G\"olgeleyen, M. Yamamoto.<p>We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, anddiscuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$.Our results are conditional stability of H\"older type in a subdomain $D$ provided that the outward normal component of $H(x)$ is positive on $\ppp D \cap \ppp\OOO$.The proofs are based on a Carleman estimate where the weight function dependson $H$.</p>http://cvgmt.sns.it/paper/4246/Inverse coefficient problems for a transport equation by local Carleman estimatehttp://cvgmt.sns.it/paper/4245/P. Cannarsa, F. F. G\"olgeleyen, G. Floridia, M. Yamamoto.<p>We consider the transport equation $\ppp_tu(x,t) + (H(x)\cdot \nabla u(x,t)) + p(x)u(x,t) = 0$ in $\OOO \times (0,T)$ where $\OOO \subset \R^n$ is a bounded domain, anddiscuss two inverse problems which consist of determining a vector-valued function $H(x)$ or a real-valued function $p(x)$ by initial values and data on a subboundary of $\OOO$.Our results are conditional stability of H\"older type in a subdomain $D$ provided that the outward normal component of $H(x)$ is positive on $\ppp D \cap \ppp\OOO$.The proofs are based on a Carleman estimate where the weight function dependson $H$.</p>http://cvgmt.sns.it/paper/4245/Finite crystallization and wulff shape emergence for ionic compounds in the square latticehttp://cvgmt.sns.it/paper/4244/M. Friedrich, L. Kreutz.<p>We present two-dimensional crystallization results in the square lattice for finite particle systems consisting of two different atomic types. We identify energy minimizers of configurational energies featuring two-body short-ranged particle interactions which favor some reference distance between different atomic types and contain repulsive contributions for atoms of the same type. We first prove that ground states are connected subsets of the square lattice with alternating arrangement of the two atomic types in the crystal lattice, and address the emergence of a square macroscopic Wulff shape for an increasing number of particles. We then analyze the signed difference of the number of the two atomic types, the so-called net charge, for which we prove the sharp scaling n to 1 over 4 in terms of the particle number n. Afterwards, we investigate the model under prescribed net charge. We provide a characterization for the minimal energy and identify a critical net charge beyond which crystallization in the square lattice fails. Finally, for this specific net charge we prove a crystallization result and identify a diamond-like Wulff-shape of energy minimizers which illustrates the sensitivity of the macroscopic geometry on the net charge.</p>http://cvgmt.sns.it/paper/4244/