cvgmt Papershttp://cvgmt.sns.it/papers/en-usSun, 17 Feb 2019 11:39:43 +0000Pairings between bounded divergence-measure vector fields and BV functionshttp://cvgmt.sns.it/paper/4226/G. Crasta, V. De Cicco, A. Malusa.<p>We introduce a family of pairings between a bounded divergence-measure vector field A and a function u of bounded variation, dependingon the choice of the pointwise representative of u.We prove that these pairings inherit from the standard one,introduced by Anzellotti and Chen-Frid, all the mainproperties and features (e.g.\ coarea, Leibniz and Gauss-Green formulas).We also characterize the pairings making the correspondingfunctionals semicontinuous with respect to the strict convergence in BV.We remark thatthe standard pairing in general does not share this property.</p>http://cvgmt.sns.it/paper/4226/Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformationhttp://cvgmt.sns.it/paper/4225/N. Ansini, S. Fagioli.<p>We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density $u$. In case of <i>fast-decay</i> mobilities, namely mobilities functions under a Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained.We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process.We show that the rescaled density $\rho$ is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density $\rho$ allow us to motivate the aforementioned change of variable and to state the results in terms of the original density $u$ without prescribing any boundary conditions.</p>http://cvgmt.sns.it/paper/4225/Point interactions for 3D sub-Laplacianshttp://cvgmt.sns.it/paper/4224/R. Adami, U. Boscain, V. Franceschi, D. Prandi.<p>In this paper we show that, for a sub-Laplacian $\Delta$ on a $3$-dimensional manifold $M$, no point interaction centered at a %contact point $q_0\in M$ exists. When $M$ is complete w.r.t.\ the associated sub-Riemannian structure, this means that $\Delta$ acting on $C^\infty_0(M\setminus\{q_0\})$ is essentially self-adjoint. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold $N$, whose associated Laplace-Beltrami operator is never essentially self-adjoint on $C^\infty_0(N\setminus\{q_0\})$, if $\operatorname{dim}N\le 3$. We then apply this result to the SchrÃ¶dinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.</p>http://cvgmt.sns.it/paper/4224/{$L^1$-{P}oincar\'e inequalities for differential forms on {E}uclidean spaces and {H}eisenberg groups}http://cvgmt.sns.it/paper/4223/A. Baldi, B. Franchi, P. Pansu.<p>https:/hal.archives-ouvertes.fr<i>hal-0201</i></p>http://cvgmt.sns.it/paper/4223/Smoothing operators in multi-marginal Optimal Transporthttp://cvgmt.sns.it/paper/4222/U. Bindini.<p> Given $N$ absolutely continuous probabilities $\rho_1, \dotsc, \rho_N$ over$\mathbb{R}^d$ which have Sobolev regularity, and given a transport plan $P$with marginals $\rho_1, \dotsc, \rho_N$, we provide a universal technique toapproximate $P$ with Sobolev regular transport plans with the same marginals.Moreover, we prove a sharp control of the energy and some continuity propertyof the approximating family.</p>http://cvgmt.sns.it/paper/4222/New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEshttp://cvgmt.sns.it/paper/4221/M. Bardi, A. Goffi.<p>We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the H\"ormander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bellman form, such as those involving the Pucci's extremal operators over H\"ormander vector fields.</p>http://cvgmt.sns.it/paper/4221/Degenerate nonlocal Cahn-Hilliard equations: well-posedness, regularity and local asymptoticshttp://cvgmt.sns.it/paper/4220/E. Davoli, H. Ranetbauer, L. Scarpa, L. Trussardi.<p>Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.</p>http://cvgmt.sns.it/paper/4220/Pointwise gradient estimates for a class of singular quasilinear equation with measure datahttp://cvgmt.sns.it/paper/4219/C. P. Nguyen, Q. H. Nguyen.<p>Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=\mu$in a bounded and possibly nonsmooth domain $\Omega$ in $\mathbb{R}^n$. Here $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian. Our results extend earlier known results to the singular case in which $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$.</p>http://cvgmt.sns.it/paper/4219/Direct and inverse limits of normed moduleshttp://cvgmt.sns.it/paper/4218/E. Pasqualetto.<p>The aim of this note is to study existence and main properties of direct and inverse limits in the category of normed $L^0$-modules (in the sense of Gigli) over a metric measure space.</p>http://cvgmt.sns.it/paper/4218/Sharp Cheeger-Buser type inequalities in $RCD(K,\infty)$ spaceshttp://cvgmt.sns.it/paper/4217/N. De Ponti, A. Mondino.<p>The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian. \\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.\\ The goal of the paper is two fold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-\'Emery 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 $RCD(K,\infty)$ spaces.</p>http://cvgmt.sns.it/paper/4217/Existence of differentiable curves in convex sets and the concept of direction of the flow in mass transportationhttp://cvgmt.sns.it/paper/4215/R. Rios-Zertuche.<p> In this paper we consider convex subsets of locally-convex topological vectorspaces. Given a fixed point in such a convex subset, we show that there existsa curve completely contained in the convex subset and leaving the point in agiven direction if and only if the direction vector is contained in thesequential closure of the tangent cone at that point. We apply this result to the characterization of the existence of weaklydifferentiable families of probability measures on a smooth manifold and of thedistributions that can arise as their derivatives. This gives us a way toconsider the mass transport equation in a very general context, in which thenotion of direction turns out to be given by an element of a Colombeau algebra.</p>http://cvgmt.sns.it/paper/4215/Characterization of minimizable Lagrangian action functionals and a dual Mather theoremhttp://cvgmt.sns.it/paper/4216/R. Rios-Zertuche.<p> We show that a necessary and sufficient condition for a smooth function onthe tangent bundle of a manifold to be a Lagrangian density whose action can beminimized is, roughly speaking, that it be the sum of a constant, a nonnegativefunction vanishing on the support of the minimizers, and an exact form. We show that this exact form corresponds to the differential of a Lipschitzfunction on the manifold that is differentiable on the projection of thesupport of the minimizers, and its derivative there is Lipschitz. This functiongeneralizes the notion of subsolution of the Hamilton-Jacobi equation thatappears in weak KAM theory, and the Lipschitzity result allows for the recoveryof Mather's celebrated 1991 result as a special case. We also show that ourresult is sharp with several examples. Finally, we apply the same type of reasoning to an example of a finitehorizon Legendre problem in optimal control, and together with the Lipschitzityresult we obtain the Hamilton-Jacobi-Bellman equation and the MaximumPrinciple.</p>http://cvgmt.sns.it/paper/4216/Weak KAM theory in higher-dimensional holonomic measure flowshttp://cvgmt.sns.it/paper/4213/R. Rios-Zertuche.<p> We construct a weak KAM theory for higher-dimensional holonomic measures. Wedefine their slices and curves of those slices. We find a weak KAM solution inthat context, and we show that in many cases it corresponds to an exact formthat satisfies a version of the Hamilton-Jacobi equation. Along the way, wegive a characterization of minimizable Lagrangians, as well as some abstractweak KAM machinery.</p>http://cvgmt.sns.it/paper/4213/Deformations of closed measures and variational characterization of measures invariant under the Euler-Lagrange flowhttp://cvgmt.sns.it/paper/4214/R. Rios-Zertuche.<p> The set of closed (or holonomic) measures provides a useful setting forstudying optimization problems because it contains all curves, while alsoenjoying good compactness and convexity properties. We study the way to do variational calculus on the set of closed measures.Our main result is a full description of the distributions that arise as thederivatives of variations of such closed measures. We give examples of how thiscan be used to extract information about the critical closed measures. The condition of criticality with respect to variations leads, in certaincircumstances, to the Euler-Lagrange equations. To understand when thishappens, we characterize the closed measures that are invariant under theEuler-Lagrange flow. Our result implies Ricardo Ma\~n\'e's statement that allminimizers are invariant.</p>http://cvgmt.sns.it/paper/4214/Riemann curvature tensor on RCD spaces and possible applicationshttp://cvgmt.sns.it/paper/4212/N. Gigli.<p>We show that on every RCD spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor.Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are RCD spaces, our construction applies in particular to the Alexan- drov setting. We conjecture that an RCD space is Alexandrov if and only if the sectional curvature - defined in terms of such abstract Riemann tensor - is bounded from below.</p>http://cvgmt.sns.it/paper/4212/Rigidity for Perimeter Inequalities under symmetrization: state of the art and open problemshttp://cvgmt.sns.it/paper/4211/F. Cagnetti.<p>We review some classical results in symmetrization theory, some recent progress in understanding rigidity, and indicate some open problems</p>http://cvgmt.sns.it/paper/4211/Adaptive image processing: first order PDE constraint regularizers and a bilevel training schemehttp://cvgmt.sns.it/paper/4210/E. Davoli, I. Fonseca, P. Liu.<p>A bilevel training scheme is used to introduce a novel class of regularizers, providing a unified approach to standard regularizers $TV$, $TGV^2$ and $NsTGV^2$. Optimal parameters and regularizers are identified, and the existence of a solution for any given set of training imaging data is proved by $\Gamma$-convergence. Explicit examples and numerical results are given.</p>http://cvgmt.sns.it/paper/4210/The nodal set of solutions to some elliptic problems: sublinear equations, and unstable two-phase membrane problemhttp://cvgmt.sns.it/paper/4209/N. Soave, S. Terracini.<p>We are concerned with the nodal set of solutions to equations of the form \begin{equation<b>}-\Delta u = \lambda<sub>+</sub> \left(u<sup>+\right)</sup><sup>{q</sup>-1} - \lambda<sub></sub>- \left(u<sup></sup>-\right)<sup>{q</sup>-1} \quad \text{in $B_1$}\end{equation</b>}where $\lambda_+,\lambda_- > 0$, $q \in [1,2)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. This class includes, the \emph{unstable two-phase membrane problem} ($q=1$), as well as \emph{sublinear} equations for $1<q<2$. </p><p>We prove the following main results: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$); (d) a partial stratification theorem.</p><p>Ultimately, the main features of the nodal set are strictly related with those of the solutions to linear (or superlinear) equations, with two remarkable differences. First of all, the admissible vanishing orders can not exceed the critical value $2/(2-q)$. At threshold, we find a multiplicity of homogeneous solutions, yielding the \emph{non-validity} of any estimate of the $(N-1)$-dimensional measure of the nodal set of a solution in terms of the vanishing order.</p><p>As a byproduct, we also prove the strong unique continuation property for the \emph{unstable obstacle problem}, corresponding to the case $\lambda_-=0$.</p><p>The proofs are based on monotonicity formul\ae \ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.</p>http://cvgmt.sns.it/paper/4209/Homogenization in $BV$ of a model for layered composites in finite crystal plasticityhttp://cvgmt.sns.it/paper/4208/E. Davoli, R. Ferreira, C. Kreisbeck.<p>In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials arranged into periodically alternating thin horizontal strips of an elastically rigid component and a softer one with one active slip system. The energies arising from these modeling assumptions are of integral form, featuring linear growth and non-convex differential constraints. We approach this non-standard homogenization problem via Gamma-convergence. A crucial first step in the asymptotic analysis is the characterization of rigidity properties of limits of admissible deformations in the space $BV$ of functions of bounded variation. In particular, we prove that, under suitable assumptions, the two-dimensional body may split horizontally into finitely many pieces, each of which undergoes shear deformation and global rotation. This allows us to identify a potential candidate for the homogenized limit energy, which we show to be a lower bound on the Gamma-limit. In the framework of non-simple materials, we present a complete Gamma-convergence result, including an explicit homogenization formula, for a regularized model with an anisotropic penalization in layer direction.</p>http://cvgmt.sns.it/paper/4208/The sharp quantitative isocapacitary inequalityhttp://cvgmt.sns.it/paper/4207/G. De Philippis, M. Marini, E. Mukoseeva.<p>We prove a sharp quantitative form of the classical isocapacitary inequality. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. This provides a positive answer to a conjecture of Hall, Hayman, and Weitsman (J. d' Analyse Math. '91).</p>http://cvgmt.sns.it/paper/4207/