CVGMT Papershttp://cvgmt.sns.it/papers/en-usTue, 21 Nov 2017 16:43:56 -0000Sensitivity of the compliance and of the Wasserstein distance with respect to a varying sourcehttp://cvgmt.sns.it/paper/3666/G. Bouchitté, I. Fragalà, I. Lucardesi.
<p>e show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover,
we provide an integral representation formula for the derivative as a linear functional of the deformation vector field.
The result holds true as well for the $p$-compliance in the scalar case of conductivity.
Then we study the limit problem as $p \to + \infty$, which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans.
When the latter contains only one element, we prove that
the derivative of the $p$-compliance converges to the derivative of the Wasserstein distance in the limit as $p \to + \infty$.</p>
http://cvgmt.sns.it/paper/3666/On the curvature energy of Cartesian surfaceshttp://cvgmt.sns.it/paper/3665/D. Mucci.
<p>We analyze the lower semicontinuous envelope of the curvature functional of Cartesian surfaces in codimension one.
To this aim, following the approach by Anzellotti-Serapioni-Tamanini, we study the class of currents that naturally arise as weak limits of Gauss graphs of smooth functions.
The curvature measures are then studied in the non-parametric case. Concerning homogeneous functions, some model examples are studied in detail. Finally, a new gap phenomenon is observed.</p>
http://cvgmt.sns.it/paper/3665/Introduction to Riemannian and Sub-Riemannian geometryhttp://cvgmt.sns.it/paper/3664/A. Agrachev, D. Barilari, U. Boscain.
<p>Lecture notes "Introduction to Riemannian and Sub-Riemannian geometry".
</p>
<p><b>New updated version 17.11.2017</b> (Ch. 13 added + Revision Ch. 8, 10, 20 + New sections added in Ch. 3, 12)
</p>
<p>Table of Contents: 1 - Geometry of surfaces in R3. 2 - Vector fields and vector bundle. 3 - Sub-Riemannian structures. 4 - Characterization and local minimality of Pontryagin extremals. 5 - Integrable systems. 6 - Chronological calculus. 7 - Lie groups and left-invariant sub-Riemannian structures 8 - End-point and exponential map. 9 - 2D Almost-Riemannian structures. 10 - Nonholonomic tangent space. 11 - Regularity of the sub-Riemannian distance. 12 - Abnormal extremals and second variation. 13 - Some model spaces 14 - Curves in the Lagrange Grassmannian 15 - Jacobi curves. 16 - Riemannian curvature. 17 - Curvature of 3D contact sub-Riemannian structures. 18 - Asymptotic expansion of the 3D contact exponential map. 19 - The volume in sub-Riemannian geometry. 20 - The sub-Riemannian heat equation.</p>
http://cvgmt.sns.it/paper/3664/Some Sphere Theorems in Linear Potential Theoryhttp://cvgmt.sns.it/paper/3663/S. Borghini, G. Mascellani, L. Mazzieri.
<p> In this paper we analyze the capacitary potential due to a charged body in
order to deduce sharp analytic and geometric inequalities, whose equality cases
are saturated by domains with spherical symmetry. In particular, for a regular
bounded domain $\Omega \subset \mathbb{R}^n$, $n\geq 3$, we prove that if the
mean curvature $H$ of the boundary obeys the condition $ - \bigg[
\frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} \leq \frac{H}{n-1} \leq
\bigg[ \frac{1}{\text{Cap}(\Omega)} \bigg]^{\frac{1}{n-2}} $, then $\Omega$
is a round ball.
</p>
http://cvgmt.sns.it/paper/3663/Higher Holder regularity for the fractional $p-$Laplacian in the superquadratic casehttp://cvgmt.sns.it/paper/3662/L. Brasco, E. Lindgren, A. Schikorra.
<p>We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$,
when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$,
which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.</p>
http://cvgmt.sns.it/paper/3662/Splitting schemes & segregation in reaction-(cross-)diffusion systemshttp://cvgmt.sns.it/paper/3661/J. A. Carrillo, S. Fagioli, F. Santambrogio, M. Schmidtchen.
<p>One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, \emph{i.e.} population densities with disjoint supports. We analyse such a reaction cross-diffusion system in 1D. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.
</p>
http://cvgmt.sns.it/paper/3661/A counterexample to gluing theorems for MCP metric measure spaceshttp://cvgmt.sns.it/paper/3660/L. Rizzi.
<p> Perelman's doubling theorem asserts that the metric space obtained by gluing
along their boundaries two copies of an Alexandrov space with curvature $\geq
\kappa$ is an Alexandrov space with the same dimension and satisfying the same
curvature lower bound. We show that this result cannot be extended to metric
measure spaces satisfying synthetic Ricci curvature bounds in the
$\mathrm{MCP}$ sense. The counterexample is given by the Grushin half-plane,
which satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 4$, while its
double satisfies the $\mathrm{MCP}(0,N)$ if and only if $N\geq 5$.
</p>
http://cvgmt.sns.it/paper/3660/Low energy configurations of topological singularities in two dimensions: A $\Gamma$-convergence analysis of dipoleshttp://cvgmt.sns.it/paper/3659/L. De Luca, M. Ponsiglione.
<p>This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the {core radius approach} and the Ginzburg-Landau energy.
Denoting by $\varepsilon$ the length scale parameter in such models, we focus on the $\log\frac{1}{\varepsilon}$ energy regime.
It is well known that, for configurations whose energy is bounded by $c \log \frac{1}{\varepsilon}$,
the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying $\pi \log \frac{1}{\varepsilon}$ energy,
plus a measure supported on small zero-average sets.
Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses.
</p>
<p>Here we perform a compactness and $\Gamma$-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale $\varepsilon^s$, for $0<s<1$), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical
$\Gamma$-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order $c \log \frac{1}{\varepsilon}$ with $c<\pi$.
</p>
http://cvgmt.sns.it/paper/3659/On the codimension of the abnormal set in step two Carnot groupshttp://cvgmt.sns.it/paper/3658/A. Ottazzi, D. Vittone.
<p>In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups. </p>
http://cvgmt.sns.it/paper/3658/Generalized crystalline evolutions as limits of flows with smooth anisotropieshttp://cvgmt.sns.it/paper/3657/A. Chambolle, M. Morini, M. Novaga, M. Ponsiglione.
<p>We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.</p>
http://cvgmt.sns.it/paper/3657/Nonlocal problems with critical Hardy nonlinearityhttp://cvgmt.sns.it/paper/3656/W. Chen, S. J. N. Mosconi, M. Squassina.
<p>By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.</p>
http://cvgmt.sns.it/paper/3656/On the optimal shape of a rigid body supported by an elastic membranehttp://cvgmt.sns.it/paper/3655/G. Buttazzo, A. Wagner.
<p>We consider an elastic membrane which supports a rigid body. Under height or volume constraints on the rigid body, we study the elastic deformation of the membrane and we determine the shape of the body which minimizes the total energy.</p>
http://cvgmt.sns.it/paper/3655/Optimal location of support points in the Kirchhoff platehttp://cvgmt.sns.it/paper/3654/G. Buttazzo, S. Nazarov.
<p>The Dirichlet problem for the bi-harmonic equation is considered as the Kirchhoff model of an isotropic elastic plate clamped at its edge. The plate is supported at certain points $P^1,\dots,P^J$, that is the deflexion $u(x)$ satisfies the Sobolev point conditions $u(P^1)=\dots=u(P^J)=0$. The optimal location of the support points is discussed such that either the compliance functional, or the minimal deflexion functional attains its minimum.</p>
http://cvgmt.sns.it/paper/3654/An obstacle problem for conical deformations of thin elastic sheetshttp://cvgmt.sns.it/paper/3653/A. Figalli, C. Mooney.
<p>A developable cone (``d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $\epsilon$.
Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $\Gamma$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle
problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $\epsilon$ is small. In particular, we rigorously justify previous results in the physics literature.</p>
http://cvgmt.sns.it/paper/3653/Equilibria configurations for epitaxial crystal growth with adatomshttp://cvgmt.sns.it/paper/3652/M. Caroccia, R. Cristoferi, L. Dietrich.
<p>The behavior of a surface energy $\mathcal{F}(E,u)$, where $E$ is a set of finite perimeter and $u\in L^1(\partial^* E, \mathbb{R}_+)$ is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process.
Regular critical points, existence and uniqueness of minimizers are discussed and the relaxation of $\mathcal{F}$ in a general setting under the $L^1$ convergence of sets and the vague convergence of measures is characterized.
</p>
http://cvgmt.sns.it/paper/3652/Analysis and geometry of RCD spaces via the Schrödinger problemhttp://cvgmt.sns.it/paper/3651/L. Tamanini.
<p>Main aim of this manuscript is to present a new interpolation technique for probability measures, different from Brenier-McCann's classical one, which is strongly inspired by the Schrödinger problem, an entropy minimization problem deeply related to optimal transport. By means of the solutions to the Schrödinger problem, we build an efficient approximation scheme, robust up to the second order. Such scheme allows us to prove the second order differentiation formula along geodesics in finite-dimensional $RCD^*$ spaces. This formula is new even in the context of Alexandrov spaces and we provide some applications.
</p>
<p>The proof relies on new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:
</p>
<p>- equiboundedness of the densities along the entropic interpolations,
</p>
<p>- local equi-Lipschitz continuity of the Schrödinger potentials,
</p>
<p>- a uniform weighted $L^2$ control of the Hessian of such potentials.
</p>
<p>These tools are very useful in the investigation of the geometric information encoded in entropic interpolations. The techniques used in this work can be also used to show that the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case.
</p>
<p>Throughout the whole manuscript, several remarks on the physical interpretation of the Schrödinger problem and connections with the better known Schrödinger equation are pointed out. Hopefully, this will allow the reader to better understand the physical and probabilistic motivations of the problem as well as to connect them with the analytical and geometric nature of the dissertation.</p>
http://cvgmt.sns.it/paper/3651/Stable $s$-minimal cones in $\mathbb R^3$ are flat for $s\sim 1$ http://cvgmt.sns.it/paper/3650/X. Cabrè, E. Cinti, J. Serra.
<p>We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb R^3$, for $s\in (0,1)$ sufficiently close to 1. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its proof can not rely on a compactness argument perturbing from $s= 1$. In fact, our proof gives a quantifiable value for the required closeness of $s$ to 1. We use the geometric formula for the second variation of the fractional $s$-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.</p>
http://cvgmt.sns.it/paper/3650/Harmonic mappings valued in the Wasserstein spacehttp://cvgmt.sns.it/paper/3649/H. Lavenant.
<p>We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings $\mathbf{\mu} : \Omega \to (\mathcal{P}(D), W_2)$ defined over a subset $\Omega$ of $\mathbf{R}^p$ and valued in the space $\mathcal{P}(D)$ of probability measures on a compact convex subset $D$ of $\mathbf{R}^q$ endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary $\partial \Omega$ are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for $\Omega$ a segment of $\mathbf{R}$. As the Wasserstein space $(\mathcal{P}(D), W_2)$ is positively curved in the sense of Alexandrov we cannot apply the theory of Koorevaar, Schoen and Jost and we use instead optimal transport based arguments. We manage to get existence of a harmonic mapping provided that the boundary values are Lipschitz on $\partial \Omega$, uniqueness is an open question. If $\Omega$ is a segment of $\mathbf{R}$, it is known that a curve valued in the Wasserstein space can be seen as a superposition of curves valued in $D$. We show that it is no longer the case in higher dimensions: a generic mapping $\Omega \to \mathcal{P}(D)$ cannot be represented as the superposition of mappings $\Omega \to D$. We are able to show a Ishihara-type property: the composition $F \circ \mathbf{\mu}$ of a function $F : \mathcal{P}(D) \to \mathbf{R}$ convex along generalized geodesics and an harmonic mapping $\mathbf{\mu} : \Omega \to \mathcal{P}(D)$ is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given location-scatter family (a finite-dimensional and geodesically convex submanifold of $(\mathcal{P}(D), W_2)$ which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport. </p>
http://cvgmt.sns.it/paper/3649/A variational proof of partial regularity for optimal transportation maps between setshttp://cvgmt.sns.it/paper/3648/M. Goldman, F. Otto.
<p>This is a simplified version of the paper
"A variational proof of partial regularity for optimal transportation maps"
(see link above)
in the case of constant densities.</p>
http://cvgmt.sns.it/paper/3648/Stationarity of the crack-front for the Mumford-Shah problem in 3Dhttp://cvgmt.sns.it/paper/3647/A. Lemenant, H. Mikayelyan.
<p>In this paper we exhibit a family of stationary solutions of the Mumford-Shah problem in $\mathbb{R}^3$, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet.
</p>
<p>We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set.
</p>
<p>A self-contained proof of the stationarity of the crack-tip function for the Mumford-Shah problem in 2D is presented. </p>
http://cvgmt.sns.it/paper/3647/