cvgmt Papershttp://cvgmt.sns.it/papers/en-usSat, 15 Dec 2018 22:53:28 +0000Some functional inequalities and spectral properties of metric measure spaces with curvature bounded belowhttp://cvgmt.sns.it/paper/4136/D. Tewodrose.http://cvgmt.sns.it/paper/4136/Sharp decay estimates for critical Dirac equationshttp://cvgmt.sns.it/paper/4134/W. Borrelli, R. L. Frank.<p>We prove sharp pointwise decay estimates for critical Dirac equations on ℝ<sup>n</sup> with n≥2. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states.</p>http://cvgmt.sns.it/paper/4134/Embedding of $RCD^*(K,N)$ spaces in $L^2$ via eigenfunctionshttp://cvgmt.sns.it/paper/4133/L. Ambrosio, S. Honda, J. Portegies, D. Tewodrose.<p>In this paper we study the family of embeddings $\Phi_t$ of a compact $\RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$via eigenmaps. Extending part of the classical results by Berard, ,Berard-Besson-Gallot, known for closed Riemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaled pull-back metrics $\Phi_t^*g_{L^2}$ in $L^2(X,m)$ induced by $\Phi_t$.Moreover we discuss the behavior of $\Phi_t^*g_{L^2}$ with respect to measured Gromov-Hausdorff convergence and $t$.Applications include the quantitative $L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result new even for closed Riemannian manifolds and Alexandrov spaces.</p>http://cvgmt.sns.it/paper/4133/H-type foliationshttp://cvgmt.sns.it/paper/4132/F. Baudoin, E. Grong, G. Molino, L. Rizzi.<p>With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound, we prove on those structures sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann, we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of those spaces in codimension more than 2.</p>http://cvgmt.sns.it/paper/4132/Infinitesimal Hilbertianity of locally ${\rm Cat}(\kappa)$-spaceshttp://cvgmt.sns.it/paper/4131/S. Di Marino, N. Gigli, E. Pasqualetto, E. Soultanis.<p>We show that, given a metric space $({\rm Y},{\sf d})$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $\mu$ on ${\rm Y}$ giving finite mass to bounded sets, the resulting metric measure space $({\rm Y},{\sf d},\mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}({\rm Y},{\sf d},\mu)$ is a Hilbert space. </p><p>The result is obtained by constructing an isometric embedding of the <i>abstract and analytical</i> space of derivations into the <i>concrete and geometrical</i> bundle whose fibre at $x\in{\rm Y}$ is the tangent cone at $x$ of ${\rm Y}$. The conclusion then follows from the fact that for every $x\in{\rm Y}$ such a cone is a ${\rm Cat}(0)$ space and, as such, has a Hilbert-like structure.</p>http://cvgmt.sns.it/paper/4131/Symmetric self-shrinkers for the fractional mean curvature flowhttp://cvgmt.sns.it/paper/4130/A. Cesaroni, M. Novaga.<p>We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.</p>http://cvgmt.sns.it/paper/4130/A variational approach to single crystals with dislocationshttp://cvgmt.sns.it/paper/4129/R. Scala, N. Van Goethem.<p>We study the graphs of maps $u:\Omega\rightarrow\mathbb R^3$ whose curl is an integral $1$-current with coefficients in $\mathbb Z^3$. We characterize the graph boundary of such maps under suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered. </p><p>The problem we address is the joint minimization of the free energy with respect to the deformation field and the dislocation lines. We apply closedness results for graphs of torus-valued maps, seen as integral currents and, from the characterization of their graph boundaries we are able to prove existence of minimizers.</p>http://cvgmt.sns.it/paper/4129/Variational evolution of dislocations in single crystalshttp://cvgmt.sns.it/paper/4128/R. Scala, N. Van Goethem.<p>In this paper we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored-elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters.</p>http://cvgmt.sns.it/paper/4128/Dissociating limit in Density Functional Theory with Coulomb optimal transport costhttp://cvgmt.sns.it/paper/4127/G. Bouchitté, G. Buttazzo, T. Champion, L. De Pascale.<p>In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of $N=2$ electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak<b> convergence of measures.</b></p>http://cvgmt.sns.it/paper/4127/Weak formulation of elastodynamics in domains with growing crackshttp://cvgmt.sns.it/paper/4126/E. Tasso.<p>In this paper we formulate and study the system of elastodynamics on domains with arbitrary growing cracks. This includes homogeneous Neumann conditions on the crack sets and mixed general Dirichlet-Neumann conditions on the boundary. The only assumptions on the crack sets are to be $(n-1)$-rectifiable with finite surface measure, and increasing in the sense of set inclusions. In particular they might be dense, hence the weak formulation must fall outside the usual context of Sobolev spaces and Korn's inequality. We prove existence of a solution both for the damped and undamped systems, while in the damped case we are also able to prove uniqueness and an energy balance.</p>http://cvgmt.sns.it/paper/4126/Monotonicity formulae for smooth extremizers of integral functionalshttp://cvgmt.sns.it/paper/4124/E. Fried, L. Lussardi.<p>A general monotonicity formula for smooth constrained local extremizers of first-order integral functionals subject to non-holonomic constraints is established. The result is then applied to recover some known monotonicity formulae and to discover some new monotonicity formulae of potential value.</p>http://cvgmt.sns.it/paper/4124/Convex sets evolving by volume preserving fractional mean curvature flowshttp://cvgmt.sns.it/paper/4123/E. Cinti, C. Sinestrari, E. Valdinoci.<p>We consider the volume preservinggeometric evolution of the boundary of a setunder fractional mean curvature. We show thatsmooth convex solutions maintain their fractional curvatures boundedfor all times, and the long time asymptoticsapproach round spheres.The proofs are based on apriori estimateson the inner and outer radii of the solutions.</p>http://cvgmt.sns.it/paper/4123/New estimates on the regularity of the pressure in density-constrained Mean Field Gameshttp://cvgmt.sns.it/paper/4122/H. Lavenant, F. Santambrogio.<p>We consider variational Mean Field Games endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player is augmented by a pressure effect, i.e. a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optimal transport techniques after time-discretization, the results obtained in a paper of the second author with Cardaliaguet and Mészáros. We prove $H^1$ and $L^\infty$ regularity under very mild assumptions on the data, and explain the consequences for the MFG, in terms of the value function and of the Lagrangian equilibrium formulation.</p>http://cvgmt.sns.it/paper/4122/Elastic networks, statics and dynamicshttp://cvgmt.sns.it/paper/4121/M. Novaga, A. Pluda.<p>We consider planar networks minimizing the elastic energy, we state an existence and regularity result, and we discuss some geometric properties of minimal configurations. We also consider the evolutionof networks by the gradient flow of the energy, and we give a well-posedness result in the case of natural boundary conditions.</p><p>Proceedings of the "11th Mathematical Society of Japan Seasonal Institute", held in Sapporo in July 2018.</p>http://cvgmt.sns.it/paper/4121/Existence of strong solutions to the Dirichlet problem for the Griffith energyhttp://cvgmt.sns.it/paper/4120/A. Chambolle, V. Crismale.<p> In this paper we continue the study of the Griffith brittle fracture energyminimisation under Dirichlet boundary conditions, suggested by Francfort andMarigo in 1998. In a recent paper, we proved the existence of weak minimisersof the problem. Now we show that these minimisers are indeed strong solutions,namely their jump set is closed and they are smooth away from the jump set andcontinuous up to the Dirichlet boundary. This is obtained by extending up tothe boundary the recent regularity results of Conti, Focardi, Iurlano, and ofChambolle, Conti, Iurlano.</p>http://cvgmt.sns.it/paper/4120/Hölder regularity for nonlocal double phase equationshttp://cvgmt.sns.it/paper/4119/C. De Filippis, G. Palatucci.<p>We prove some regularity estimates for viscosity solutions to a class of possible degenerate and singular integro-differential equations whose leading operator switches between two different types of fractional elliptic phases, according to the zero set of a modulating coefficient $a=a(\cdot,\cdot)$. The model case is driven by the following nonlocal double phase operator,\[\int \frac{<br>u(x)-u(y)<br>^{p-2}(u(x)-u(y))}{<br>x-y<br>^{n+sp}}\,{\rm d}y + \int a(x,y)\frac{<br>u(x)-u(y)<br>^{q-2}(u(x)-u(y))}{<br>x-y<br>^{n+tq}}\,{\rm d}y,\]where $q\geq p$ and $a(\cdot,\cdot)\geqq 0$.Our results do also apply for inhomogeneous equations, for very general classes of measurable kernels. By simply assuming the boundedness of the modulating coefficient, we are able to prove that the solutions are Hölder continuous, whereas similar sharp results for the classical local case do require $a$ to be Hölder continuous. To our knowledge, this is the first (regularity) result for nonlocal double phase problems.</p>http://cvgmt.sns.it/paper/4119/Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation typehttp://cvgmt.sns.it/paper/4118/A. Cesaroni, S. Dipierro, M. Novaga, E. Valdinoci.<p>We consider an energy functional combiningthe square of the local oscillation of a one--dimensionalfunction with a double well potential.We establish the existence of minimal heteroclinic solutionsconnecting the two wells of the potential.</p><p>This existence result cannot be accomplished by standard methods,due to the lack of compactness properties.</p><p>In addition, we investigate the main properties of theseheteroclinic connections. We show that these minimizersare monotone, and therefore they satisfy a suitable Euler-Lagrange equation.</p><p>We also prove that,differently from the classical cases arising in ordinary differentialequations, in this context the heteroclinic connectionsare not necessarily smooth, and not even continuous(in fact, they can be piecewise constant). Also, we show thatheteroclinics are not necessarily unique upto a translation, which is also in contrast with the classical setting.</p><p>Furthermore, we investigate the associated Dirichlet problem,studying existence, uniqueness and partial regularity properties,providing explicit solutions in terms of the external data and of the forcingsource, and exhibiting an example of discontinuous solution.</p>http://cvgmt.sns.it/paper/4118/Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentationhttp://cvgmt.sns.it/paper/4117/P. R. A. S. A. N. T. A. K. U. M. A. R. BARIK.<p> In this paper we study the continuous coagulation and multiple fragmentationequation for the mean-field description of a system of particles taking intoaccount the combined effect of the coagulation and the fragmentation processesin which a system of particles growing by successive mergers to form a biggerone and a larger particle splits into a finite number of smaller pieces. Wedemonstrate the global existence of mass-conserving weak solutions for a wideclass of coagulation rate, selection rate and breakage function. Here, both thebreakage function and the coagulation rate may have algebraic singularity onboth the coordinate axes. The proof of the existence result is based on a weakL<sup>1</sup> compactness method for two different suitable approximations to theoriginal problem, i.e. the conservative and non-conservative approximations.Moreover, the mass-conservation property of solutions is established for bothapproximations.</p>http://cvgmt.sns.it/paper/4117/The planning problem in Mean Field Games as regularized mass transporthttp://cvgmt.sns.it/paper/4116/P. J. Graber, A. R. Mészáros, F. Silva, D. Tonon.<p>In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. As an alternative way, we show that solutions of the planning problem can be approximated, via a $\Gamma$-convergence procedure, by solutions of standard mean field games with suitable penalized final couplings. In the same time (relying on the techniques developed recently by Graber and M\'esz\'aros), under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of mean field games with general final couplings and the planning problem as well, both for space and time derivatives.</p>http://cvgmt.sns.it/paper/4116/Existence of a Lens–Shaped Cluster of Surfaces Self–Shrinking by Mean Curvaturehttp://cvgmt.sns.it/paper/4115/P. Baldi, E. Haus, C. Mantegazza.http://cvgmt.sns.it/paper/4115/