cvgmt Papershttp://cvgmt.sns.it/papers/en-usMon, 22 Apr 2019 16:11:38 +0000Variational approximation of functionals defined on 1-dimensional connected sets in $\mathbb{R}^n$http://cvgmt.sns.it/paper/4284/M. Bonafini, G. Orlandi, E. Oudet.<p>In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert-Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in $\mathbb{R}^n$. Following the analysis developed for the planar case, we provide a variational approximation through Ginzburg-Landau type energies proving a $\Gamma$-convergence result for $n \geq 3$.</p>http://cvgmt.sns.it/paper/4284/On the umbilic set of immersed surfaces in three-dimensional space formshttp://cvgmt.sns.it/paper/4283/G. Catino, A. Roncoroni, L. Vezzoni.<p>We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.</p>http://cvgmt.sns.it/paper/4283/Variational analysis of a two-dimensional frustrated spin system: emergence and rigidity of chirality transitionshttp://cvgmt.sns.it/paper/4282/M. Cicalese, M. Forster, G. Orlando.<p>We study the discrete-to-continuum variational limit of the $J_{1}$-$J_{3}$spin model on the square lattice in the vicinity of the helimagnet-ferromagnettransition point as the lattice spacing vanishes. Carrying out the$\Gamma$-convergence analysis of proper scalings of the energy, we prove theemergence and characterize the geometric rigidity of the chirality phasetransitions.</p>http://cvgmt.sns.it/paper/4282/Euler's optimal profile problemhttp://cvgmt.sns.it/paper/4281/F. Maddalena, E. Mainini, D. Percivale.<p>We study an old variational problem formulated by Euler as Proposition 53 of his “Scientia Navalis" by means of the direct method of the calculus of variations. Precisely, through relaxation arguments, we prove the existence of minimizers. We fully investigate the analytical structure of the minimizers in dependence of the geometric parameters and we identify the ranges of uniqueness and non-uniqueness.</p>http://cvgmt.sns.it/paper/4281/A formula for the anisotropic total variation of SBV functionshttp://cvgmt.sns.it/paper/4280/F. Farroni, N. Fusco, S. Guarino Lo Bianco, R. Schiattarella.<p>The purpose of this article is to present the relation between certain BMO–type seminorms and thetotal variation of SBV functions. Following a paper by Ambrosio and Comi, we give a representation formula of the anisotropic total variation of SBV functions which does not make use of the distributional derivatives.</p>http://cvgmt.sns.it/paper/4280/External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-developmenthttp://cvgmt.sns.it/paper/4279/M. Carioni, J. Fischer, A. Schlömerkemper.<p>This paper is concerned with equilibrium configurations of one-dimensional particle system with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a Γ-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend onLagrangian or Eulerian coordinates.Our result is based on a new technique for deriving compactness results which are requiredfor calculating the first-order Γ-limit: instead of comparing a configuration of n atoms to aglobal minimizer of the Γ-limit, we compare the configuration to a minimizer in some subclassof functions which in some sense are “close to” the configuration. This new technique isrequired due to the additional presence of forces with non-convex potentials. The paper iscomplemented with the study of the minimizers of the Γ–limit.</p>http://cvgmt.sns.it/paper/4279/Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergencehttp://cvgmt.sns.it/paper/4278/M. Friedrich, F. Solombrino.<p>We analyse integral representation and $\Gamma$-convergence properties of functionals defined on \emph{piecewise rigid functions}, i.e., functions which are piecewise affine on a Caccioppoli partition whose derivative in each component is constant and lies in a set without rank-one connections. Such functionals are customary in the variational modeling of materials which locally show a rigid behavior, and account for interfacial energies, e.g., for polycrystals or in fracture mechanics. Our results are based on localization techniques for $\Gamma$-convergence and a careful adaption of the global method for relaxation (Bouchitt\'eet al. 1998, 2001) to this new setting, under rather general assumptions. They constitute a first step towards the investigation of lower semicontinuity, relaxation, and homogenization for free-discontinuity problems in spaces of (generalized) functions of bounded deformation.</p>http://cvgmt.sns.it/paper/4278/$\Gamma$-convergence for functionals depending on vector fields I. Integral representation and compactness.http://cvgmt.sns.it/paper/4277/A. Maione, A. Pinamonti, F. Serra Cassano.<p>Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$.We prove an integral representation for local functionals with respect to $X$ and a result of $\Gamma$-compactness for a class of integral functionals depending on $X$.</p>http://cvgmt.sns.it/paper/4277/On the Sobolev quotient of three-dimensional CR manifoldshttp://cvgmt.sns.it/paper/4276/J. H. Cheng, A. Malchiodi, P. Yang.<p>We exhibit examples of compact three-dimensional CR manifolds of positive Webster class,Rossi spheres, for which the pseudo-hermitian mass as defined in <a href='CMY17'>CMY17</a> is negative, and for which theinfimum of the CR-Sobolev quotient is not attained. To our knowledge, this is the first geometric contexton smooth closed manifolds where this phenomenon arises, in striking contrast to the Riemannian case.</p>http://cvgmt.sns.it/paper/4276/A gap theorem for $\alpha$-harmonic maps between two-sphereshttp://cvgmt.sns.it/paper/4275/T. Lamm, A. Malchiodi, M. Micallef.<p>In this paper we consider approximations à la Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in <a href='6'>6</a> about limits of $\alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity in <a href='7'>7</a>, we obtain an optimal gap theorem for the $\alpha$-harmonic maps of degree $-1, 0$ or $1$.</p>http://cvgmt.sns.it/paper/4275/Gradient potential estimates on the Heisenberg Grouphttp://cvgmt.sns.it/paper/4274/S. Mukherjee, Y. Sire.<p>We establish pointwise estimates for the horizontal gradient of solutions to quasi-linear p-Laplacian type non-homogeneous equations with measure data in the Heisenberg Group.</p>http://cvgmt.sns.it/paper/4274/On the well-posedness of branched transportationhttp://cvgmt.sns.it/paper/4273/M. Colombo, A. De Rosa, A. Marchese.<p>We show in full generality the stability of optimal traffic paths in branched transport: namely we prove that any limit of optimal traffic paths is optimal as well.This solves an open problem in the field (cf. Open problem 1 in the book Optimal transportation networks, by Bernot, Caselles and Morel), which has been addressed up to now only under restrictive assumptions.</p>http://cvgmt.sns.it/paper/4273/A nonlocal supercritical Neumann problemhttp://cvgmt.sns.it/paper/4272/E. Cinti, F. Colasuonno.<p> We establish existence of positive non-decreasing radial solutions for anonlocal nonlinear Neumann problem both in the ball and in the annulus. Thenonlinearity that we consider is rather general, allowing for supercriticalgrowth (in the sense of Sobolev embedding). The consequent lack of compactnesscan be overcome, by working in the cone of non-negative and non-decreasingradial functions. Within this cone, we establish some a priori estimates whichallow, via a truncation argument, to use variational methods for provingexistence of solutions. As a side result, we prove a strong maximum principlefor nonlocal Neumann problems, which is of independent interest.</p>http://cvgmt.sns.it/paper/4272/Geometric Patterns and Microstructures in the study of Material Defects and Compositeshttp://cvgmt.sns.it/paper/4271/S. Fanzon.<p>The main focus of this PhD thesis is the study of microstructures and geometric patterns in materials, in the framework of the Calculus of Variations. This thesis is divided into two main parts. In the first part we present the results obtained in <a href='22, 23'>22, 23</a>. In these two works geometric patterns have to be understood as patterns of dislocations in crystals. The second part is devoted to <a href='21'>21</a>, where suitable microgeometries are needed as a mean to produce gradients that display critical integrability properties.</p>http://cvgmt.sns.it/paper/4271/Variational problems involving unequal dimensional optimal transporthttp://cvgmt.sns.it/paper/4270/L. Nenna, B. Pass.<p>This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.</p>http://cvgmt.sns.it/paper/4270/$L^\infty$ bounds of Steklov eigenfunctions and spectrum stability under domain variationhttp://cvgmt.sns.it/paper/4268/D. Bucur, A. Giacomini, P. Trebeschi.<p>We give a practical tool to control the $L^\infty$-norm of the Steklov eigenfunctions in a Lipschitz domain in terms of the norm of the $BV$-trace operator. The norm of this operator has the advantage to be characterized by purely geometric quantities. As a consequence, we give a spectral stability result for the Steklov eigenproblem under geometric domain perturbations and several examples where stability occurs. In particular we deal with geometric domains which are not equi-Lipschitz, like vanishing holes, merging sets, approximations of inner peaks.</p>http://cvgmt.sns.it/paper/4268/Perturbed minimizing movements of families of functionalshttp://cvgmt.sns.it/paper/4267/A. Braides, A. Tribuzio.<p>We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($\Gamma$-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $\varepsilon$ and $\tau$, governing energy and time scales, respectively. We characterize the extreme cases when $\varepsilon/\tau$ and $\tau/\varepsilon$ converges to $0$ sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of $\varepsilon$ and $\tau$. We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.</p>http://cvgmt.sns.it/paper/4267/The Plateau problem in the Calculus of Variationshttp://cvgmt.sns.it/paper/4266/L. Lussardi.<p>This is a survey paper written for a course held for the Ph. D. program in Pure and Applied Mathematics at Politecnico di Torino during autumn 2018. The course has been dedicated to an overview of the main techniques for solving the Plateau problem, that is to find a surface with minimal area that spans a given boundary curve in the space. This problem dates back to the physical experiments of Plateau who tried to understand the possible configurations of soap films. From the mathematical point of view the problem is very hard and a lot of possible formulations are available: perhaps still today none of these answers is the answer to the original formulation by Plateau. In this paper first of all we will briefly introduce the problem showing that, at least in the smooth case, if the first variation of the area vanishes then the surface must have zero mean curvature. Then we will describe how the classical solution by Douglas and Rado works, and we will pass to modern formulations of the problem in the context of Geometric Measure Theory: finite perimeter sets, currents and minimal sets.</p>http://cvgmt.sns.it/paper/4266/On the optimal map in the 2-dimensional random matching problemhttp://cvgmt.sns.it/paper/4265/L. Ambrosio, F. Glaudo, D. Trevisan.<p>We show that, on a $2$-dimensional compact manifold, the optimal transportmap in the semi-discrete random matching problem is well-approximated in the$L^2$-norm by identity plus the gradient of the solution to the Poisson problem$-\Delta f^{n,t} = \mu^{n,t}-1$, where $\mu^{n,t}$ is an appropriateregularization of the empirical measure associated to the random points. Thisshows that the ansatz of Caracciolo et al. (Scaling hypothesis for theEuclidean bipartite matching problem) is strong enough to capture the behaviorof the optimal map in addition to the value of the optimal matching cost. As part of our strategy, we prove a new stability result for the optimaltransport map on a compact manifold.</p>http://cvgmt.sns.it/paper/4265/New Directions in Harmonic Analysis on $L^1$http://cvgmt.sns.it/paper/4264/D. Spector.<p> The study of what we now call Sobolev inequalities has now been consideredfor more than a century by physicists, while it has been eighty years sinceSobolev's seminal mathematical contributions. Yet there are still things wedon't understand about the action of integral operators on functions. This isno more apparent than in the $L^1$ setting, where only recently have optimalinequalities been obtained on the Lebesgue and Lorentz scale for scalarfunctions, while the full resolution of similar estimates for vector-valuedfunctions is incomplete. The purpose of this paper is to discuss how some oftenoverlooked estimates for the classical Poisson equation give an entry intothese questions, to the present state of the art of what is known, and tosurvey some open problems on the frontier of research in the area.</p>http://cvgmt.sns.it/paper/4264/