cvgmt Papershttp://cvgmt.sns.it/papers/en-usThu, 20 Sep 2018 00:12:36 +0000Sparsity of solutions for variational inverse problems with finite-dimensional datahttp://cvgmt.sns.it/paper/4051/K. Bredies, M. Carioni.<p>In this paper we characterize sparse solutions for variational problems of the form minu∈X φ(u) + F (Au), where X is a locally convex space, A is a linear continuous oper- ator that maps into a finite dimensional Hilbert space and φ is a seminorm. More precisely, we prove that there exists a minimizer that is “sparse” in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer φ (possibly translated by an element in the null space of φ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear dif- ferential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in <a href='31'>31</a> under weaker hypotheses.</p>http://cvgmt.sns.it/paper/4051/$C^{1,\alpha} isometric embeddings of polar capshttp://cvgmt.sns.it/paper/4050/C. De Lellis, D. Inauen .<p>We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"olderspace $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the Levi-Civita connectionof any isometric immersion is induced by the Euclidean connection, whereas for any $\alpha < \frac{1}{2}$ we construct $C^{1,\alpha}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.</p>http://cvgmt.sns.it/paper/4050/On the continuity of the trace operator in $GSBV(\Omega)$ and $GSBD(\Omega)$http://cvgmt.sns.it/paper/4049/E. Tasso.<p>In this paper we present a new result of continuity for the trace operator acting on functions that might jump on a prescribed $(n-1)$-dimensional set $\Gamma$, with the only hypothesis of being rectifiable and of finite measure. We also show an application of our result in relation to the variational model of elasticity with cracks, when the associated minimum problems are coupled with Dirichlet and Neumann boundary conditions.</p><p>Preprint SISSA 39$/$2018$/$MATE</p>http://cvgmt.sns.it/paper/4049/On the asymptotic behavior of the solutions to parabolic variational inequalitieshttp://cvgmt.sns.it/paper/4048/M. Colombo, L. Spolaor, B. Velichkov.<p>We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new \emph{constrained \L ojasiewicz inequality}. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L.\,Simon (Ann. of Math. 118(3), 1983) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the convergence at infinity of the strong global solutions to the parabolic obstacle and thin-obstacle problems to a unique stationary solution with a rate. Secondly, we give an abstract proof, based on a parabolic approach, of the epiperimetric inequality, which we then apply to the singular points of the obstacle and thin-obstacle problems.</p>http://cvgmt.sns.it/paper/4048/Lipschitz regularity for orthotropic functionals with nonstandard growth conditionshttp://cvgmt.sns.it/paper/4047/P. Bousquet, L. Brasco.<p>We consider a model convex functional with orthotropic structure and super-quadratic nonstandard growth conditions. We prove that bounded local minimizers are locally Lipschitz, with no restrictions on the ratio between the highest and the lowest growth rate.</p>http://cvgmt.sns.it/paper/4047/Sharp decay estimates for critical Dirac equationshttp://cvgmt.sns.it/paper/4046/W. Borrelli.<p>We prove sharp decay estimates for critical Dirac equations on$\mathbb{R}^{n}$, with $n\geq 2$. They appear, e.g., in the study of criticalDirac equations on compact spin manifolds, describing blow-up profiles (theso-called \emph{bubbles}) in the associated variational problem. We establishregularity and integrability properties of $L^{2^{\sharp}}$-solutions (where$2^{\sharp}$ is the Sobolev critical exponent of the embedding of$H^{\frac{1}{2}}(\mathbb{R}^{n} ,\mathbb{C}^{N})$ into Lebesgue spaces) andprove decay estimates, which are shown to be optimal proving the existence of afamily of solutions having the prescribed asymptotic behavior.</p>http://cvgmt.sns.it/paper/4046/Infinitesimal Hilbertianity of weighted Riemannian manifoldshttp://cvgmt.sns.it/paper/4045/D. Lučić, E. Pasqualetto.<p>The main result of this paper is the following: any `weighted' Riemannianmanifold $(M,g,\mu)$ - i.e. endowed with a generic non-negative Radon measure $\mu$- is `infinitesimally Hilbertian', which means that its associatedSobolev space $W^{1,2}(M,g,\mu)$ is a Hilbert space.</p><p>We actually prove a stronger result: the abstract tangentmodule (à la Gigli) associated to any weighted reversible Finslermanifold $(M,F,\mu)$ can be isometrically embedded into the space ofall measurable sections of the tangent bundle of $M$ that are $2$-integrablewith respect to $\mu$.</p>http://cvgmt.sns.it/paper/4045/A note on topological dimension, Hausdorff measure, and rectifiabilityhttp://cvgmt.sns.it/paper/4044/G. David, E. Le Donne.<p>The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $H^n(X)$, is finite. Suppose further that the lower $n$-density of the measure $H^n(X)$ is positive, $H^n(X)$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $H^n(X)$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csörnyei-Jones.</p>http://cvgmt.sns.it/paper/4044/Restricting open surjectionshttp://cvgmt.sns.it/paper/4043/J. Á. Jaramillo, E. Le Donne, T. Rajala.<p>We show that any continuous open surjection from a complete metric space to another metric space can be restricted to a surjection for which the domain has the same density character as the target. This improves a recent result of Aron, Jaramillo and Le Donne.</p>http://cvgmt.sns.it/paper/4043/The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg grouphttp://cvgmt.sns.it/paper/4042/S. Nicolussi Golo, F. Serra Cassano.<p>We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locallyLipschitz intrinsic graph admitting vanishing first variation of itssub-Riemannian area and non-negative second variation must be an intrinsicplane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreovertwo examples are given for stressing result's sharpness.</p>http://cvgmt.sns.it/paper/4042/On the Alexandrov Topology of sub-Lorentzian Manifoldshttp://cvgmt.sns.it/paper/4041/I. Markina, S. Wojtowytsch.<p> It is commonly known that in Riemannian and sub-Riemannian Geometry, themetric tensor on a manifold defines a distance function. In LorentzianGeometry, instead of a distance function it provides causal relations and theLorentzian time-separation function. Both lead to the definition of theAlexandrov topology, which is linked to the property of strong causality of aspace-time. We studied three possible ways to define the Alexandrov topology onsub-Lorentzian manifolds, which usually give different topologies, but agree inthe Lorentzian case. We investigated their relationships to each other and themanifold's original topology and their link to causality.</p>http://cvgmt.sns.it/paper/4041/On the Boundary Regularity of Phase-Fields for Willmore's Energyhttp://cvgmt.sns.it/paper/4040/P. Dondl, S. Wojtowytsch.<p> We demonstrate that Radon measures which arise as the limit of theModica-Mortola measures associated to phase-fields with uniformly boundeddiffuse area and Willmore energy may be singular at the boundary of a domainand discuss implications for practical applications. We furthermore givepartial regularity results for the phase-fields $u_\epsilon$ at the boundary interms of boundary conditions and counterexamples without boundary conditions.</p>http://cvgmt.sns.it/paper/4040/The Effect of Forest Dislocations on the Evolution of a Phase-Field Model for Plastic Sliphttp://cvgmt.sns.it/paper/4039/P. W. Dondl, Matthias W. Kurzke, S. Wojtowytsch.<p> We consider the gradient flow evolution of a phase-field model for crystaldislocations in a single slip system in the presence of forest dislocations.The model consists of a Peierls-Nabarro type energy penalizing non-integer slipand elastic stress. Forest dislocations are introduced as a perforation of thedomain by small disks where slip is prohibited. The $\Gamma$-limit of thisenergy was deduced by Garroni and M\"uller (2005 and 2006). Our main resultshows that the gradient flows of these $\Gamma$-convergent energy functionalsdo not approach the gradient flow of the limiting energy. Indeed, the gradientflow dynamics remains a physically reasonable model in the case of non-monotoneloading. Our proofs rely on the construction of explicit sub- andsuper-solutions to a fractional Allen-Cahn equation on a flat torus or in theplane, with Dirichlet data on a union of small discs. The presence of theseobstacles leads to an additional friction in the viscous evolution whichappears as a stored energy in the $\Gamma$-limit, but it does not act as adriving force. Extensions to related models with soft pinning and non-viscousevolutions are also discussed. In terms of physics, our results explain how inthis phase field model the presence of forest dislocations still allows forplastic as opposed to only elastic deformation.</p>http://cvgmt.sns.it/paper/4039/Keeping it together: a phase field version of path-connectedness and its implementationhttp://cvgmt.sns.it/paper/4038/P. Dondl, S. Wojtowytsch.<p> We describe the implementation of a topological constraint in finite elementsimulations of phase field models which ensures path-connectedness of preimagesof intervals in the phase field variable. Two main applications of our methodare presented. First, a discrete steepest decent of a phase field version of abending energy with spontaneous curvature and additional surface area penaltyis shown, which leads to disconnected surfaces without our topologicalconstraint but connected surfaces with the constraint. The second applicationis the segmentation of an image into a connected component and its exterior.Numerically, our constraint is treated using a suitable geodesic distancefunction which is computed using Dijkstra's algorithm.</p>http://cvgmt.sns.it/paper/4038/The Γ-limit of traveling waves in FitzHugh-Nagumo systemshttp://cvgmt.sns.it/paper/4037/C. N. Chen, Y. S. Choi, N. Fusco.http://cvgmt.sns.it/paper/4037/On the existence and regularity of non-flat profiles for a Bernoulli free boundary problemhttp://cvgmt.sns.it/paper/4036/G. Gravina, G. Leoni.<p> In this paper we consider a large class of Bernoulli-type free boundaryproblems with mixed periodic-Dirichlet boundary conditions. We show thatsolutions with non-flat profile can be found variationally as global minimizersof the classical Alt-Caffarelli energy functional.</p>http://cvgmt.sns.it/paper/4036/Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a $1$-Poincaré inequalityhttp://cvgmt.sns.it/paper/4035/E. Durand-Cartagena, S. Eriksson-Bique, R. Korte, N. Shanmugalingam.<p>We consider two notions of functions of bounded variation in complete metric measure spaces,one due to Martio and the other due to Miranda Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincaré inequality,then the metric space supports a Semmes family of curves structure.</p>http://cvgmt.sns.it/paper/4035/Travelling waves in the discrete stochastic Nagumo equationhttp://cvgmt.sns.it/paper/4034/C. Geldhauser, C. Kuehn.<p>Many physical, chemical and biological systems have an inherent discrete spatial structure that strongly influences their dynamical behaviour. Similar remarks apply to internal or external noise, as well as to nonlocal coupling. In this paper we study the combined effect of nonlocal spatial discretization and stochastic perturbations on travelling waves in the Nagumo equation. We prove that under suitable parameter conditions, various discrete-stochastic variants of the Nagumo equation have solutions, which stay close on long time scales to the classical monotone Nagumo front with high probabilityif the noise level and spatial discretization are sufficiently small.</p>http://cvgmt.sns.it/paper/4034/On the fine structure of the free boundary for the classical obstacle problemhttp://cvgmt.sns.it/paper/4033/A. Figalli, J. Serra.<p>In the classical obstacle problem,the free boundary can be decomposed into "regular'' and "singular'' points. As shown by Caffarelli in his seminal papers, regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of $C^1$ manifolds of varying dimension. In two dimensions, this $C^1$ result has been improved to $C^{1,\alpha}$ by Weiss.</p><p>In this paper we prove that, for $n=2$ singular points are locally contained in a $C^2$ curve. In higher dimension $n\ge 3$, we show that the same result holds with $C^{1,1}$ manifolds (or with countably many $C^2$ manifolds), up to the presence of some ``anomalous'' points of higher codimension.In addition, we prove that the higher dimensional stratum is always contained in a $C^{1,\alpha}$ manifold, thus extending to every dimension the result of Weiss.</p><p>We note that, in terms of density decay estimates for the contact set, our result is optimal.In addition, for $n\ge3$ we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.</p>http://cvgmt.sns.it/paper/4033/Upscaling of dislocation walls in finite domainshttp://cvgmt.sns.it/paper/4028/M. A. Peletier, A. Muntean, P. van Meurs.<p> We wish to understand the macroscopic plastic behaviour of metals byupscaling the micro-mechanics of dislocations. We consider a highly simplifieddislocation network, which allows our microscopic model to be a one dimensionalparticle system, in which the interactions between the particles (dislocationwalls) are singular and non-local. As a first step towards treating realistic geometries, we focus onfinite-size effects rather than considering an infinite domain as typicallydiscussed in the literature. We derive effective equations for the dislocationdensity by means of \Gamma-convergence on the space of probability measures.Our analysis yields a classification of macroscopic models, in which the sizeof the domain plays a key role.</p>http://cvgmt.sns.it/paper/4028/