cvgmt Papershttp://cvgmt.sns.it/papers/en-usSun, 25 Aug 2019 06:54:21 +0000Least gradient problem on annulihttp://cvgmt.sns.it/paper/4441/S. Dweik, W. Górny .<p>We consider the two dimensional BV least gradient problem on an annulus with given boundary data $g \in BV(\partial\Omega)$. Firstly, we prove that this problem is equivalent to the optimal transport problem with source and target measures located on the boundary of the domain. Then, under some admissibility conditions on the trace, we show that there exists a unique solution for the BV least gradient problem. Moreover, we prove some $L^p$ estimates on the corresponding minimal flow of the Beckmann problem, which implies directly $W^{1,p}$ regularity for the solution of the BV least gradient problem.</p>http://cvgmt.sns.it/paper/4441/A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Grouphttp://cvgmt.sns.it/paper/4440/M. Capolli , A. Pinamonti, G. Speight.<p>We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.</p>http://cvgmt.sns.it/paper/4440/Characterization of generalized Young measures generated by $\mathcal A$-free measureshttp://cvgmt.sns.it/paper/4439/A. Arroyo-Rabasa.<p> We organize a robust analytical framework in terms of the space$\mathrm{BV}^{\mathcal A}(\mathbb R^d)$ of functions with bounded $\mathcalA$-variation, where $\mathcal A$ is a partial differential operator satisfyingMurat's constant rank property. This perspective enable us to carryconstructions available for gradients into the $\mathcal A$-free framework(introduced by Dacorogna and Fonseca & M\"uller). In particular, this allowsthe gluing and localization of $\mathcal A$-free measures without modifying theunderlying $\mathcal A$-free constraint. We combine these advances withdelicate geometric constructions to give a full characterization of the classof generalized Young measures generated by sequences of $\mathcal A$-freemeasures (where $\mathcal A$ is an operator of arbitrary order). The maincharacterization result is stated in terms of a well-known separation propertyinvolving the class of $\mathcal A$-quasiconvex integrands. We give a secondcharacterization in terms of the tangent Young measures being $\mathcal A$-freeYoung measures. Lastly, we show that the inclusion \[ \mathrm L^1(\Omega) \cap\ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A \] isdense with respect to the area-strict convergence of measures.</p>http://cvgmt.sns.it/paper/4439/Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometryhttp://cvgmt.sns.it/paper/4438/R. Jones, P. Lahti, N. Shanmugalingam.<p>We generalize a result of Kelly~\cite{Kelly} to the setting of Ahlfors $Q$-regular metric measure spacessupporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supportinga $1$-Poincar\'e inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collectionof measures $\mathcal{H}^{Q-1}\vert_{\Sigma E}$ corresponding to any collection of sets $E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $\Sigma E$, $f(E)$ is also of finite perimeter. Even in the standard Euclidean setting our results are more general than that of Kelly, and hence are new even in there.</p>http://cvgmt.sns.it/paper/4438/Rigidity for perimeter inequality under spherical symmetrisationhttp://cvgmt.sns.it/paper/4437/F. Cagnetti, M. Perugini, D. Stöger.<p>Necessary and sufficient conditions forrigidity of the perimeter inequality under spherical symmetrisation are given.That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided.This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by P\'olya in 1950.</p>http://cvgmt.sns.it/paper/4437/Uniform distribution of dislocations in Peierls-Nabarro models for semi-coherent interfaceshttp://cvgmt.sns.it/paper/4436/S. Fanzon, M. Ponsiglione, R. Scala.<p>In this paper we introduce Peierls-Nabarro type models for edge dislocationsat semi-coherent interfaces between two heterogeneous crystals, and prove theoptimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy $\Gamma$-converges to a limitfunctional comprised of two contributions: one is given by a constant$c_\infty>0$ gauging the minimal energy induced by dislocations at theinterface, and corresponding to a uniform distribution of edge dislocations;the other one accounts for the far field elastic energy induced by the presenceof further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering thelimit from semi-coherent to coherent interfaces, we show that $c_\infty$ isreached when dislocations are evenly-spaced on the one dimensional circle.</p>http://cvgmt.sns.it/paper/4436/Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problemhttp://cvgmt.sns.it/paper/4435/M. Colombo, L. Spolaor, B. Velichkov.<p>We answer a question left open in <a href='Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184'>Arch. Rat. Mech. Anal. 230 (1) (2018), 125-184</a> and <a href='Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784'>Arch. Rat. Mech. Anal. 230 (2) (2018), 783-784</a>, by proving that the blow-up of minimizers u of the lower dimensional obstacle problem is unique at generic point of the free-boundary.</p>http://cvgmt.sns.it/paper/4435/Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded belowhttp://cvgmt.sns.it/paper/4434/N. De Ponti, M. Muratori, C. Orrieri.<p> Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Riccicurvature bounded from below, we discuss the stability of the solutions of aporous medium-type equation with respect to the 2-Wasserstein distance. Weproduce (sharp) stability estimates under negative curvature bounds, which tosome extent generalize well-known results by Sturm and Otto-Westdickenberg. Thestrategy of the proof mainly relies on a quantitative $L^1-L^\infty$ smoothingproperty of the equation considered, combined with the Hamiltonian approachdeveloped by Ambrosio, Mondino and Savar\'e in a metric-measure setting.</p>http://cvgmt.sns.it/paper/4434/Singular periodic solutions to a critical equation in the Heisenberg grouphttp://cvgmt.sns.it/paper/4433/C. Afeltra.<p>We construct positive solutions to the equation\[-\Delta_{\mathbf{H}^n} u =u^{\frac{Q+2}{Q-2}}\]on the Heisenberg group, singular in the origin, similarto the Fowler solutions of the Yamabe equations on $\mathbf{R}^n$. Thesesatisfy the homogeneity property $u\circ\delta_T=T^{-\frac{Q-2}{2}}u$ for some$T$ large enough, where $Q=2n+2$ and $\delta_T$ is the natural dilation in$\mathbf{H}^n$. We use the Lyapunov-Schmidt method applied to a family ofapproximate solutions built by periodization from the global regular solutionclassified by Jerison and Lee.</p>http://cvgmt.sns.it/paper/4433/From the $N$-clock model to the $XY$ model: emergence of concentration effects in the variational analysishttp://cvgmt.sns.it/paper/4432/M. Cicalese, G. Orlando, M. Ruf.<p>We investigate the relationship between the $N$-clock model (also known as planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero temperature) through a $\Gamma$-convergence analysis as both the number of particles and $N$ diverge. By suitably rescaling the energy of the $N$-clock model, we illustrate how its thermodynamic limit strongly depends on the rate of divergence of~$N$ with respect to the number of particles. The $N$-clock model turns out to be a good approximation of the $XY$ model only for $N$ sufficiently large; in other regimes of $N$, we show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of various dimensions.</p>http://cvgmt.sns.it/paper/4432/Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torushttp://cvgmt.sns.it/paper/4431/G. A. Pavliotis, J. A. Carrillo, R. S. Gvalani, A. Schlichting.<p> We study the McKean-Vlasov equation \[ \partial_t \varrho= \beta^{-1} \Delta\varrho + \kappa \nabla \cdot (\varrho \nabla (W \star \varrho)) \, , \] withperiodic boundary conditions on the torus. We first study the global asymptoticstability of the homogeneous steady state. We then focus our attention on thestationary system, and prove the existence of nontrivial solutions branchingfrom the homogeneous steady state, through possibly infinitely manybifurcations, under appropriate assumptions on the interaction potential. Wealso provide sufficient conditions for the existence of continuous anddiscontinuous phase transitions. Finally, we showcase these results by applyingthem to several examples of interaction potentials such as the noisy Kuramotomodel for synchronisation, the Keller--Segel model for bacterial chemotaxis,and the noisy Hegselmann--Krausse model for opinion dynamics.</p>http://cvgmt.sns.it/paper/4431/Barriers of the McKean--Vlasov energy via a mountain pass theorem in the space of probability measureshttp://cvgmt.sns.it/paper/4430/Rishabh S. Gvalani, A. Schlichting.<p> We show that the empirical process associated to a system of weaklyinteracting diffusion processes exhibits a form of noise-induced metastability.The result is based on an analysis of the associated McKean--Vlasov freeenergy, which for suitable attractive interaction potentials has at least twodistinct global minimisers at the critical parameter value $\beta=\beta_c$. Onthe torus, one of these states is the spatially homogeneous constant state andthe other is a clustered state. We show that a third critical point exists atthis value. As a result, we obtain that the probability of transition of theempirical process from the constant state scales like $\exp(-N \Delta)$, with$\Delta$ the energy gap at $\beta=\beta_c$. The proof is based on a version ofthe mountain pass theorem for lower semicontinuous and $\lambda$-geodesicallyconvex functionals on the space of probability measures $\mathcal{P}(M)$equipped with the $W_2$ Wasserstein metric, where $M$ is a Riemannian manifoldor $\mathbb{R}^d$.</p>http://cvgmt.sns.it/paper/4430/Axisymmetric critical points of a nonlocal isoperimetric problem on the two-spherehttp://cvgmt.sns.it/paper/4429/R. Choksi, I. Topaloglu, G. Tsogtgerel.<p> On the two dimensional sphere, we consider axisymmetric critical points of anisoperimetric problem perturbed by a long-range interaction term. When theparameter controlling the nonlocal term is sufficiently large, we prove theexistence of a local minimizer with arbitrary many interfaces in theaxisymmetric class of admissible functions. These local minimizers in thisrestricted class are shown to be critical points in the broader sense (i.e.,with respect to all perturbations). We then explore the rigidity, due tocurvature effects, in the criticality condition via several quantitativeresults regarding the axisymmetric critical points.</p>http://cvgmt.sns.it/paper/4429/Convergence of regularized nonlocal interaction energieshttp://cvgmt.sns.it/paper/4427/K. Craig, I. Topaloglu.<p> Inspired by numerical studies of the aggregation equation, we study theeffect of regularization on nonlocal interaction energies. We consider energiesdefined via a repulsive-attractive interaction kernel, regularized byconvolution with a mollifier. We prove that, with respect to the 2-Wassersteinmetric, the regularized energies $\Gamma$-converge to the unregularized energyand minimizers converge to minimizers. We then apply our results to prove$\Gamma$-convergence of the gradient flows, when restricted to the space ofmeasures with bounded density.</p>http://cvgmt.sns.it/paper/4427/Existence of Ground States of Nonlocal-Interaction Energieshttp://cvgmt.sns.it/paper/4428/R. Simione, D. Slepčev, I. Topaloglu.<p> We investigate which nonlocal-interaction energies have a ground state(global minimizer). We consider this question over the space of probabilitymeasures and establish a sharp condition for the existence of ground states. Weshow that this condition is closely related to the notion of stability (i.e.$H$-stability) of pairwise interaction potentials. Our approach uses the directmethod of the calculus of variations.</p>http://cvgmt.sns.it/paper/4428/Sharp interface limit of an energy modelling nanoparticle-polymer blendshttp://cvgmt.sns.it/paper/4426/S. Alama, L. Bronsard, I. Topaloglu.<p> We identify the $\Gamma$-limit of a nanoparticle-polymer model as the numberof particles goes to infinity and as the size of the particles and the phasetransition thickness of the polymer phases approach zero. The limiting energyconsists of two terms: the perimeter of the interface separating the phases anda penalization term related to the density distribution of the infinitely manysmall nanoparticles. We prove that local minimizers of the limiting energyadmit regular phase boundaries and derive necessary conditions of localminimality via the first variation. Finally we discuss possible critical andminimizing patterns in two dimensions and how these patterns vary from globalminimizers of the purely local isoperimetric problem.</p>http://cvgmt.sns.it/paper/4426/Droplet phase in a nonlocal isoperimetric problem under confinementhttp://cvgmt.sns.it/paper/4424/S. Alama, L. Bronsard, R. Choksi, I. Topaloglu.<p> We address small volume-fraction asymptotic properties of a nonlocalisoperimetric functional with a confinement term, derived as the sharpinterface limit of a variational model for self-assembly of diblock copolymersunder confinement by nanoparticle inclusion. We introduce a small parameter$\eta$ to represent the size of the domains of the minority phase, and studythe resulting droplet regime as $\eta\to 0$. By considering confinementdensities which are spatially variable and attain a nondegenerate maximum, wepresent a two-stage asymptotic analysis wherein a separation of length scalesis captured due to competition between the nonlocal repulsive and confiningattractive effects in the energy. A key role is played by a parameter $M$ whichgives the total volume of the droplets at order $\eta^3$ and its relation toexistence and non-existence of Gamow's Liquid Drop model on $\mathbb{R}^3$. Forlarge values of $M$, the minority phase splits into several droplets at anintermediate scale $\eta^{1/3}$, while for small $M$ minimizers form a singledroplet converging to the maximum of the confinement density.</p>http://cvgmt.sns.it/paper/4424/Nonlocal Shape Optimization via Interactions of Attractive and Repulsive Potentialshttp://cvgmt.sns.it/paper/4425/A. Burchard, R. Choksi, I. Topaloglu.<p> We consider a class of nonlocal shape optimization problems for sets of fixedmass where the energy functional is given by an attractive<i>repulsiveinteraction potential in power-law form. We find that the existence ofminimizers of this shape optimization problem depends crucially on the value ofthe mass. Our results include existence theorems for large mass andnonexistence theorems for small mass in the class where the attractive part ofthe potential is quadratic. In particular, for the case where the repulsion isgiven by the Newtonian potential, we prove that there is a critical value forthe mass, above which balls are the unique minimizers, and below whichminimizers fail to exist. The proofs rely on a relaxation of the variationalproblem to bounded densities, and recent progress on nonlocal obstacleproblems.</i></p>http://cvgmt.sns.it/paper/4425/Droplet breakup in the liquid drop model with background potentialhttp://cvgmt.sns.it/paper/4422/S. Alama, L. Bronsard, R. Choksi, I. Topaloglu.<p> We consider a variant of Gamow's liquid drop model, with a general repulsiveRiesz kernel and a long-range attractive background potential with weight $Z$.The addition of the background potential acts as a regularization for theliquid drop model in that it restores the existence of minimizers for arbitrarymass. We consider the regime of small $Z$ and characterize the structure ofminimizers in the limit $Z\to 0$ by means of a sharp asymptotic expansion ofthe energy. In the process of studying this limit we characterize allminimizing sequences for the Gamow model in terms of "generalized minimizers".</p>http://cvgmt.sns.it/paper/4422/Ground-states for the liquid drop and TFDW models with long-range attractionhttp://cvgmt.sns.it/paper/4423/S. Alama, L. Bronsard, R. Choksi, I. Topaloglu.<p> We prove that both the liquid drop model in $\mathbb{R}^3$ with an attractivebackground nucleus and the Thomas-Fermi-Dirac-von Weizs\"{a}cker (TFDW) modelattain their ground-states \emph{for all} masses as long as the externalpotential $V(x)$ in these models is of long range, that is, it decays slowerthan Newtonian (e.g., $V(x)\gg <br>x<br>^{-1}$ for large $<br>x<br>$.) For the TFDW modelwe adapt classical concentration-compactness arguments by Lions, whereas forthe liquid drop model with background attraction we utilize a recentcompactness result for sets of finite perimeter by Frank and Lieb.</p>http://cvgmt.sns.it/paper/4423/