cvgmt Papershttps://cvgmt.sns.it/papers/en-usTue, 10 Sep 2024 07:39:40 +0000A numerical method for regularized transportation problemshttps://cvgmt.sns.it/paper/6774/J. D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyré.<p>In this short note, we discuss several results we have obtained in our Frontiers of Science Award (FSA) paper. In particular we present a numerical method based on iterative Bregman projections and its impact on computational optimal transport.</p>https://cvgmt.sns.it/paper/6774/Strong unique continuation and local asymptotics at the boundary for fractional elliptic equationshttps://cvgmt.sns.it/paper/6772/A. De Luca, V. Felli, S. Vita.<p>We study local asymptotics of solutions to fractional elliptic equations atboundary points, under some outer homogeneous Dirichlet boundary condition. Ouranalysis is based on a blow-up procedure which involves some Almgren typemonotonicity formulae and provides a classification of all possible homogeneitydegrees of limiting entire profiles. As a consequence, we establish a strongunique continuation principle from boundary points.</p>https://cvgmt.sns.it/paper/6772/Unique continuation from the edge of a crackhttps://cvgmt.sns.it/paper/6773/A. De Luca, V. Felli.<p>In this work we develop an Almgren type monotonicity formula for a class ofelliptic equations in a domain with a crack, in the presence of potentialssatisfying either a negligibility condition with respect to the inverse-squareweight or some suitable integrability properties. The study of the Almgrenfrequency function around a point on the edge of the crack, where the domain ishighly non-smooth, requires the use of an approximation argument, based on theconstruction of a sequence of regular sets which approximate the crackeddomain. Once a finite limit of the Almgren frequency is shown to exist, ablow-up analysis for scaled solutions allows us to prove asymptotic expansionsand strong unique continuation from the edge of the crack.</p>https://cvgmt.sns.it/paper/6773/Strong unique continuation from the boundary for the spectral fractional Laplacianhttps://cvgmt.sns.it/paper/6770/A. De Luca, V. Felli, G. Siclari.<p>We investigate unique continuation properties and asymptotic behaviour atboundary points for solutions to a class of elliptic equations involving thespectral fractional Laplacian. An extension procedure leads us to study adegenerate or singular equation on a cylinder, with a homogeneous Dirichletboundary condition on the lateral surface and a non homogeneous Neumanncondition on the basis. For the extended problem, by an Almgren-typemonotonicity formula and a blow-up analysis, we classify the local asymptoticprofiles at the edge where the transition between boundary conditions occurs.Passing to traces, an analogous blow-up result and its consequent strong uniquecontinuation property is deduced for the nonlocal fractional equation.</p>https://cvgmt.sns.it/paper/6770/Nonlocal capillarity for anisotropic kernelshttps://cvgmt.sns.it/paper/6771/A. De Luca, S. Dipierro, E. Valdinoci.<p>We study a nonlocal capillarity problem with interaction kernels that arepossibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance will be modeled via two differentfractional exponents $s_1, s_2\in (0,1)$ which take into account thepossibility that the container and the environment present different featureswith respect to particle interactions. We determine a nonlocal Young's law for the contact angle and discuss theunique solvability of the corresponding equation in terms of the interactionkernels and of the relative adhesion coefficient.</p>https://cvgmt.sns.it/paper/6771/Unique continuation from conical boundary points for fractional equationshttps://cvgmt.sns.it/paper/6768/A. De Luca, V. Felli, S. Vita.<p> We provide fine asymptotics of solutions of fractional elliptic equations atboundary points where the domain is locally conical; that is, corner typesingularities appear. Our method relies on a suitable smoothing of the cornersingularity and an approximation scheme, which allow us to provide a Pohozaevtype inequality. Then, the asymptotics of solutions at the conical point followby an Almgren type monotonicity formula, blow-up analysis and Fourierdecomposition on eigenspaces of a spherical eigenvalue problem. A strong uniquecontinuation principle follows as a corollary.</p>https://cvgmt.sns.it/paper/6768/Layered patterns in reaction-diffusion models with Perona-Malik diffusionshttps://cvgmt.sns.it/paper/6769/A. De Luca, R. Folino, M. Strani.<p> In this paper we deal with a reaction-diffusion equation in a boundedinterval of the real line with a nonlinear diffusion of Perona-Malik's type anda balanced bistable reaction term. Under very general assumptions, we study thepersistence of layered solutions, showing that it strongly depends on thebehavior of the reaction term close to the stable equilibria $\pm1$, describedby a parameter $\theta>1$. If $\theta\in(1,2)$, we prove existence of steadystates oscillating (and touching) $\pm1$, called $compactons$, while in thecase $\theta=2$ we prove the presence of $metastable$ $solutions$, namelysolutions with a transition layer structure which is maintained for anexponentially long time. Finally, for $\theta>2$, solutions with an unstabletransition layer structure persist only for an algebraically long time.</p>https://cvgmt.sns.it/paper/6769/A note on geometric assumptions for unique continuation from the edge of a crackhttps://cvgmt.sns.it/paper/6767/A. De Luca.<p> The present paper aims at representing an improvement of the result in <a href='2'>2</a>,where a strong unique continuation property and a description of the localbehaviour around the edge of a crack for solutions to an elliptic problem areestablished, by relaxing the star-shapedness condition on the complement of thecrack. More specifically, this assumption will be dropped off by applying asuitable diffeomorphism which straightens the boundary of the crack, beforeperforming the approximation procedure developed in <a href='2'>2</a> in order to derive asuitable monotonicity formula. This will yield the appearence of a matrix inthe equation, which shall be handled appropriately: for this we will take ahint from <a href='4'>4</a>.</p>https://cvgmt.sns.it/paper/6767/A Note on Ricci-pinched three-manifoldshttps://cvgmt.sns.it/paper/6766/L. Benatti, C. Mantegazza, F. Oronzio, A. Pluda.<p>Let $(M, g)$ be a complete, connected, non-compact Riemannian $3$-manifold. Suppose that $(M,g)$ satisfies the <i>Ricci-pinching condition</i> $\mathrm{Ric}\geqslant\varepsilon\mathrm{R} g$ for some $\varepsilon>0$, where $\mathrm{Ric}$ and $\mathrm{R}$ are the Ricci tensor and scalar curvature, respectively. In this short note, we give an alternative proof based on potential theory of the fact that if $(M,g)$ has Euclidean volume growth, then it is flat. Deruelle-Schulze-Simon and by Huisken-Koerber have already shown this result and together with the contributions by Lott and Lee-Topping led to a proof of the so-called <i>Hamilton's pinching conjecture</i>.</p>https://cvgmt.sns.it/paper/6766/Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stabilityhttps://cvgmt.sns.it/paper/6765/A. Kristály, A. Mondino.<p>We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature and dimension bounded above by N>1 in the synthetic sense. The initial conjecture -- an isoperimetric inequality for the principal frequency of clamped plates -- has been formulated in 1877 by Lord Rayleigh in the Euclidean case and solved affirmatively in dimensions 2 and 3 by Ashbaugh and Benguria <a href='Duke Math. J., 1995'>Duke Math. J., 1995</a> and Nadirashvili <a href='Arch. Rat. Mech. Anal., 1995'>Arch. Rat. Mech. Anal., 1995</a>. The main contribution of the present work is a new isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces whenever N is close enough to 2 or 3. The inequality contains the so-called ``asymptotic volume ratio", and turns out to be sharp under the subharmonicity of the distance function, a condition satisfied in metric measure cones. In addition, rigidity (i.e., equality in the isoperimetric inequality) and stability results are established in terms of the cone structure of the RCD(0,N) space as well as the shape of the eigenfunction for the principal frequency, given by means of Bessel functions. These results are new even for Riemannian manifolds with non-negative Ricci curvature. We discuss examples of both smooth and non-smooth spaces where the results can be applied.</p>https://cvgmt.sns.it/paper/6765/Liquid drop with capillarity and rotating traveling waveshttps://cvgmt.sns.it/paper/6764/P. Baldi, V. Julin, D. A. La Manna.<p> We consider the free boundary problem for a 3-dimensional, incompressible,irrotational liquid drop of nearly spherical shape with capillarity. We studythe problem from scratch, extending some classical results from the flat case(capillary water waves) to the spherical geometry: the reduction to a problemon the boundary, its Hamiltonian structure, the analyticity and tame estimatesfor the Dirichlet-Neumann operator in Sobolev class, and a linearizationformula for it, both with the method of the good unknown of Alinhac and by adifferential geometry approach. Then we prove the bifurcation of travelingwaves, which are nontrivial (i.e., nonspherical) fixed profiles rotating withconstant angular velocity.</p>https://cvgmt.sns.it/paper/6764/Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaceshttps://cvgmt.sns.it/paper/6763/M. Santilli, P. Valentini.<p>In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.</p>https://cvgmt.sns.it/paper/6763/Quantum optimal transport with convex regularizationhttps://cvgmt.sns.it/paper/6762/E. Caputo, A. Gerolin, N. Monina, L. Portinale.<p> The goal of this paper is to settle the study of non-commutative optimaltransport problems with convex regularization, in their static andfinite-dimensional formulations. We consider both the balanced and unbalancedproblem and show in both cases a duality result, characterizations ofminimizers (for the primal) and maximizers (for the dual). An important tool wedefine is a non-commutative version of the classical $(c,\psi)$-transformsassociated with a general convex regularization, which we employ to prove theconvergence of Sinkhorn iterations in the balanced case. Finally, we show theconvergence of the unbalanced transport problems towards the balanced one, aswell as the convergence of transforms, as the marginal penalization parametersgo to $+\infty$.</p>https://cvgmt.sns.it/paper/6762/Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currentshttps://cvgmt.sns.it/paper/6761/P. Minter, D. Parise, A. Skorobogatova, L. Spolaor.<p>We prove that the singular set of an $m$-dimensional integral current $T$ in $\mathbb{R}^{n + m}$, semicalibrated by a $C^{2, \kappa_0}$ $m$-form $\omega$ is countably $(m - 2)$-rectifiable. Furthermore, we show that there is a unique tangent cone at $\mathcal{H}^{m - 2}$-a.e. point in the interior singular set of $T$. Our proof adapts techniques that were recently developed in \cite{DLSk1, DLSk2, DMS} for area-minimizing currents to this setting.</p>https://cvgmt.sns.it/paper/6761/On De Giorgi's lemma for variational interpolants in metric and Banach spaceshttps://cvgmt.sns.it/paper/6760/A. Mielke, R. Rossi.<p> Variational interpolants are an indispensable tool for the construction ofgradient-flow solutions via the Minimizing Movement Scheme. De Giorgi's lemmaprovides the associated discrete energy-dissipation inequality. It wasoriginally developed for metric gradient systems. Drawing from this theory westudy the case of generalized gradient systems in Banach spaces, where arefined theory allows us to extend the validity of the discreteenergy-dissipation inequality and to establish it as an equality. For thelatter we have to impose the condition of radial differentiability of thedissipation potential. Several examples are discussed to show how sharp theresults are.</p>https://cvgmt.sns.it/paper/6760/Existence and weak-strong uniqueness for damage systems in viscoelasticityhttps://cvgmt.sns.it/paper/6759/R. Lasarzik, E. Rocca, R. Rossi.<p> In this paper we investigate the existence of solutions and their weak-stronguniqueness property for a PDE system modelling damage in viscoelasticmaterials. In fact, we address two solution concepts, weak and strongsolutions. For the former, we obtain a global-in-time existence result, but thehighly nonlinear character of the system prevents us from proving theiruniqueness. For the latter, we prove local-in-time existence. Then, we showthat the strong solution, as long as it exists, is unique in the class of weaksolutions. This weak-strong uniqueness statement is proved by means of asuitable relative energy inequality.</p>https://cvgmt.sns.it/paper/6759/Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Fieldhttps://cvgmt.sns.it/paper/6758/E. Bruè, M. Colombo, A. Kumar.<p>Given a divergence-free vector field ${\bf u} \in L^\infty_t W^{1,p}_x(\mathbb R^d)$ and a nonnegative initial datum $\rho_0 \in L^r$, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of $L^\infty_t L^r_x$ densities for $\frac{1}{p} + \frac{1}{r} \leq 1$. This range was later improved in \cite{BrueColomboDeLellis21} to $\frac{1}{p} + \frac{d-1}{dr} \leq 1$.We prove that this range is sharp by providing a counterexample to uniqueness when $\frac{1}{p} + \frac{d-1}{dr} > 1$.</p><p>To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization.</p>https://cvgmt.sns.it/paper/6758/Flexibility of Two-Dimensional Euler Flows with Integrable Vorticityhttps://cvgmt.sns.it/paper/6757/E. Bruè, M. Colombo, A. Kumar.<p>We propose a new convex integration scheme in fluid mechanics, and we provide an application to the two-dimensional Euler equations. We prove the flexibility and nonuniqueness of $L^\infty L^2$ weak solutions with vorticity in $L^\infty L^p$ for some $p>1$, surpassing for the first time the critical scaling of the standard convex integration technique.</p><p>To achieve this, we introduce several new ideas, including:\begin{itemize} \item<a href='(i)'>(i)</a> A new family of building blocks built from the Lamb-Chaplygin dipole. \item<a href='(ii)'>(ii)</a> A new method to cancel the error based on time averages and non-periodic, spatially-anisotropic perturbations.\end{itemize}</p>https://cvgmt.sns.it/paper/6757/Thermo-elastodynamics of nonlinearly viscous solidshttps://cvgmt.sns.it/paper/6756/S. Almi, R. Badal, M. Friedrich, S. Schwarzacher.<p>In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity <a href='35, 2'>35, 2</a> with a variational approach to hyperbolic PDEs developed in <a href='5'>5</a>, our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order p-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.</p>https://cvgmt.sns.it/paper/6756/Stability of the Von Kármán regime for thin plates under Neumann boundary conditionshttps://cvgmt.sns.it/paper/6755/E. G. Tolotti.<p> We analyze the stability of the Von K\'arm\'an model for thin plates subjectto pure Neumann conditions and to dead loads, with no restriction on theirdirection. We prove a stability alternative, which extends previous results byLecumberry and M\"uller in the Dirichlet case. Because of the rotationinvariance of the problem, their notions of stability have to be modified andcombined with the concept of optimal rotations due to Maor and Mora. Finally,we prove that the Von K\'arm\'an model is not compatible with some specifictypes of forces. Thus, for such, only the Kirchoff model applies.</p>https://cvgmt.sns.it/paper/6755/