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Simone Di Marino

Publications and source records attributed to Simone Di Marino.

31 records · Page 2Linked to original sources
arXiv2018

Infinitesimal Hilbertianity of locally CAT($κ$)-spaces

We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $μ$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,μ)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,μ)$ is a Hilbert space. The result is obtained by constructing an isometric embedding of the `abstract and analytical' space of derivations into the `concrete and geometrical' bundle whose fibre at $x\in Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $x\in Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.

math.MG↗
arXiv2018

The entropic regularization of the Monge problem on the real line

We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem, the most 'diffuse' one on the zones where it may have a density.

math.OC↗
arXiv2018

New characterizations of Sobolev metric spaces

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.

math.AP↗
arXiv2017

A tumor growth model of Hele-Shaw type as a gradient flow

In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity assumption on the initial condition. We also derive a numerical scheme based on the discretization of the gradient flow and the idea of entropic regularization. We assess the convergence of the scheme on explicit solutions. In doing this analysis, we prove several new properties of the optimal transport-growth metric, which generally have a known counterpart for the Wasserstein metric.

math.AP↗
arXiv2017

Uniqueness issues for evolution equations with density constraints

In this paper we present some basic uniqueness results for evolutive equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first order systems modeling crowd motion with hard congestion effects, introduced recently by \emph{Maury et al.} The monotonicity of the velocity field implies that the $2-$Wasserstein distance along two solutions is $λ$-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^1-$contraction property. In this case, by the regularization effect of the non-degenerate diffusion, the result follows even if the given velocity field is only $L^\infty$ as in the standard Fokker-Planck equation.

math.AP↗
arXiv2017

The strictly-correlated electron functional for spherically symmetric systems revisited

The strong-interaction limit of the Hohenberg-Kohn functional defines a multimarginal optimal transport problem with Coulomb cost. From physical arguments, the solution of this limit is expected to yield strictly-correlated particle positions, related to each other by co-motion functions (or optimal maps), but the existence of such a deterministic solution in the general three-dimensional case is still an open question. A conjecture for the co-motion functions for radially symmetric densities was presented in Phys.~Rev.~A {\bf 75}, 042511 (2007), and later used to build approximate exchange-correlation functionals for electrons confined in low-density quantum dots. Colombo and Stra [Math.~Models Methods Appl.~Sci., {\bf 26} 1025 (2016)] have recently shown that these conjectured maps are not always optimal. Here we revisit the whole issue both from the formal and numerical point of view, finding that even if the conjectured maps are not always optimal, they still yield an interaction energy (cost) that is numerically very close to the true minimum. We also prove that the functional built from the conjectured maps has the expected functional derivative also when they are not optimal.

cond-mat.str-el↗
arXiv2015

Optimal Transportation Theory with Repulsive Costs

This paper intents to present the state of art and recent developments of the optimal transportation theory with many marginals for a class of repulsive cost functions. We introduce some aspects of the Density Functional Theory (DFT) from a mathematical point of view, and revisit the theory of optimal transport from its perspective. Moreover, in the last three sections, we describe some recent and new theoretical and numerical results obtained for the Coulomb cost, the repulsive harmonic cost and the determinant cost.

math.AP↗
arXiv2014

On the duality between p-Modulus and probability measures

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and probability measures with barycenter in $L^q(X,\mathfrak m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)

math.FA↗
arXiv2014

The p-Weak Gradient Depends on p

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for small p but non trivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on the real line.

math.FA↗
arXiv2012

Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope

In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on $Γ$-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of $\mm$. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.

math.AP↗