A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.

Metric completion of $Diff([0,1])$ with the $H1$ right-invariant metric

We consider the group of smooth increasing diffeomorphisms Diff on the unit interval endowed with the right-invariant $H^1$ metric. We compute the metric completion of this space which appears to be the space of increasing maps of the unit interval with boundary conditions at $0$ and $1$. We compute the lower-semicontinuous envelope associated with the length minimizing geodesic variational problem. We discuss the Eulerian and Lagrangian formulation of this relaxation and we show that smooth solutions of the EPDiff equation are length minimizing for short times.

Infinitesimal Hilbertianity of locally CAT($kappa$)-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 $mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,mu)$ 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 $xin Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $xin Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.

An entropy minimization approach to second-order variational mean-field games

We propose a new viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We show the equivalence of such mean-field games with a relative entropy minimization at the level of probabilities on curves. We also address the time-discretization of such problems, establish $Gamma$-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

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.

Linear Lipschitz and $C^1$ extension operators through random projection

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces.

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.

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.

Perimeter as relaxed Minkowski content in metric measure spaces

In this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the Minkowski content w.r.t. convergence in measure

The adiabatic strictly-correlated-electrons functional: kernel and exact properties

We investigate a number of formal properties of the adiabatic strictly-correlated electrons (SCE) functional, relevant for time-dependent potentials and for kernels in linear response time-dependent density functional theory. Among the former, we focus on the compliance to constraints of exact many-body theories, such as the generalised translational invariance and the zero-force theorem. Within the latter, we derive an analytical expression for the adiabatic SCE Hartree exchange-correlation kernel in one dimensional systems, and we compute it numerically for a variety of model densities. We analyse the non-local features of this kernel, particularly the ones that are relevant in tackling problems where kernels derived from local or semi-local functionals are known to fail.