arXiv subjects

Simone Vincini

Publications and source records attributed to Simone Vincini.

3 recordsLinked to original sources

arXiv2026

Pyramids and Extended Metric Measure Spaces

A pyramid is a generalisation of metric measure spaces (mm-spaces) introduced by M.~Gromov (Birkh\"auser 1999) to establish a geometric framework for measure-concentration problems. An extended metric measure space (emm-space) is another generalisation of mm-spaces introduced by Ambrosio--Gigli--Savar\'e (Invent.~Math.~2014) to extend Sobolev calculus and optimal transport theory to a broader extent. We prove that every pyramid has a representation by an emm-space through $1$-Lipschitz order. Furthermore, if pyramids are concentrated, this correspondence is unique up to isomorphism. This shows, for the first time, that all pyramids can be realised by concrete geometric spaces. Based on this representation, we introduce a new notion, a concentrated emm-space. We then define the observable distance for concentrated emm-spaces and establish three equivalent characterisations of concentration in terms of pyramids, Lipschitz observables and the observable distance. Furthermore, by developing an fibration approach, we show the $\Gamma$-$\limsup$ inequality of Cheeger energies as well as the stability of the log-Sobolev inequality and the Poincar\'e inequality under the weak convergence of pyramids associated with emm-spaces. Our results provide a new geometric approach for studying both the convergence of emm-spaces and concentration-of-measure phenomena across a broad class of infinite-dimensional models that have so far remained largely beyond the reach of existing geometric methods. Many of the significant examples arise in probability theory, including the Wiener space, the configuration space, Gaussian fields such as massive Gaussian free fields, spatial white noise and massive bi-Laplacian fields.

math.MG↗

arXiv2024

Stability of the heat flow under convergence in concentration and consequences

We extend to the framework of convergence in concentration virtually all the results concerning stability of Sobolev functions and differential operators known to be in place under the stronger measured-Gromov-Hausdorff convergence. These include, in particular: i) A general $\Gamma$--$\varlimsup$ inequality for the Cheeger energy, ii) Convergence of the heat flow under a uniform ${\sf CD}(K,\infty)$ condition on the spaces. As we will show, building on ideas developed for mGH-convergence, out of this latter result we can obtain clean stability statements for differential, flows of vector fields, Hessian, eigenvalues of the Laplacian and related objects in presence of uniform lower Ricci bounds. At the technical level, one of the tools we develop to establish these results is the notion of convergence of maps between metric measure spaces converging in concentration. Previous results in the field concerned the stability of the ${\sf CD}$ (Funano-Shioya '13) and ${\sf RCD}$ (Ozawa-Yokota '19) conditions. The latter was obtained proving $\Gamma$-convergence for the Cheeger energies. This is not sufficient to pass to the limit in the heat flow; one of the byproducts of our work is the improvement of this to Mosco-convergence.

math.MG↗

arXiv2024

Elastic bodies with kinematic constraints on many small regions

We study the equilibrium of hyperelastic solids subjected to kinematic constraints on many small regions, which we call perforations. Such constraints on the displacement $u$ are given in the quite general form $u(x) \in F_x$, where $F_x$ is a closed set, and thus comprise confinement conditions, unilateral constraints, controlled displacement conditions, etc., both in the bulk and on the boundary of the body. The regions in which such conditions are active are assumed to be so small that they do not produce an overall rigid constraint, but still large enough so as to produce a non-trivial effect on the behaviour of the body. Mathematically, this is translated in an asymptotic analysis by introducing two small parameters: $\varepsilon$, describing the distance between the elements of the perforation, and $\delta$, the size of the element of the perforation. We find the critical scale at which the effect of the perforation is non-trivial and express it in terms of a $\Gamma$-limit in which the constraints are relaxed so that, in their place, a penalization term appears in the form of an integral of a function $\varphi(x,u)$. This function is determined by a blow-up procedure close to the perforation and depends on the shape of the perforation, the constraint $F_x$, and the asymptotic behaviour at infinity of the strain energy density $\sigma$. We give a concise proof of the mathematical result and numerical studies for some simple yet meaningful geometries.

math.AP↗