Optimal Transport: A Semi-Discrete Approach

Sets of Finite Perimeter and Geometric Variational Problems: Equilibrium shapes of liquids and sessile drops

When writing can change your life, when writing can enrich you by offering much money, why don't you try it? Are you still very confused of where getting the ideas? Do you still have no idea with what you are going to write? Now, you will need reading. A good writer is a good reader at once. You can define how you write depending on what books to read. This functions of bounded variation and free discontinuity problems can help you to solve the problem. It can be one of the right sources to develop your writing skill.

書評 L.Ambrosio, N.Fusco and D.Pallara: Functions of Bounded Variation and Free Discontinuity Problems

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces. 
In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence.

/pdf/non-collapsed-spaces-with-ricci-curvature-bounded-from-below-4agdkmx62v.pdf

Non-collapsed spaces with Ricci curvature bounded from below

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable versions of Gargliardo's seminorms.

Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows

We continue the study of the space $BV^\alpha(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha\in(0,1)$ introduced in arXiv:1809.08575, by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of our previous work, we prove that the fractional $\alpha$-variation converges to the standard De Giorgi's variation both pointwise and in the $\Gamma$-limit sense as $\alpha\to1^-$. We also prove that the fractional $\beta$-variation converges to the fractional $\alpha$-variation both pointwise and in the $\Gamma$-limit sense as $\beta\to\alpha^-$ for any given $\alpha\in(0,1)$.

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

This note is devoted to the study of sets of finite perimeter over RCD$(K,N)$ metric measure spaces. Its aim is to complete the picture about the generalization of De Giorgi's theorem within this framework. Starting from the results of [2] we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss-Green integration by parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.

Elia Bruè

Papers

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

Constancy of the dimension for RCD(K,N) spaces via regularity of Lagrangian flows

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

Rectifiability of the reduced boundary for sets of finite perimeter over $\RCD(K,N)$ spaces.