Weak formulation of the MTW condition and convexity properties of potentials

c-凸函数的一个替代定义及MTW条件的另一综合表述

An alternative definition for c-convex functions and another synthetic statement of MTW condition

本文提出c-凸函数的替代定义，证明其与标准c-凸性等价当且仅当代价函数满足MTW条件，并研究了相关性质。

Time-convexity of the entropy in the multiphasic formulation of the incompressible euler equation

We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global in-compressibility constraint. We are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savar{é} to study first order in time PDE, namely the …

A variational principle for problems with a hint of convexity

A variational principle is introduced to provide a new formulation and resolution for several boundary value problems with a variational structure. This principle allows one to deal with problems well beyond the weakly compact structure. As a result, we study several super-critical semilinear Elliptic problems.

Convexity and the Hele-Shaw equation

Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the Dirichlet-to-Neumann operator, normal forms transformations. In this paper, we introduce several identities for the Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we study convex changes of unknowns and obtain a large class of strong Lyapounov functions; in addition to be non-increasing, these Lyapounov functions are convex functions of time. The analysis relies on a simple elliptic formul…

Hidden convexity in a problem of nonlinear elasticity

We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.

Hellinger–Kantorovich 距离的测地线精细性质与测地 $\lambda$-凸性

Fine properties of geodesics and geodesic $λ$-convexity for the Hellinger-Kantorovich distance

本文研究 HK 问题最优势与对偶动态方程的正则性，构造输运–膨胀流刻画测地插值，并给出泛函沿 HK 测地线 $\lambda$-凸的精确条件。