On the convexity theory of generating functions

Displacement convexity for first-order mean-field games

Here, we consider the planning problem for first-order mean-field games (MFG). When there is no coupling between players, MFG degenerate into optimal transport problems. Displacement convexity is a fundamental tool in optimal transport that often reveals hidden convexity of functionals and, thus, has numerous applications in the calculus of variations. We explore the similarities between the Benamou-Brenier formulation of optimal transport and MFG to extend displacement convexity methods from to MFG. In particular, we identify a class of functions, that depend on solutions of MFG, that are con…

On strict convexity and $C^1$ regularity of potential functions in optimal transportation

This note concerns the relationship between conditions on cost functions and domains and the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if the cost function satisfies the condition (A3), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function. Our result extends, by d…

Heisenberg群中水平拟凸性的二阶算子及其在水平曲率流保凸性中的应用

A second-order operator for horizontal quasiconvexity in the Heisenberg group and application to convexity preserving for horizontal curvature flow

本文提出基于非线性二阶椭圆算子的Heisenberg群水平拟凸函数PDE方法，给出粘性下解的充分必要条件，并证明水平曲率流在演化中保持h-凸性。

Observation estimate for the heat equations with Neumann boundary condition via logarithmic convexity

We prove an inequality of Hölder type traducing the unique continuation property at one time for the heat equation with a potential and Neumann boundary condition. The main feature of the proof is to overcome the propagation of smallness by a global approach using a refined parabolic frequency function method. It relies with a Carleman commutator estimate to obtain the logarithmic convexity property of the frequency function.

Convexity criteria and uniqueness of absolutely minimizing functions

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$

Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

It is known that the $L^{2}$-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the $L^{2}$-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular functions and obtain a differential inequality for the $L^{2}$-norms of harmonic functions over these hypersurfaces. To illustrate our result, we consider ellipses with constant eccentricity and growing tori in $\mathbf{R}^3.$ Moreover, we give a new proof of the convexity resul…

Convexity estimate for translating solitons of concave fully nonlinear extrinsic geometric flows in $\mathbb{R}^{n+1}$

The main result of this paper is a convexity estimate for translating solitons of extrinsic geometric flows which evolve under a $1$-homogeneous concave function in the principal curvatures. In addition, we show examples of these hypersurfaces in $\mathbb{R}^{n+1}$ for particular functions.

The effect of curvature on convexity properties of harmonic functions and eigenfunctions

We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.

Wasserstein空间子流形中的测地凸性与强化泛函不等式

Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space

本文利用EVI梯度流保持子流形不变的度量空间一般原理，研究Wasserstein空间子流形上能量与熵泛函的测地凸性，得到若干新凸性结果并改进Talagrand和HWI不等式。

关于热力学函数凸性与凹性关系的注记

A note on relations between convexity and concavity of thermodynamic functions

本文证明了不同变量集合下热力学函数凸性、凹性等性质的等价关系，该关系不依赖状态方程，可用于验证Euler方程熵密度的凹性等，为热力学相关数学分析提供支撑。