Well-posedness of Wasserstein Gradient Flow Solutions of Higher Order Evolution Equations

A relaxed notion of displacement convexity is defined and used to establish short time existence and uniqueness of Wasserstein gradient flows for higher order energy functionals. As an application, local and global well-posedness of different higher order non-linear evolution equations are derived. Examples include the thin-film equation and the quantum drift diffusion equation in one spatial variable

Vector-valued optimal Lipschitz extensions

Consider a bounded open set $U$ in $R^n$ and a Lipschitz function g from the boundary of $U$ to $R^m$. Does this function always have a canonical optimal Lipschitz extension to all of $U$? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case $n=m=2$, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.Comment: 24 pages, 10 figure

Existence and uniqueness for a crystalline mean curvature flow

An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach

Embedding SnS^n into Rn+1R^{n+1} with given integral Gauss curvature and optimal mass transport on SnS^n

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations