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A general class of free boundary problems for fully nonlinear elliptic equations
A. Figalli, H. Shahgholian
In this paper we study the fully nonlinear free boundary problem $$ {{array}{ll} F(D^2u)=1 & \text{a.e. in}B_1 \cap Ω|D^2 u| \leq K & \text{a.e. in}B_1\setminusΩ, {array}. $$ where…
Sobolev regularity for Monge-Ampère type equations
Guido De Philippis, Alessio Figalli
In this note we prove that, if the cost function satisfies some necessary structural conditions and the densities are bounded away from zero and infinity, then strictly -convex…
A sharp stability result for the relative isoperimetric inequality inside convex cones
Alessio Figalli, Emanuel Indrei
The relative isoperimetric inequality inside an open, convex cone states that, at fixed volume, minimizes the perimeter inside . Star…
Partial Regularity for optimal transport maps
Guido De Philippis, Alessio Figalli
We prove that, for general cost functions on , or for the cost on a Riemannian manifold, optimal transport maps between smooth densities are always smooth out…
On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces
Luigi Ambrosio, Alessio Figalli, Eris Runa
We study sets of finite perimeter in Wiener space, and prove that at almost every point (with respect to the perimeter measure) a set of finite perimeter blows-up to a halfspace.
A global existence result for the semigeostrophic equations in three dimensional convex domains
Luigi Ambrosio, Maria Colombo, Guido De Philippis +1
Exploiting recent regularity estimates for the Monge-Ampère equation, under some suitable assumptions on the initial data we prove global-in-time existence of Eulerian distribution…