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regularity for solutions of the Monge-Ampère equation
Guido De Philippis, Alessio Figalli
In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, is .…
Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case
Luigi Ambrosio, Maria Colombo, Guido De Philippis +1
In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global…
Total Variation Flow and Sign Fast Diffusion in one dimension
Matteo Bonforte, Alessio Figalli
We consider the dynamics of the Total Variation Flow (TVF) and of the Sign Fast Diffusion Equation (SFDE) $u_t=Δ\sign(u)$ in one spatial dimension. We find the exp…
Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller-Segel equation
Eric A. Carlen, Alessio Figalli
Starting from the quantitative stability result of Bianchi and Egnell for the 2-Sobolev inequality, we deduce several different stability results for a Gagliardo-Nirenberg-Sobolev…
On supporting hyperplanes to convex bodies
Alessio Figalli, Young-Heon Kim, Robert J. McCann
Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quan…
Hölder continuity and injectivity of optimal maps
Alessio Figalli, Young-Heon Kim, Robert J. McCann
Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth…