199 citations · 522 across the 42 of their papers we have counts for
5 papers · 1 filter
The Monge-Ampère equation and its link to optimal transportation
Guido De Philippis, Alessio Figalli
We survey the (old and new) regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general clas…
Rigidity of equality cases in Steiner's perimeter inequality
Filippo Cagnetti, Maria Colombo, Guido De Philippis +1
Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equalit…
Faber-Krahn inequalities in sharp quantitative form
Lorenzo Brasco, Guido De Philippis, Bozhidar Velichkov
The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. In this paper we prove a s…
Higher integrability for minimizers of the Mumford-Shah functional
Guido De Philippis, Alessio Figalli
We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi.
Existence and regularity of minimizers for some spectral functionals with perimeter constraint
Guido De Philippis, Bozhidar Velichkov
In this paper we prove that the shape optimization problem has a solution for any and…