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Regularity of interfaces in phase transitions via obstacle problems
Alessio Figalli
The aim of this note is to review some recent developments on the regularity theory for the stationary and parabolic obstacle problems. After a general overview, we present some re…
Free boundary regularity in obstacle problems
Alessio Figalli
These notes record and expand the lectures for the `Journées Équations aux Dérivées Partielles 2018' held by the author during the week of June 11-15, 2018. The aim is to give a ov…
An overview of unconstrained free boundary problems
Alessio Figalli, Henrik Shahgholian
In this paper we present a survey concerning unconstrained free boundary problems of type $$ \left\{ \begin{array}{ll} F_1(D^2u,\nabla u,u,x)=0 & \text{in }B_1 \cap Ω,\\ F_2 (D^2 u…
On the Monge-Ampère equation
Alessio Figalli
This text contains the material discussed by the author in the Bourbaki seminar of June 2018, on the recent developments in the theory of the Monge-Ampère equation.
On the Continuity of Center-Outward Distribution and Quantile Functions
Alessio Figalli
To generalize the notion of distribution function to dimension , in the recent papers it was proposed a concept of center-outward distribution function based on optimal tr…