The so-called Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase change, for example ice passing to water, and one aims to describe the regularity of the interface separating the two phases. In its stationary version, the Stefan problem can be reduced to the classical obstacle problem, which consists in finding the equilibrium position of an elastic membrane whose boundary is held fixed and that is constrained to lie above a given obstacle. The aim of this talk is to give a general overview of the classical theory of the obstacle problem, and then discuss some recent developments on the optimal regularity of interface both in the static and the parabolic setting.
Alessio Figalli received his master's degree in mathematics from the Scuola Normale Superiore di Pisa in 2006, and earned his doctorate in 2007 under the supervision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa and Cédric Villani at the École Normale Supérieure de Lyon. In 2007 he was appointed Chargé de recherche at the French National Centre for Scientific Research, in 2008 he went to the École polytechnique as Professeur Hadamard. In 2009 he moved to the University of Texas at Austin as Associate Professor. Then he became Full Professor in 2011, and R. L. Moore Chair holder in 2013. Since 2016, he is a chaired professor at ETH Zürich.
Amongst his several recognitions, Alessio Figalli has won an EMS Prize in 2012, he has been awarded the Peccot-Vimont Prize 2011 and Cours Peccot 2012 of the Collège de France and has been appointed Nachdiplom Lecturer in 2014 at ETH Zürich. He has won the 2015 edition of the Stampacchia Medal, and the 2017 edition of the Feltrinelli Prize for Mathematics.
Alessio Figalli works in the broad areas of Calculus of Variations and Partial Differential Equations, with particular emphasis on optimal transportation, Monge-Ampère equations, functional and geometric inequalities, elliptic PDEs of local and non-local type, free boundary problems, Hamilton-Jacobi equations, transport equations with rough vector-fields, and random matrix theory. Amongst the several fundamental results that he obtained in this areas, stand out the ones on the regularity of the second derivatives of solutions to the Monge–Ampère equations (together with Guido De Philippis), a sharp quantitative version of the anisotropic isoperimetric inequality (together with Francesco Maggi and Aldo Pratelli), the proof of De Giorgi's conjecture for boundary reaction terms in dimension ≤ 5 (together with Joaquim Serra), the proof of the generic hyperbolicity of Aubry sets on compact surfaces (in collaboration with Gonzalo Contreras and Ludovic Rifford).
Alessio has been invited speaker at the at the European Congress of Mathematics (ECM) at Krakow, in 2012, and at the International Congress of Mathematicians (ICM) at Seoul (2014). He serves in the editorial board of many leading publications in the field.
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