Stable solutions to semilinear elliptic PDEs appear in several problems. It is known since the 1970’s that, in dimension n > 9, there exist singular stable solutions. In this talk I will describe a recent work with Cabré, Ros-Oton, and Serra, where we prove that stable solutions in dimension n ≤ 9 are smooth. This answers also a famous open problem, posed by Brezis, concerning the regularity of extremal solutions to the Gelfand problem.
Alessio Figalli 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, 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.
In 2018 he won the Fields Medal “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability”.
Image credit: © ETH Zürich – Giulia Marthaler