Xavier is an ICREA Research Professor at the University of Barcelona since 2020. Previously, he has been Assistant Professor at Universität Zürich, as well as R. H. Bing Instructor at the University of Texas at Austin. He is a mathematician who works on Partial Differential Equations (PDEs). Specifically, he studies the regularity of solutions to elliptic and parabolic PDEs, and he is mostly known for his results on free boundary problems and integro-differential equations. He is the PI of an ERC Starting Grant (2019-2024), has received several awards for young mathematicians in Spain, as well as the Scientific Research Award from the Fundación Princesa de Girona in 2019.
My research interests are on Partial Differential Equations (PDE), a vast and very active field of research in both pure and applied mathematics. PDE are used in essentially all sciences and engineering, and have important connections with several branches of pure mathematics. I work mainly on topics related to the regularity of solutions to nonlinear elliptic/parabolic PDE. This is one of the most basic and important question in PDE theory: to understand whether all solutions to a given PDE are smooth or if, instead, they may have singularities. Some of my main contributions have been in the context of free boundary problems. These are PDE problems that involve unknown/moving interfaces, such as ice melting to water (phase transitions). From the mathematical point of view, they give rise to extremely challenging questions, and their study is closely connected to geometric measure theory. In particular, the study of free boundary problems has a strong geometrical flavor.