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Plenary speaker

Xavier Cabré

Polytechnic University of Catalonia

Personal webpage

Biosketch

Born in Barcelona. PhD in Mathematics, area: Partial Differential Equations, adviser: Louis Nirenberg, Courant Institute, New York University, 1994. Kurt Friedrichs Prize, New York University, 1995. Member of the Institute for Advanced Study, Princeton, 1994-95. Habilitation à diriger des recherches, Université Pierre et Marie Curie-Paris VI, 1998. Harrington Faculty Fellow, The University of Texas at Austin, 2001-02. Tenure Associate Professor, The University of Texas at Austin, 2002-03. ICREA Research Professor since 2003 at the Universitat Politècnica de Catalunya. Fellow of the American Mathematical Society, inaugural class, 2012. Former editor of the Journal of the European Mathematical Society, 2014-17, and currently of “Calculus of Variations and Partial Differential Equations” and “Publicacions Matemàtiques”.

Research area

My research area is Partial Differential Equations (PDEs) and its relations with the Calculus of Variations. An important example are Hilbert's 19th and 23rd problems, which concerned the regularity of minimizing solutions to elliptic functionals. An essential progress on Hilbert's 19th problem was made in 1957 by Ennio De Giorgi and, independently, by John Forbes Nash. This result allowed attacking several central problems in PDEs and in Geometry –-a very important one being the regularity of minimal hypersurfaces of Euclidean space which are area minimizing. However, closely related questions to the theory of minimal surfaces remain still to be answered and are main goals of my research. I have been focused on three of such problems:

  1. The regularity of stable solutions to semilinear elliptic equations up to dimension 9 has been an open problem since it was raised by H. Brezis in the mid-nineties. The result is expected to be true due to a number of partial contributions, for instance its validity in the radial case [Cabré & Capella, JFA 2006]. Its validity in general domains up to dimension 4 was set in [Cabré, CPAM 2010]. These results required new developments in other topics, such as some sharp isoperimetric and Sobolev inequalities with weights [Cabré, Ros-Oton & Serra JEMS 2016].
  2. The regularity properties of minimizers to the Allen-Cahn equation in dimension 8 is still to be understood. Recent results have established, almost totally, a conjecture of Ennio De Giorgi that was opened since 1978: [Ambrosio & Cabré, JAMS 2000], [Savin, Ann of Math 2009], and [del Pino, Kowalczyk & Wei, Ann of Math 2011]. The saddle-shaped solution vanishing on the Simons cone is expected to be a minimizer in dimension n=8, but this is still an open problem.
  3. A third topic is the regularity of nonlocal minimal surfaces and of stable solutions to elliptic nonlocal PDEs with fractional diffusion. Nonlinear equations with fractional diffusion, or nonlinear integrodifferential equations, have been a very active area within PDEs for about fifteen years and still now. They have applications, among others, to American options in Mathematical Finance and propagation of diseases and plagues in the Earth. Indeed, these equations arise naturally in the presence of stochastic processes with jumps, more precisely of Lévy flights. These types of processes, well studied in Probability, are of great interest in Finance, Ecology, and Physics. In addition, these equations also appear naturally in other contexts such as Image Processing, Fluid Mechanics, and Geometry [D. Applebaum, Lévy Processes – From Probability to Finance and Quantum Groups, Notices of the AMS 2004]. My research in the area focuses now on extending the theory of classical minimal surfaces to the nonlocal case (Simons’ classification of minimal cones is still widely open), an example being the gradient bound for nonlocal minimal graphs in [Cabré & Cozzi, Duke Math J 2019]. Another open question is to establish the optimal dimension of space to guarantee regularity of stable solutions to fractional semilinear equations ---point a) above in the fractional case. Questions as in b), now for the fractional Allen- Cahn equation, are also of my interest and are connected to nonlocal minimal surfaces.