Xavier Cabré

Xavier Cabré

Universitat Politècnica de Catalunya

Experimental Sciences & Mathematics

Born in 1966 in Barcelona. PhD in Mathematics (area: Partial Differential Equations), 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 at the Universitat Politècnica de Catalunya, since 2003. Fellow of the American Mathematical Society, inaugural class, 2013. Plenary speaker at the 8th European Congress of Mathematics, 2021.

Research interests

My research field is the mathematical analysis of Partial Differential Equations. These equations arise in mathematical physics, differential geometry, finance, and biology. My focus is on elliptic and parabolic equations, and on the analytical understanding of the regularity, symmetry, and other qualitative properties of their solutions. This often involves the use of geometric tools such as isoperimetric inequalities, whose study is an important part of my research. My main current project concerns a recently flourishing area: reaction problems for fractional diffusions associated to jump or Lévy processes. These are the so called 'anomalous diffusions', well noticed in the last decades in some reaction and biological fronts, as well as in mathematical finance.

Selected publications

- Cabre X & Sanz-Perela T 2022, 'A universal Holder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations', Journal Of Differential Equations, 317, pp 153-195.

- Cabré X & Miraglio P 2022, 'Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space', Communications in Contemporary Mathematics, 24, 05, 2150063.

- Cabre X,  Miraglio P & Sanchon M 2022, 'Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian', Advances in Calculus of Variations, 15, pp 749-785.  

Selected research activities

Lecturer of a 5 hours minicourse, 'Geometric aspects of nonlinear partial differential equations', Sept 2022, Institute Mittag-Leffler, Sweden

Editorial Advisory Board of PNLDE, Book series 'Progress in Nonlinear Differential Equations and Their Applications', Springer

Editor of the journal 'Publicacions Matemàtiques'