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.

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

Cabré X, Fall MM & Weth T 2018, ‘Delaunay hypersurfaces with constant nonlocal mean curvature’, Journal de Mathématiques Pures et Appliquées, vol. 110, no. 9, pp 32-70.

Cabré X, Fall MM & Weth T 2018, ‘Near-sphere lattices with constant nonlocal mean curvature’, Mathematische Annalen, vol. 370, no. 3-4, pp 1513-1569.

Cabré X, Lucia M,  Sanchon M & Villegas S 2018, ‘Antisymmetry of solutions for some weighted elliptic problems’, Communications In Partial Differential Equations, vol. 43, no. 3, pp. 506-547.

Cabré X & Poggesi G 2018, ‘Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions’, in Geometry of PDEs and Related Problems, Lecture Notes in Mathematics 2220, CIME Foundation Subseries, Springer.

Cabré X, Fall MM, Solà-Morales J & Weth T 2018, ‘Curves and surfaces with constant nonlocal mean curvature: meeting Alexandrov and Delaunay’, J. Reine Angew. Math. vol. 745 (2018), pp 253-280.