Xavier
CabréXavier
Cabré
ICREA Research Professor and Catedrático de Matemática Aplicada
at the Universitat Politècnica de Catalunya
Institució
Catalana de Recerca i Estudis Avançats (ICREA) and
Universitat
Politècnica de Catalunya
Departament de Matemàtica
Aplicada I
Av. Diagonal, 647
08028 Barcelona. SPAIN
Phone:
+34 93 401 09 74
Fax: +34 93 401 17 13
Email:
xavier.cabre
upc.edu
Curriculum Vitae vita.pdf
Research Topics
Fully
nonlinear elliptic equations. Maximum principles and regularity
theory
Semilinear elliptic and
parabolic equations. Symmetry and qualitative properties
Geometric
inequalities. Minimal surfaces and phase transitions
Coordinator of
"TOPICS IN PDE'S AND APPLICATIONS 2008"
1st edition: GRANADA, APRIL 7-11, 2008.Courses by: Luigi Ambrosio, Luis Caffarelli, Francois Golse, Pierre-Louis Lions, Horng-Tzer Yau
2nd edition: CRM (Bellaterra, BARCELONA) MAY 5-9, 2008. Courses by: Henri Berestycki, Haim Brezis, Carlos Kenig, Robert V. Kohn, Gang Tian
Editorial Boards
Communications on Pure and Applied Analysis
Annales de la Faculté des Sciences de Toulouse
Research Networks
Member of the Steering Committee of Global and geometrical aspects of nonlinear partial differential equations (GLOBAL)
http://www.math.kth.se/global/index.html and http://www.esf.org/
Book
Fully Nonlinear Elliptic Equations (with L. Caffarelli). Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.
Articles and Preprints
Saddle-shaped solutions of bistable diffusion equations in all of R^{2m} (with J. Terra), arXiv:0801.3379. To appear in Jour. of the European Math. Society.
Regularity of radial minimizers of reaction equations involving the p-Laplacian (with A. Capella and M. Sanchón), arXiv:0712.2788. To appear in Calculus of Variations and Partial Differential Equations.
Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20 (2008), 425-457.
Semi-stable and extremal solutions of reaction equations involving the p-Laplacian (with M. Sanchón). Comm. Pure Appl. Anal. 6 (2007), 43-67.
Regularity of minimizers for three elliptic problems:minimal cones, harmonic maps, and semilinear equations (with A. Capella). Pure and Applied Math Quarterly 3 (2007), 801-825.
Regularity of radial extremal solutions of semilinear elliptic equations. Bol. Soc. Esp. Mat. Apl. 34 (2006), 92-98.
Regularity of radial minimizers and extremal solutions of semilinear elliptic equations (with A. Capella). J. Functional Analysis 238 (2006), 709-733.
Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems. To appear in Contemporary Mathematics, American Math. Soc. 2007, in: Perspectives in Nonlinear Partial Differential Equations: In honor of Haim Brezis.
The parameterization method for invariant manifolds III: overview and applications (with E. Fontich and R. de la Llave). J. Differential Equations 218 (2005), 444-515.
Layer solutions in a half-space for boundary reactions (with J. Solà-Morales). Comm. Pure Appl. Math. 58 (2005), 1678-1732.
A mean field equation on a torus: one-dimensional symmetry of solutions (with M. Lucia and M. Sanchón). Comm. Partial Differential Equations 30 (2005), 1315-1330.
On the stability of radial solutions of semilinear elliptic equations in all of $R^n$ (with A. Capella). C. R. Acad. Sci. Paris, Ser. I 338 (2004), 769-774.
Interior $C^{2,\alpha}$ regularity theory for a class of nonconvex fully nonlinear elliptic equations (with L. Caffarelli). J. Math. Pures Appl. 82 (2003), 573-612.
The parameterization method for invariant manifolds II: regularity with respect to parameters (with E. Fontich and R. de la Llave). Indiana Univ. Math. J. 52 (2003), 329-360.
The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces (with E. Fontich and R. de la Llave). Indiana Univ. Math. J. 52 (2003), 283-328.
Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. Discrete and Continuous Dynamical Systems 8 (2002), 331-359.
A conjecture of De Giorgi on symmetry for elliptic equations in $\bold \R^n$. European Congress of Math. Progr. Math. I, vol. 201, 259-265. Birkhäuser, 2001.
On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property (with G. Alberti and L. Ambrosio). Acta Applicandae Mathematicae 65 (2001), 9-33.
Entire solutions of semilinear elliptic equations in $\bold R\sp 3$ and a conjecture of De Giorgi (with L. Ambrosio). J. Amer. Math. Soc. 13 (2000), 725-739.
Partial differential equations, geometry and stochastic control (Catalan). Butl. Soc. Catalana Mat. 15 (2000), 7-27.
Existence versus instantaneous blow-up for linear heat equations with singular potentials (with Y. Martel). C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), 973-978.
Stable solutions of semilinear elliptic problems in convex domains (with S. Chanillo). Selecta Math. (New Ser.) 4 (1998), 1-10.
Some simple nonlinear PDE's without solutions (with H. Brezis). Boll. Unione Mat. Ital. 1-B (1998), 223-262.
Weak eigenfunctions for the linearization of extremal elliptic problems (with Y. Martel). J. Funct. Anal. 156 (1998), 30-56.
Nondivergent elliptic equations on manifolds with nonnegative curvature. Comm. Pure Appl. Math. 50 (1997), 623-665.
Truncation of nonlinearities in some supercritical elliptic problems (with P. Majer). C. R. Acad. Sci. Paris Ser. I Math. 322 (1996), 1157-1162.
Regularity for viscosity solutions of fully nonlinear equations $F(D^2u)=0$ (with L. Caffarelli). Topol. Methods Nonlinear Anal. 6 (1995), 31-48.
On the Alexandroff-Bakelman-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Comm. Pure Appl. Math. 48 (1995), 539-570.
Habilitation à diriger les recherches
Contributions à l'étude des équations aux dérivées partielles elliptiques et paraboliques. Research work done before 1998.
Doctorate Thesis
Estimates for Solutions of Elliptic and Parabolic Problems. Ph. D. Thesis, Courant Institute (NYU), May 1994.