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
My former PhD students
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
Editorial Boards
Communications on Pure and Applied Analysis
Annales de la Faculté des Sciences de Toulouse
Book
Fully Nonlinear Elliptic Equations (with L. Caffarelli). Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995.
Articles and Preprints
Regularity of stable solutions up to dimension 7 in domains of double revolution, (with X. Ros-Oton), arxiv.org/abs/1202.1220
Geometric-type Sobolev inequalities and applications to the regularity of minimizers, (with M.Sanchón), arxiv.org/abs/1111.2801
Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions,(with Y. Sire), arxiv.org/abs/1111.0796
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, (with Y. Sire), arxiv.org/abs/1012.0867
Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation, arxiv.org/abs/1102.3111, to appear in Journal de Mathématiques Pures et Appliquées..
Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian (with E. Cinti), arxiv.org/abs/1004.2866, Discrete and Continuous Dynamical Systems 28 (2010), 1179-1206. A special issue Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday.
Regularity of minimizers of semilinear elliptic problems up to dimension four, arxiv.org/abs/0909.4696, Comm. Pure Applied Mathematics 63 (2010), 1362-1380.
Qualitative properties of saddle-shaped solutions to bistable diffusion equations, (with J. Terra), arxiv.org/abs/0907.3008, Communications in Partial Differential Equations 35 (2010), 1923-1957.
Front propagation in Fisher-KPP equations with fractional diffusion (with J.-M. Roquejoffre), arxiv.org/abs/0905.1299 C. R. Math. Acad. Sci. Paris 347 (2009), 1361-1366.
Positive solutions of nonlinear problems involving the square root of the Laplacian (with Jinggang Tan), arxiv.org/abs/0905.1257 Advances in Mathematics 224 (2010), 2052-2093.
Saddle-shaped solutions of bistable diffusion equations in all of R^{2m} (with J. Terra), arXiv:0801.3379 Jour. of the European Math. Society 11 (2009), no. 4, 819-843.
Regularity of radial minimizers of reaction equations involving the p-Laplacian (with A. Capella and M. Sanchón), arXiv:0712.2788. Calculus of Variations and Partial Differential Equations 34 (2009), 475-494.
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. Birkhauser, 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 Holder 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.