Research area Partial Differential Equations (PDEs) Click here for my Lines of Research. I study PDEs of the type:

Topics:

Grants Click here
Articles and Preprints Click on the Articles Tab on top of this page
Slides from some of my talks
 Nonlocal minimal cones and a gradient estimate for nonlocal minimal graphs. 2018 slides on recent works with M. Cozzi, E. Cinti, and J. Serra on nonlocal minimal cones and graphs, with a fast introduction to fractional elliptic operators
 Uniqueness and stability of saddleshaped solutions to the AllenCahn equation. 2016 slides on AllenCahn and a conjecture of De Giorgi
 Nonlinear elliptic and parabolic equations with fractional diffusion. 2015 slides on recent works with J. SolàMorales, Y. Sire, E. Cinti, J.M. Roquejoffre, A.C. Coulon, N. Consul, J.V. Mandé, M.M. Fall, and T. Weth
 Sharp isoperimetric inequalities with densities via the ABP method. 2014 slides on recent joint works with X. RosOton and J. Serra. Slides start with an open problem on ReactionDiffusion equations that motivated the new isoperimetric inequalities
 What do heating your living room, financial investments, and image processing have in common? Divulgation. Elementary introduction to diffusion and PDEs for a general scientific public or for undergraduate students
 Louis Nirenberg (my Ph.D. director) 2015 Abel Prize. Link to The Abel Prize webpage. Slides of a presentation on Nirenberg's works and career
Videos from some of my talks
 Videos on Stable solutions to some elliptic problems. Five lecures at "Calculus of Variations and Nonlinear Partial Differential Equations, week one: minicourses". Columbia University (USA), May 2016.
 Video on The saddleshaped solution to the AllenCahn equation, a conjecture of De Giorgi, and minimal surfaces (the slides are above in this page). Talk at "A Mathematical Tribute to Ennio De Giorgi". Centro De Giorgi, Pisa (Italy), September 2016.
 Video on Sharp isoperimetric inequalities and on ReactionDiffusion equations (the slides are above in this page). Talk at the "29º Colóquio Brasileiro de Matemática", IMPA, Rio de Janeiro, 21 de julho a 02 de agosto de 2013.
Lecture notes and expository papers
 Stable solutions to some elliptic problems: minimal cones, the AllenCahn equation, and blowup solutions, with G. Poggesi. 2018. They contain proofs of the Simons and the BombieriDe GiorgiGiusti theorems on minimal cones
 Boundedness of stable solutions to semilinear elliptic equations: a survey. 2017
 Isoperimetric, Sobolev, and eigenvalue inequalities via the ABP method: a survey. 2015
 Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions. 2008. Tutorial, with an introductory section for non experts and students
 Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations, with A. Capella. 2007
 The parameterization method for invariant manifolds III: overview and applications, with E. Fontich and R. de la Llave. 2005. Expository notes on invariant manifolds for Dynamical Systems
 Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations. 2002. Based in a Doctorate course organized by J.M. Roquejoffre in 1997 at l'Université Paul Sabatier (Toulouse)
 Equacions en derivades parcials, geometria i control estocàstic, in Catalan. 2002
Book
Fully Nonlinear Elliptic Equations, with Luis Caffarelli. Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995
Fully Nonlinear Elliptic Equations, with Luis Caffarelli. Colloquium Publications 43, American Mathematical Society, Providence, RI, 1995
Habilitation à diriger les recherches
Contributions à l'étude des équations aux dérivées partielles elliptiques et paraboliques. Université Pierre et Marie Curie, Paris 6
Research work done before 1998
Contributions à l'étude des équations aux dérivées partielles elliptiques et paraboliques. Université Pierre et Marie Curie, Paris 6
Research work done before 1998
Doctorate Thesis
Estimates for Solutions of Elliptic and Parabolic Problems. PhD Thesis. Courant Institute, New York University, 1994. Adviser: Louis Nirenberg
Estimates for Solutions of Elliptic and Parabolic Problems. PhD Thesis. Courant Institute, New York University, 1994. Adviser: Louis Nirenberg