Research area Partial Differential Equations (PDEs) Click here for my Lines of Research. I study PDEs of the type:
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Topics:
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Grants Click here
Articles and Preprints Click on the Articles Tab on top of this page
Slides from some of my talks
- Stable solutions to semilinear elliptic equations are smooth up to dimension 9. 2019 slides on a recent work with A. Figalli, X. Ros-Oton, and J. Serra on the regularity theory of stable solutions to semilinear elliptic equations
- 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 saddle-shaped solutions to the Allen-Cahn equation. 2016 slides on Allen-Cahn and a conjecture of De Giorgi
- Nonlinear elliptic and parabolic equations with fractional diffusion. 2015 slides on recent works with J. Solà-Morales, 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. Ros-Oton and J. Serra. Slides start with an open problem on Reaction-Diffusion 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
- Video of my 8ECM Plenary Lecture "Stable solutions to semilinear elliptic equations are smooth up to dimension 9"
- 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 saddle-shaped solution to the Allen-Cahn 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
Lecture notes and expository papers
- Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutions, with G. Poggesi. 2018. They contain proofs of the Simons and the Bombieri-De Giorgi-Giusti 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