Click Here to see a summary of all my papers with a brief abstract (current as of October 2010).

• $C^{1,\alpha}$ regularity of solutions of degenerate fully non-linear elliptic equations.
  Preprint. Joint work with Cyril Imbert.
• Holder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side
  Communications in Partial Differential Equations. Accepted for publication. Joint work with Pierre Cardaliaguet.
• Partial regularity of solutions of fully nonlinear uniformly elliptic equations
  Communications on Pure and Applied Mathematics Accepted for publication. Joint work with Scott Armstrong and Charles Smart.
• Holder continuity for a drift-diffusion equation with pressure
  Preprint. Joint work with Vlad Vicol.
• Unique continuation for fully nonlinear elliptic equations
  Mathematical Research Letters. Accepted for publication. Joint work with Scott Armstrong.
• On the differentiability of the solution to an equation with drift and fractional diffusion.
  Indiana University Mathematical Journal. Accepted for publication.
• On divergence free drifts.
  Journal of Differential Equations. Accepted for publication. Joint work with Gregory Seregin, Vladimir Sverak and Andrej Zlatos.
• Upper bounds for multiphase composites in any dimension.
  Preprint. (work in progress)
• Holder estimates for advection fractional-diffusion equations.
  Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Accepted for publication.
• Full regularity of a free boundary problem with two phases.
  Calculus of Variations and PDE. Accepted for publication. Joint work with Huiqiang Jiang and Christopher Larsen.
• Holder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient
  Discrete and Continuous Dynamical Systems Volume: 28, Number: 3, November 2010.
  A special issue Dedicated to Louis Nirenberg on the Occasion of his 85th Birthday Part II
• Eventual regularization of the slightly supercritical fractional Burgers equation.
  Discrete and Continuous Dynamical Systems Volume: 27, Number: 2, June 2010.
  A special issue Trends and Developments in DE/Dynamics Part I
  Joint work with Chi Hin Chan and Magdalena Czubak.
• On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion.
  Advances in Mathematics. 226 (2011), no. 2, 2020-2039
• Smooth approximations to solutions of nonconvex fully nonlinear elliptic equations.
  American Mathematical Society Translations--Series 2 Advances in the Mathematical Sciences 2010; Volume: 229.
  Nonlinear Partial Differential Equations and Related Topics: Dedicated to Nina N. Uraltseva.
  Joint work with Luis Caffarelli.
• On the Evans-Krylov theorem.
  Proceedings of the AMS. 138 (2010), 263-265. Joint work with Luis Caffarelli.
• The Evans-Krylov theorem for non local fully non linear equations.
  Annals of Mathematics. Accepted for publication. Joint work with Luis Caffarelli.
• Regularity results for nonlocal equations by approximation
  Archive of Rational Mechanics and Analysis. Volume 200, Issue 1 (2011), Page 59. Joint work with Luis Caffarelli.
• Eventual regularization for the slightly supercritical quasi-geostrophic equation
  Annales de l'Institut Henri Poincare (C) Anal. Non Lineaire 27 (2010), Issue 2, Pages 693-704
• The Dirichlet Problem for the Convex Envelope
  Transactions of the AMS. Accepted for publication. Joint work with Adam Oberman.
• Regularity theory for fully nonlinear integro-differential equations.
  Communications on Pure and Applied Mathematics. 62 (2009) Issue 5, 597--638. Joint work with L. Caffarelli.
• A characterization of optimal two-phase multifunctional composite designs
  Proc. R. Soc. A 463, Number 2086 (2007). The source code of the numerical tests relevant to this paper can be downloaded here.
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian
  Inventiones Mathematicae. 171, Number 2 (2008). Joint work with Luis Caffarelli and Sandro Salsa.
• Regularity for the nonlinear Signorini problem
  Advances in Mathematics. 217, Issue 3 (2008). Joint work with Manolis Milakis.
• An extension problem related to the fractional Laplacian
  Communications in Partial Differential Equations, 32 (2007) Issue 8, 1245. Joint work with Luis Caffarelli.
• Issues in homogenization for problems with nondivergence structure.
  
  Lecture notes of Luis Caffarelli's course in a CIME summer school. Published as a chapter of the book:
  Calculus of variations and nonlinear partial differential equations, 43--74, Lecture Notes in Math., 1927, Springer, Berlin, 2008.
• Regularity of the obstacle problem for a fractional power of the laplace operator.
  
  Communications on Pure and Applied Mathematics. 60 (2007), no. 1, 67--112.
  This article is based on My PhD Thesis
• Regularity for fully nonlinear elliptic equations with Neumann boundary data.
  Communications in Partial Differential Equations 31 (2006), No. 8. Joint work with Manolis Milakis.
• Holder estimates for solutions of integro differential equations like the fractional Laplacian.
  Indiana University Mathematical Journal 55 (2006), 1155-1174.
• The two membranes problem.
  
  Communications in Partial Differential Equations 30 (2005), no. 1-3, 245--257
• Weak matrix majorization.
  
  Linear Algebra Appl. 403 (2005), 343--368. Joint work with Francisco Martínez Pería and Pedro Massey