37 (2012), no. 9, 1668-1688. Joint work with Pierre Cardaliaguet.
Partial regularity of solutions of fully nonlinear uniformly elliptic equations Communications on Pure and Applied Mathematics 65 (2012), no.8, 1169-1184. Joint work with Scott Armstrong and Charles Smart.
Holder continuity for a drift-diffusion equation with pressure Annales de l'Institut Henri Poincare (C) Anal. Non Lineaire. 29 (2012), no. 4, 637-652. Joint work with Vlad Vicol.
Unique continuation for fully nonlinear elliptic equations Mathematical Research Letters. Volume 18, Issue 5, September 2011 pp. 921-926. Joint work with Scott Armstrong.
On the differentiability of the solution to an equation with drift and fractional diffusion. Indiana University Mathematical Journal. 61 (2012), no. 2, 557-584.
On divergence free drifts. Journal of Differential Equations 252 (2012), no. 1, 505-540. Joint work with Gregory Seregin, Vladimir Sverak and Andrej Zlatos.
Holder estimates for advection fractional-diffusion equations. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. 11 (2012), no. 4, 843–855.
Full regularity of a free boundary problem with two phases. Calculus of Variations and PDE 42 (2011), no. 3-4, 301-321. 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.174 (2011), no. 2, 1163-1187. 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. 363 (2011), no. 11, 5871-5886. 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
Unpublished articles:
The articles below will not be published but might be of interest to some people.
- A non obvious estimate for the pressure.
The paper gives a somewhat surprising estimate in the Holder space $C^{2\alpha}$ for the pressure in terms of the Holder norm $C^\alpha$ of the velocity. Here, the velocity and pressure are related by either the incompressible Euler or NS equation. I came up with this estimate in 2010. I told several people about it and I spent some time looking for an interesting application which I couldn't find. The result was never sent for publication but it was cited in this paper and this paper as a personal communication.
- On the superharmonicity of the first eigenfunction of the fractional Laplacian for certain exponents. Joint work with Moritz Kassmann.
This note originated from a brief conversation we had with Moritz in 2012. We were interested in the problem of the concavity of the first eigenfunction of the fractional Laplacian in a bounded interval. We came up with a simple proof in the case that the power of the Laplacian is of the form $1/m$ for some positive integer $m$. That proof is written in this note. We tried to extend the result to all exponents of the Laplacian in $(0,1)$ but we were never able to make progress in that direction. We left the problem behind and this short note was never sent for publication. An alternative proof of the same result is given in this paper.
- Upper bounds for multiphase composites in any dimension.
This article gives a non optimal upper bound for multiphase composites which improves the Hashin-Shtrikman bounds in some range of parameters. I abandomed the project when I found out about these other bounds which are similar (although not exactly the same).
- On the regularity of a singular variational problem. Joint work with Erik Lindgren.
After initially submiting this paper for publication, we learned that the result follows from the work of Giusti and Giaquinta. In particular from the paper Sharp estimates for the derivatives of local minima of variational integrals. (Bollettino della unione matematica Italiana 3.2 (1984): 239-248) which in turn applies the techniques developped in here, here and here. Our article has the modest advantage of providing a self contained proof.
Partially supported by NSF.
Articles: