Publications

- The slides of my talk about the obstacle problem for the fractional Laplacian and the thin obstacle problem.
- The slides of my talk about optimal design of composites.
*(For the animatios to work well, you have to open the slides with Adobe Reader in a separate window and press Ctrl-L)* - The slides of my talk about integro-differential equations.
- The slides of my talk about the Hamilton-Jacobi equation with critical fractional diffusion.
- The slides of my talk about partial regularity for fully nonlinear elliptic PDE.
- The slides of my talk about drift-diffusion equations.
- The slides of my talk about degenerate elliptic equations.
- The slides of my talk about the parabolic p-Laplacian equation.

- Talk at the IAS (2009) about fully nonlinear integro-differential equations.
- Talk at MSRI (2011) about unique continuation and partial regularity for fully nonlinear elliptic PDE.
- Talk at the ICM (Seoul - 2014) about regularity results for integro-differential equations.
- Talk at Carnegie Mellon (2018) about regularity and structure of scalar conservation laws.
- Talks about conditional regularity for the Boltzmann equation at the One World PDE seminar and in the PDE Seminar organized by Quoc-Hung Nguyen.

- Lecture notes on nonlocal equations.
- Lecture notes on parabolic equations (with Cyril Imbert).
- Lecture notes on viscosity solutions .
- Lecture notes from my undergraduate PDE course .

- On the monotonicity of the Fisher information for the Boltzmann equation Preprint. Joint work with Cyril Imbert and Cédric Villani.
- Partial regularity in time for the space-homogeneous Boltzmann equation with very soft potentials Preprint. Joint work with François Golse and Cyril Imbert.
- The Landau equation does not blow up Preprint. Joint work with Nestor Guillen.
- Conditional L∞ estimates for the non-cutoff Boltzmann equation in a bounded domain. Arch. Ration. Mech. Anal. 248 (2024), no. 4. Joint work with Zhimeng Ouyang.
- Entropy dissipation estimates for the Boltzmann equation without cut-off Kinet. Relat. Models 16 (2023), no. 5. Joint work with Jamil Chaker.
- Holder estimates for kinetic Fokker-Planck equations up to the boundary Ars Inveniendi Analytica. 2022, Paper No. 6, 35
- Regularity estimates and open problems in kinetic equations Barrett lectures 2021.
- Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian Mathematics in Engineering 5 (2023), no. 2, Paper No. 034. Joint work with Stanley Snelson.
- Singular solutions to parabolic equations in nondivergence form.
*Annali della Scuola Normale Superiore, Classe di Scienze*2022: VOL. XXIII, ISSUE 2. - Regularity for the Boltzmann equation conditional to macroscopic bounds
*EMS Surveys in Mathematical Sciences*7 (2020), no. 1. joint with Cyril Imbert. - Global regularity estimates for the Boltzmann equation without cut-off
*Journal of the American Mathematical Society*. 35 (2022), no. 3. Joint work with Cyril Imbert. - Coercivity estimates for integro-differential operators
*Calculus of Variations and PDE.*59 (2020), no. 4. Joint work with Jamil Chaker. - Gaussian lower bounds for the Boltzmann equation without cut-off
*SIAM J. Math. Anal.*52 (2020), no. 3. Joint work with Cyril Imbert and Clement Mouhot. - The Schauder estimate for kinetic integral equations.
*Analysis and PDE*. 14 (2021), no. 1. Joint work with Cyril Imbert. - Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates
*Archive of Rational Mechanics and Analysis*. 234 (2019), no. 3. Joint work with Denis Serre. -
Decay estimates for large velocities in the Boltzmann equation without cut-off
*Journal de l'École polytechnique*. Mathématiques 7 (2020). Joint work with Cyril Imbert and Clement Mouhot. - Oscillation properties of scalar conservation laws.
*Communications on Pure and Applied Mathematics*. 72 (2019), no. 6, 1321–1348. - Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials.
*Annales de l'Institut Henri Poincare (C) Anal. Non Lineaire.*35 (2018), no. 3, 625–642. Joint work with Stephen Cameron and Stanley Snelson. - The weak Harnack inequality for the Boltzmann equation without cut-off.
*Journal of the European Mathematical Society.*22 (2020), no. 2. Joint work with Cyril Imbert. - Hölder gradient estimates for a class of singular or degenerate parabolic equations.
*Advances in Nonlinear Analysis.*8 (2019), no. 1, 845–867. Joint work with Cyril Imbert and Tianling Jin. - Upper bounds for parabolic equations and the Landau equation.
*Journal of Differential Equations*262 (2017), no. 3, 3034–3055. - An integro-differential equation without continuous solutions.
*Mathematical Research Letters*23 (2016), no. 4, 1157–1166. Joint work with Stanley Snelson. - Hölder gradient estimates for parabolic homogeneous p-Laplacian equations.
*Journal de Mathématiques Pures et Appliquées*. (9) 108 (2017), no. 1, 63–87. Joint work with Tianling Jin. - Regularity estimates for fully non linear elliptic equations which are asymptotically convex.
*Progress in Nonlinear Differential Equations and their Applications.*Contributions to nonlinear elliptic equations and systems, 425–438, 86, Birkhäuser/Springer, Cham, 2015. Joint work with Eduardo Teixeira. - A new regularization mechanism for the Boltzmann equation without cut-off.
*Communications in Mathematical Physics*348 (2016), no. 1, 69–100. - Regularity for parabolic integro-differential equations with very irregular kernels.
*Analysis and PDE*9 (2016), no. 3, 727–772. Joint work with Russell Schwab. - Propagation in a non local reaction diffusion equation with spatial and genetic trait structure.
*Nonlinearity*29 (2016), no. 4, 1434–1466. Joint work with Henri Berestycki and Tianling Jin. - Regularity estimates for parabolic integro-
differential equations and applications.
*Proceedings of the ICM 2014.* - On a transport equation with nonlocal drift
*Transactions of the AMS*. 368 (2016), no. 9, 6159–6188. Joint work with Vlad Vicol. The article is about this equation. - A non local Monge-Ampere equation.
*Communications in Analysis and Geometry*. 24 (2016), no. 2, 307–335. Joint work with Luis Caffarelli. - On Landis' conjecture in the plane
*Communications in Partial Differential Equations.*40 (2015), no. 4, 766–789. Joint work with Carlos Kenig and Jenn-Nan Wang. - Overdetermined problems for fully nonlinear elliptic equations
*Calculus of Variations and PDE.*54 (2015), no. 1, 989–1007. Joint work with Boyan Sirakov. - Boundary regularity for viscosity solutions of fully nonlinear elliptic equations
*Communications in Partial Differential Equations*. 39 (2014), no. 9, 1694-1717. Joint work with Boyan Sirakov. - Estimates on elliptic equations that hold only where the gradient is large
*Journal of the European Mathematical Society.*18 (2016), no. 6, 1321–1338. Joint work with Cyril Imbert. - Uniqueness of radial solutions for the fractional Laplacian
*Communications on Pure and Applied Mathematics*. 69 (2016), no. 9, 1671–1726. Joint work with Rupert Frank and Enno Lenzmann. - Holder regularity for generalized master equations with rough kernelsPublished as chapter 4 in the book
*Advances in Analysis: The Legacy of Elias M. Stein.*Princeton University Press, 2014. Joint work with Luis Caffarelli. - On the loss of continuity for super-critical drift-diffusion equations
*Archive of Rational Mechanics and Analysis*. 207 (2013), no. 3, 845-877. Joint work with Vlad Vicol and Andrej Zlatos. - Global well-posedness of slightly supercritical active scalar equations.
*Analysis and PDE*. 7 (2014), no. 1, 43-72. Joint work with Michael Dabkowski, Alexander Kiselev and Vlad Vicol. - $C^{1,\alpha}$ regularity of solutions of degenerate fully non-linear elliptic equations.
*Advances in Mathematics.*233 (2013), 196-206. 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.*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

- A non obvious estimate for the pressure.
- On the superharmonicity of the first eigenfunction of the fractional Laplacian for certain exponents. Joint work with Moritz Kassmann.
- Upper bounds for multiphase composites in any dimension.
- On the regularity of a singular variational problem. Joint work with Erik Lindgren.

Partially supported by NSF.

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