Publications

Articles

• C^1+α regularity for fractional p-harmonic functions
  Preprint. Joint work with Davide Giovagnoli and David Jesus.
• The Landau equation and Fisher information
  Preprint. Joint work with Nestor Guillen.
• On the monotonicity of the Fisher information for the Boltzmann equation
  Inventiones Mathematicae. 243, 127–179 (2026). 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
  Acta Mathematica. 234 (2025), no. 2, 315–375. 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 work 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 Poincaré (C) Anal. Non Linéaire. 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-Ampère 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 kernels
  Published as chapter 4 in 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,α 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 Poincaré (C) Anal. Non Linéaire. 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. 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. 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 Poincaré (C) Anal. Non Linéaire 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 Luis Caffarelli.
• A characterization of optimal two-phase multifunctional composite designs
  Proc. R. Soc. A 463, Number 2086 (2007). Source code of the numerical tests 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. Chapter of: 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 Hölder space $C^{2\alpha}$ for the pressure in terms of the Hölder 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 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 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 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 abandoned 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 submitting 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 developed here, here and here. Our article has the modest advantage of providing a self-contained proof.