De Giorgi’s regularity theory for elliptic, parabolic and kinetic equations. Univ. Berkeley Chancellor’s professor Lecture notes, fall 2025.hal | arxiv
Preprints
With J. Guerand and C. Mouhot. Gehring’s Lemma for kinetic Fokker-Planck equations. hal | arxiv | video
With N. Forcadel and R. Monneau. Germs for scalar conservation laws: the Hamilton-Jacobi equation point of view. hal | arxiv
With P. Auscher and L. Niebel. Weak solutions to Kolmogorov-Fokker-Planck equations: regularity, existence and uniqueness. hal | arxiv
With F. Golse and L. Silvestre. Partial regularity in time for the space-homogeneous Boltzmann equation with very soft potentials. hal | arxiv
With A. Loher. Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded domain. Annales de l’IHP. analyse non-linéaire (to appear). hal | arxiv
Articles
With L. Silvestre and C. Villani. On the monotonicity of the Fisher information for the Boltzmann equation. Inventiones Mathematicae, Volume 243, pages 127–179, (2026). hal | arxiv | doi
With P. Auscher and L. Niebel. Fundamental solutions to Kolmogorov-Fokker-Planck equations with rough coefficients: existence, uniqueness, upper estimates. SIAM J. Math. Anal. (2025), 57 (2), p. 2114-2137. hal | arxiv | doi
With N. Forcadel and R. Monneau. The twin blow-up method for Hamilton-Jacobi equations in higher dimension. ESAIM: COCV. (2025) Volume 31, number 12.hal | arxiv | doi
With N. Forcadel and R. Monneau. Non-convex coercive Hamilton-Jacobi equations: Guerand’s relaxation revisited. Pure and Applied Analysis (2024) Vol. 6, No. 4, 1055–1089. hal | arxiv | doi
With N. Forcadel and R. Monneau. Coercive Hamilton-Jacobi equations in domains: the twin blow-ups method. Comptes rendus. Mathématique. Volume 362 (2024), pp. 829-839.hal | arxiv | doi
With F. Golse, Sehyun Ji and A. F. Vasseur. Local regularity for the space homogeneous Landau equation with very soft potentials. Journal of Evolution Equations. (2024) Volume 24, article number 82. hal | arxiv | doi
With J. Guerand. Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations. Journal de l’institut mathématique de Jussieu. Volume 22 , Issue 6 , November 2023 , pp. 2749 – 2774. hal | arxiv | doi
With F. Golse, M. P. Gualdani, A. F. Vasseur. Partial Regularity in Time for the Space Homogeneous Landau Equation with Coulomb Potential. Annales scientifiques de l’ENS (2022) Fascicule 6, tome 55, pp. 1575-1611.hal | arxiv | doi
With L. Silvestre. Global regularity estimates for the Boltzmann equation without cut-off. Journal of the Am. Math. Soc.(2022) Vol 35, No. 3, 625–703. hal | arxiv | doi
With L. Silvestre. The Schauder estimate for kinetic integral equations. Analysis and PDE. (2021) Vol. 14, No. 1, 171–204.hal | arxiv | doi
With C. Mouhot. The Schauder estimate in kinetic theory with application to a toy nonlinear model. Annales Henri Lebesgue (2021) Vol. 4, pp. 369-405.hal | arxiv | doi
With L. Silvestre. Regularity for the Boltzmann equation conditional to macroscopic bounds. EMS Surv. Math. Sci.7 (2020), 117–172.hal | arxiv | doi
With R. Tarhini and F. Vigneron. Regularity of solutions of a fractional porous medium equation. Interfaces and free boundaries (2020) Volume 22, Issue 4.hal | arxiv | doi
With C. Mouhot and L. Silvestre. Gaussian lower bounds for the Boltzmann equation without cut-off. SIAM J. Math. Anal. (2020)52, no. 3, 2930–2944.hal | arxiv | doi
With L. Silvestre. The weak Harnack inequality for the Boltzmann equation without cut-off. Journal of the European Mathematical Society (2020)Volume 22, Issue 2, pp. 507–592. doi | hal | arxiv | video
With C. Mouhot and L. Silvestre. Decay estimates for large velocities in the Boltzmann equation without cut-off. Journal de l’École Polytechnique — Mathématiques (2020), Volume 7, p. 143-184.Journal | doi | hal | arxiv
With T. Jin and L. Silvestre. Hölder gradient estimates for a class of singular or degenerate parabolic equations. Advances in Nonlinear Analysis (2019), 8(1), pp. 845-867. doi | hal | arxiv
With F. Golse, C. Mouhot and A. F. Vasseur. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2019). p. 253-295 | Vol. XIX, issue 1. doi | hal | arxiv | video
With G. Barles, A. Briani and E. Chasseigne. Flux-limited and classical viscosity solutions for regional control problems. ESAIM:COCV (2018), Vol. 24, pp. 1881–1906. doi | hal | arxiv
With T. Jin and R. Shvydkoy. Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation. Annales de la Faculté des Sciences de Toulouse . Sér. 6, 27 no. 4 (2018), p. 667-677. doi | hal | arxiv
With V. D. Nguyen. Generalized junction conditions for degenerate parabolic equations. Calculus of Variations and PDE’s (2017), 56: 157. doi | hal | arxiv
With R. Monneau. Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case. Discrete and continuous dynamical systems – series A (2017), Vol. 37, no. 12, p. 6405 – 6435. doi | hal | arxiv
With R. Monneau. Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks. Annales Scientifiques de l’ÉNS (2017), 50, fascicule 2, pp. 357-448.hal | arXiv
With R. Shvydkoy and F. Vigneron. Global well-posedness for a non-local Burgers’s equation: the periodic case. Annales de la faculté des sciences de Toulouse, Sér. 6, 25 no. 4 (2016), p. 723-758.doi | hal | arxiv | pdf
With L. Silvestre. Estimates on elliptic equations that hold only where the gradient is large. Journal of the European Mathematical Society (2016), Vol 18, pp. 1321–1338. doi | hal | arXiv
Finite speed of propagation for a non-local porous medium equation. Colloquium Mathematicum (2016), Vol.143, No. 2, pp. 149-157. doi | hal | arxiv
With G. Galise and R. Monneau. A junction condition by specified homogenization and application to traffic lights Analysis and PDE (2015), Vol 8, No 8, 1891–1929. doi | hal | arxiv
With A. Mellet. Self-similar solutions for a fractional thin film equation governing hydraulic fractures. Communications in Mathematical Physics (2015), vol 340, no. 3, 1187–1229. doi | hal | arxiv
With N. Forcadel and R. Monneau. Steady state and long time convergence of spirals moving by forced mean curvature motion. Communications in Partial Differential Equations (2015), 40:6, 1137-1181. doi | hal | arxiv
With P. Biler and G. Karch. Nonlocal porous medium equation: Barenblatt profiles and other weak solutions. Archive for Rational Mechanics and Analysis (2015), Vol 215, No 2, pp. 497–529. doi | hal | arxiv
With G. Barles, E. Chasseigne and A. Ciomaga. Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations. Calculus of Variations and Partial Differential Equations (2014), 50, no. 1-2, pp. 283–304. doi | hal | arxiv
With R. Monneau and H. Zidani. A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM: Control, Optim. and Calc. Var. (2013), Vol 19, No 1, pp. 129-166.doi | hal | arxiv
With L. Silvestre. C1,α regularity of solutions of degenerate fully non-linear elliptic equations. Advances in Mathematics (2013), Vol 233, pp. 196–206. doi | hal | arxiv
With A. Mellet. Electrified thin films: Global existence of non-negative solutions. Annales de l’IHP, analyse non-linéaire (2012), Vol 29, No 11, 413–433. doi | hal | arxiv
With G. Barles, E. Chasseigne and A. Ciomaga. Lipschitz regularity of solutions for mixed integro-differential equations. Journal of Differential Equations Vol 252 (2012), 6012–6060.doi | hal | arxiv
With N. Forcadel and R. Monneau. Uniqueness and existence of spirals moving by forced mean curvature motion. Interfaces and Free Boundaries (2012), Vol 14, pp. 365–400. doi | hal | arxiv
With N. Forcadel and R. Monneau. Homogenization of accelerated Frenkel-Kontorova models with n types of particles. Trans. Am. Math. Soc. 364 (2012) 6187-6227. doi | hal | arxiv
With A. Mellet. Existence of solutions for a higher order non-local equation appearing in crack dynamics. Nonlinearity 24 (2011) 3487–3514. doi | hal | arxiv
With G. Barles and E. Chasseigne. Hölder continuity of solutions of second-order elliptic integro-differential equations. Journal of the European Mathematical Society (2011), Vol 13, Issue 1, pp. 1-26. doi | hal | arxiv
Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations. Journal of Differential Equations Vol 250, Issue 3 (2011), 1325–1766doi | hal | arxiv — The proof of the Harnack inequality (Corollary 1) contained in this paper is not correct. A correct one is given here.
With S. Serfaty. Repeated games for eikonal equations, integral curvature flows and integro-differential parabolic equations. Discrete Contin. Dyn. Syst. 29, 4 (2011), 1517–1552. doi | hal | arxiv
With B. Bouchard and R. Elie. Optimal control under stochastic target constraints. SIAM J. Control Optim. (2010), Volume 48, Issue 5, pp. 3501-3531. hal
With N. Alibaud and G. Karch. Asymptotic properties of entropy solutions to fractal Burgers equations. SIAM J. Math. Anal. Volume 42, Issue 1, pp. 354-376 (2010). doi | hal | arxiv
With N. Forcadel and R. Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete and Continuous Dynamical Systems – Serie A 23 (2009), no 3, 785-826. hal
Level set approach for fractional mean curvature flows. Interfaces and Free Boundaries (2009), Vol 11, Issue 1, 153-176. doi | hal | arxiv
With N. Forcadel and R. Monneau. Homogenization of the fully overdamped Frenkel-Kontorova models. J. Differential Equations 246 (2009), 1057-1097. doi | hal
With N. Alibaud. Fractional semi-linear parabolic equations with unbounded data. Transations of the American Mathematical Society 361 (2009), 2527-2566. doi | hal
With R. Monneau and E. Rouy. Homogenization of first order equations with (u/ε)-periodic Hamiltonians. Part II: application to dislocation dynamics. Communications in PDEs (2008), Volume 33, No 3, 479 — 516. doi | hal
With G. Barles. Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions’ Theory Revisited. Annales de l’IHP (2008), Volume 25, No 3, Pages 567-585. doi | hal | arxiv
With R. Monneau. Homogenization of first order equations with (u/ε)-periodic Hamiltonians. Part I: local equations. Archive for Rational Mechanics and Analysis 187 (2008), 49-89. doi | hal
With G. Barles and E. Chasseigne. The Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. (2008), Volume 57, No 1, 213-146. doi | hal
With I. Gentil. The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium. Asymptotic Analysis (2008), Vol 59, No 3-4, 125-138. doi | hal
Convexity of solutions and C1,1 estimates for fully nonlinear elliptic equations. Journal de Mathématiques Pures and Appliquées, Vol 85, Issue 6 (2006), pp. 791-807. doi | hal
With J. Droniou. Fractal first order partial differential equations. Archive for Rational Mechanics and Analysis, Vol 182, No 2 (2006), pp. 299-331. doi | hal
A non-local regularization of first order Hamilton-Jacobi equations. Journal of Differential Equations, Vol 211, Issue 1 (2005), pp. 218-246. doi | hal
With J. Vovelle. A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications. SIAM – Mathematical Analysis, Vol 36, Issue 1 (2004), pp. 214-232. doi | hal
With J. Droniou and J. Vovelle. An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions. Annales de l’Institut Henri Poincaré – Analyse non linéaire, Vol 21, Issue 5 (2004), pp. 689-714. doi | hal
Support functions of Clarke’s generalized jacobian and of its plenary hull. Nonlinear Analysis, Theory, Methods and Applications, Vol 29, No 8 (2002) pp. 1111-1125. hal
Some regularity results for anisotropic motion of fronts. Differential and Integral Equations, Vol 15, No 10 (2002) pp. 1263-1271. hal
With Michel Volle. On vectorial Hamiton-Jacobi equations. Control and Cybernetics, Vol 31, No 3 (2002) pp. 493-506. hal | pdf
Convex Analysis techniques for Hopf-Lax formulae in Hamilton-Jacobi equations. Journal of Nonlinear and Convex Analysis Vol 2, No 3 (2001) pp. 333-343. hal
Notes and Proceedings
Fisher information for solutions of the Boltzmann equation. hal | arxiv(short review, 14 p.)
With F. Golse. Local regularity for the Landau equation (with Coulomb interaction potential). Kolmogorov operators and their applications. (2024) 1–22. Springer INdAM Ser., 56.
With P. Biler and G. Karch. Barenblatt profiles for a nonlocal porous medium equation. C. R. Acad. Sci. Paris, Ser. I 349 (2011) 641–645. doi | hal | arxiv
With I. Gentil. Logarithmic Sobolev inequalities: regularizing effect of Lévy operators and asymptotic convergence in the Lévy-Fokker-Planck equation. Stochastics An International Journal of Probability and Stochastic Processes, 81:3, 401-414. doi | arxiv
With J.-B. Hiriart-Urruty. Les fonctions d’appui de la jacobienne généralisiée de Clarke and de son enveloppe plénèire. C. R. Acad. Sci. Paris Sér. I Math., Vol 326 No 11 (1998), pp. 1275-1278. hal
Unpublished notes
With M. Koumaiha. Error estimates for finite difference schemes associated with Hamilton-Jacobi equations on a junction. hal | arxiv
With T. Souganidis. Phasefield theory for fractional diffusion-reaction equations and applications. hal | arxiv
With C. Mouhot. Hölder regularity for solutions to hypoelliptic equations with bounded measurable coefficients. hal | arxiv