PREPRINTS AND SOME RECENT PAPERS (.ps or .pdf)

      The Parallel Refractor , (with F. Tournier) Developments in Mathematics, to appear.

      Harnack inequality for a degenerate elliptic equation , (with F. Tournier) Communications in PDE, Vol. 36, Issue 12, pp. 2103-2116 (2011).

      Interior gradient estimates for solutions to the linearized Monge--Ampère equation , (with T. Nguyen) Advances in Mathematics, Vol. 228, Issue 4, pp. 2034-2070 (2011).

      Surfaces reflecting and refracting collimated beams , (with F. Tournier) Journal Optical Society of America A, Vol. 28, Issue 9, pp.1860-1863 (2011).

      Reflection, refraction, and the Legendre Transform , Journal Optical Society of America A, Vol. 28, Issue 2, pp.284-289 (2011).

      The refractor problem with loss of energy , (with H. Mawi).

      The near field refractor , (with Q. Huang), submitted.

      An Aleksandrov type estimate for alpha convex functions , (with F. Tournier), Proc. Amer. Math. Soc. 138 (2010), no. 6, 2001-2014.

      Schauder estimates for sub elliptic equations, (with E. Lanconelli), Journal of Evolution Equations 9 (2009), no. 4, 707-726.

      The refractor problem in reshaping light beams, (with Qingbo Huang), Archive for Rational Mechanics and Analysis, 2009, vol. 193, no. 2, pp. 423-443.

      Covering theorems, inequalities on metric spaces and applications to pde's, (with G. Di Fazio and E. Lanconelli), Math. Ann. (2008) 341:255-291.

      On the regularity of reflector antennas, (with L. A. Caffarelli and Qingbo Huang), Annals of Mathematics, 167 (2008), 299-323.

      On Monge-Ampere type equations arising in optimal transportation problems, (with T. Van Nguyen), Calc. Var. (2007) 28: 275-316.

      W2p estimates for the linearized Monge-Ampère equation, (with F. Tournier), Trans. Amer. Math. Soc. 358 (2006) 4843-4872.

      Homogenization and convergence of correctors in Carnot groups, (with B. Franchi and T. Van Nguyen), Comm. Partial Differential Equations, 2005, Vol. 30, Issue 12, 1817-1841.

      On the second order derivatives of convex functions on the Heisenberg group, (with A. Montanari), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 2, 349--366.

      Maximum and comparison principles for convex functions on the Heisenberg group, (with A. Montanari), Comm. Partial Differential Equations, 2004, Vol. 29, Issue 9/10, 1305-1334.

      Homogenization of Quasilinear Parabolic Equations in Periodic Media, (with J. Garcia--Azorero, and I. Peral), Comm. Partial Differential Equations, 2003, Vol. 28, Issue 11/12, 1887-1910.

      Maximum principle, Harnack inequality, and Liouville theorems for X--elliptic operators, (with E. Lanconelli), Comm. Partial Differential Equations, 2003, Vol. 28, Issue 11/12, 1833-1862.

      Regularity of weak solutions to the Monge-Ampère equation, (with D. Hartenstine), Trans. Amer. Math. Soc. 355 (2003) 2477-2500.

      W2-p-estimates for the parabolic Monge-Ampère equation, (with Q. Huang), Arch. Ration. Mech. Anal. 159 (2001), no. 2, 137--177.

      A Harmonic Analysis theorem and applications to Homogenization, (with I. Peral), Indiana Univ. Math. J. 50 (2001), no. 4, 1651--1674.

     Riesz Transforms, g-functions, and multipliers for the Laguerre semigroup , (with A. Incognito and J. L. Torrea), Houston J. Math. 27 (2001), no. 3, 579--592.

     Geometric Properties of the Sections of  Solutions to the Monge-Ampère  Equation , (with Q. Huang), Transactions of the A. M. S. 352 (2000), no. 9, 4381--4396.  

    A generalization of a theorem by Calabi to the parabolic Monge-Ampère equation, (with Q. Huang), Indiana Univ. Math. Journal 47 (1998), no. 4, 1459--1480.

    Properties of the Solutions of the Linearized Monge-Ampère Equation, (with L. A. Caffarelli), American J. of Math., vol. 119(2), 1997, 423-465.

    Singular Integrals Related to the Monge-Ampère Equation, (with L. A. Caffarelli), in "Wavelet Theory and Harmonic Analysis in Applied Sciences," edited by C. E. D'Attellis and E. M. Fernández-Berdaguer; Birkhauser, 1997, pages 3-13.
 
    Real Analysis Related to the Monge-Ampère Equation, (with L. A. Caffarelli), Transactions of A. M. S., vol. 348(3), 1996, 1075-1092.

    On the Harnack Inequality for Viscosity Solutions of Non-Divergence Equations, Sinet, Ethiopian Journal of Science, Proc. of the conference in Honor of T. Retta, edited by S. Berhanu, vol. 19, 1996, 48-72.