Preprints and some recent papers
      PREPRINTS AND SOME RECENT PAPERS (.ps or .pdf)

      Harnack's inequality for a class of non-divergent equations in the Heisenberg group, (with F. Abedin and G. Tralli), Commnunications in pdes, to appear, 2017.

      General refraction problems with phase discontinuities on nonflat metasurfaces, (with L. Pallucchini and E. Stachura), Journal of the Optical Society of America A, Vol. 34, No. 7, pp. 1160-1162, July 2017.

      An Iterative Method For Generated Jacobian Equations, (with F. Abedin), Calculus of variations and pdes, vol. 56 (online first), 2017.

      Freeform Lens Design for Scattering Data With General Radiant Fields, (with A. Sabra), Archive for Rational Mechanics and Analysis, to appear, 2017.

      Form-Invariance of Maxwell Equations in Integral Form, Harmonic Analysis, Partial Differential Equations and Applications In Honor of Richard L. Wheeden, pp. 69-78, Birkhauser, 2017.

      C1-alpha estimates for the parallel refractor, (with F. Abedin and G. Tralli), Nonlinear Analysis, vol. 142, September 2016, pp. 1-25.

      On the Numerical Solution of the Far Field Refractor Problem, (with R. De Leo and H. Mawi), Nonlinear Analysis, Vol. 157, July 2017, pp. 123-145.

      Aspherical lens design and imaging, (with A. Sabra), SIAM Journal on Imaging Sciences, Vol. 9, No.1, pp. 386-411, 2016.

      Uniform refraction in negative refractive index materials, (with E. Stachura), Journal Optical Society of America A, 32(11), 2110-2122 (2015).

      Refraction problems in geometric optics, Lecture Notes in Mathematics Volume 2087, Springer, 2014, pp 95-150.

      Local near field refractors and reflectors, (with F. Tournier) Nonlinear Analysis: Theory, Methods & Applications, Volume 108, October 2014, Pages 302-311.

      Design of pairs of reflectors, (with A. Sabra) Journal Optical Society of America A, Vol. 31, Issue 4, pp. 891-899 (2014).

      Aspherical Lens Design, Journal Optical Society of America A, Vol. 30, Issue 9, pp. 1719-1726 (2013).

      The Reflector Problem and the Inverse Square Law , (with A. Sabra), Nonlinear Analysis: Theory, Methods & Applications, vol. 96 (2014) 109-133.

      Regularity for the near field parallel refractor and reflector problems , (with F. Tournier), Calculus of Variations and PDEs, September 2015, Volume 54, Issue 1, pp 917-949.

      Interior second derivative estimates for solutions to the linearized Monge--Ampère equation , (with T. Nguyen) Trans. Amer. Math. Soc. 367 (7), 2015, 4537-4568.

      The refractor problem with loss of energy , (with H. Mawi), Nonlinear Analysis: Theory, Methods & Applications, vol. 82, pp. 12-46, 2013.

      The near field refractor , (with Q. Huang), Annales de L'Institut Henri Poincaré (C) Analyse Non Linéaire, vol. 31 (4), pp. 655-684, 2014.

      Nonsmooth hypersurfaces with smooth Levi curvature , (with E. Lanconelli and A. Montanari), Nonlinear Analysis: Theory, Methods & Applications, Vol. 76, January 2013, Pages 115-121 .

      The Parallel Refractor , (with F. Tournier) Developments in Mathematics, Vol. 28, pp. 325-334 (2013).

      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).

      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.

      The near field refractor , (with Q. Huang), Geometric Methods in PDE's, Conference for the 65th birthday of E. Lanconelli, vol. 7, Lecture notes of Seminario Interdisciplinare di Matematica, Universita degli Studi della Basilicata, Potenza, Italy, 2008, pp. 175-188.

      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. and PDEs (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.