RESEARCH PUBLICATIONS OF VLADIMIR OLIKER

    BOOKS

  1. Geometry and Nonlinear Partial Differential Equations, V. Oliker and A. Treibergs (eds.), Contemporary Mathematics, vol. 127, AMS, 1992, 1-154
  2. The Minkowski Multidimensional Problem, A.V. Pogorelov, Translated from Russian into English by V. Oliker and E. Oliker, John Wiley & Sons, 1978, 1-106.
  3. Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, M. Ghomi, J. Li, J. McCuan, V. Oliker, F. Schwartz, G. Weinstein (eds.), Contemporary Mathematics, vol. 599, AMS, 2013, 1-243

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    SPECIAL PROJECTS

  1. Rainbows circa 2000, Scientific Visualization of High-Frequency Electromagnetic Fields, E. Oliker and V. Oliker, Two computer-generated images and a computer animation presented at the exhibit "Science and Art: Shared Frontiers", Shatten Gallery, January 26-May 31, 2001, Emory University

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    PAPERS

  1. Unboundedness of certain PL-embeddings, Proc. of the Second National Symposium on Geometry in the "Large," 1967, 49-50, Petrozavodsk, USSR.
  2. A paratingency property of a continuous surface and its applications (with A.L. Verner), Vestnik Leningrad University, (13) 3 (1968), 21-27; English Transl., Vestnik Leningrad University, (3) 1 (1974), 181-188.
  3. On the structure of paratingency of a continuous surface, Proc. Third National Symposium on Geometry in the "Large," 1969, 51, Petrozavodsk, USSR.
  4. The uniqueness of the solution of Christoffel's problem for open surfaces (with Yu A. Volkov), Matem. Zametki, 8 (1970), 251-257; English transl., Mathematical Notes of the Academy of Sciences of the USSR, 9 (1/2)(1970), 611-614.
  5. The Christoffel problem for complete surfaces, Vestnik Leningrad University, (13) 3 (1970), 155-158.
  6. Existence of complete generalized envelopes with prescribed sum of the principal radii of curvature, Ucen. Zapiski Leningrad Gos. Ped. Inst. 395 (1970), 231-255, USSR.
  7. Existence of a k-dimensional plane in the paratingency of a k-surface in (n + 1)-dimensional Euclidean space, Ucen. Zapiski Leningrad Gos. Ped. Inst. 395 (1970), 256-267, USSR.
  8. The Christoffel problem for open surfaces, Summary of Ph.D. dissertation, Leningrad University Press, 1970, 1-12, USSR.
  9. Convexity of the complement to the paratingency of a continuous surface, Vestnik Leningrad University, (19) 4 (1971), 150-152, USSR.
  10. Effect of atmospheric thermal stratification on sand transportation by wind (with V.Sh. Ezhikova, V.B. Ivanov, S.V. Solonin), Problems of Desert Development, (1)(1972), 74-77, USSR.
  11. On the problem of processing the data on deflation processes (with V.Sh. Ezhikova), Notes of the Ashkabad Hydrometeorological Observatory, (7)(1972), 50-55, USSR.
  12. Uniqueness of the solution in the Christoffel and Minkowski problems for open surfaces, Matem. Zametki 1 (1973), 41-49; English transl., Mathematical Notes of the Academy of Sciences of the USSR, 13 (1/2)(1973), 26-31.
  13. On the relationship between the sample size and the number of variables in a linear regression model, Comm. in Statistics, Part A, Theory and Methods, A7(6)(1978), 509-516.
  14. On minimum variance unbiased estimations and characterization of probability densities (with J. Singh), Proc. International Conf. on Optimizing Methods in Statistics, ed. F. Rustagi, Academic Press (1979), 435-442.
  15. On certain elliptic differential equations on a hypersphere and their geometric applications, Indiana Univ. Math. J., (1) 28 (1979), 35-51.
  16. On compact submanifolds with nondegenerate parallel normal vector fields, Pacific J. Math., (2) 83 (1979), 481-493.
  17. A note on the asymptotic behavior of the sum of principal radii of curvature on noncompact complete hypersurfaces, Proc. Amer. Math. Soc., (3) 77 (1979), 381-384.
  18. Eigenvalues of the Laplacian and uniqueness in the multidimensional Minkowski problem, J. Differential Geometry, (1) 14 (1979), 93-98.
  19. Infinitesimal deformations of submanifolds preserving a parallel normal vector field, Lecture Notes in Math., Springer-Verlag, 792 (1980), 381-405.
  20. Some remarks on elliptic equations and infinitesimal deformations of submanifolds, Lecture Notes in Math., Springer-Verlag, 838 (1981), 211-220.
  21. An application of A.D. Aleksandrov's inequality to the problem of characterization of spheres, Lecture Notes in Math., Springer-Verlag, 838 (1981), 275-288.
  22. On the theory of infinitesimal deformations of submanifolds in Euclidean space, Amer. J. Math., Supplemental Volume, Contributions to Analysis and Geometry, Eds. D.N. Clark, G. Pecelli, R. Saksteder (1981), 219-238.
  23. On the linearized Monge-Ampere equations related to the boundary value Minkowski problem and its generalizations, in "Monge-Ampere Equations and Related Topics", Proc. of Seminar held in Firenze, Sep.-Oct. 1980, ed. F. Gherardelli, 1982, 79-112.
  24. The boundary value Minkowski problem. The parametric case, Annali della Scuola Normale Superiore di Pisa, Ser. IV, IX (3)(1982), 463-490.
  25. Codazzi tensors and equations of Monge-Ampere type on compact manifolds of constant sectional curvature (with U. Simon), J. fur Reine and Angewandte Mathematik, B. 342 (1983), 35-65.
  26. Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature, in the series of Seminari dell' Istituto di Mathematica Applicata "Giovanni Sansone", Universita degli Studi di Firenze, 1983, 1-39
  27. Hypersurfaces with prescribed Gaussian curvature and related equations of Monge-Ampere type, Communications in Partial Differential Equations 9 (8)(1984), 807-838.
  28. The problem of embedding $S sup n$ into $R sup n+1$ with prescribed Gauss curvature and its solution by variational methods, Transactions of the American Mathematical Society 295 (1)(1986), 291-303.
  29. Three problems in nonlinear differential geometry, in Contemporary Mathematics, Vol. 51, ed. by D. DeTurck, (1986), 105-111.
  30. A rigidity theorem for submanifolds of spheres (with U. Simon), Prace Naukowe Politechniki Szczecinskij, NR 323, Inst. Mat., NR 9, International Seminar on Differential Geometry, Szczecin, 1987, 89-91.
  31. Radially symmetric solutions of a Monge-Ampere equation arising in the reflector mapping problem (with P. Waltman), Proceedings of the UAB International Conference on Differential Equations and Mathematical Physics, ed. by I.Knowles and Y.Saito, Lecture Notes in Math., Springer-Verlag, 1285 (1987), 973-1000. 1285 (1987), 361-374.
  32. Near radially symmetric solutions of an inverse problem in geometric optics, Journal of Inverse Problems, 3 (1987), 743-756.
  33. The Christoffel problem in relative differential geometry (with U. Simon), Topics in Differential Geometry, Ed. by J. Szenthe and L. Tamassy, Colloquia Mathematica Societatis Janos Bolyai, North-Holland, 46 (1988), 973-1000.
  34. Remarks on projectively invariant nonlinear partial differential equations, Results in Mathematics, vol. 13, no. 3/4 (1988), 363-366.
  35. On the numerical solution of the equation ${\(pd sup 2 z} over {\(pd x sup 2} ~ {\(pd sup 2 z} over {\(pd y sup 2} ~-~ left ( ~ {\(pd sup 2 z} over {\(pd x \(pd y} ~ right ) sup 2 ~=~ f$ and its discretizations, I (with L. Prussner), Numerische Mathematik 54 (1988), 271-293.
  36. Sur une propriete des fonctions propres de l'operateur de Laplace Beltrami, (with M. Chipot), Bulletin Canadien de Mathematiques, 32 (1989), 29-39.
  37. Singular nonlinear boundary value problems for second order ordinary differential equations, (with J. A. Gatica and P. Waltman), Journal of Differential Equations, vol.79, no.1 (1989), 62-78.
  38. On reconstructing a reflecting surface from the scattering data in the geometric optics approximation, Journal of Inverse problems, 5 (1989), 51-65.
  39. The Gauss curvature and Minkowski problems in space forms, Contemporary Mathematics, Vol. 101, ed. by R. Greene and S.-Y. Cheng, (1989), 107-123.
  40. Iterative procedures for nolinear second order boundary value problems (with J. A. Gatica and P. Waltman), Annali di Matematica Pura ed Applicata, (IV), Vol.CLVII(1990), 1-25.
  41. A stability theorem for unbounded curves with periodic curvature and torsion (with Y.A. Volkov), Differential Geometry, ed. by B. Lawson and K. Tenenblat, Pitman Surveys in Pure and Applied Mathematics, v. 52(1991),303-310.
  42. Evolution of nonparametric surfaces with speed depending on curvature,I. The Gauss curvature case, Indiana Univ. Journal, Vol.40, No.1(1991), 237-257
  43. A priori estimates of the principal curvatures of spacelike hypersurfaces in de Sitter space with applications to hypersurfaces in hyperbolic space, American Journal of Mathematics, 114(1992), 605-626
  44. Affine geometry and polar surfaces (with U. Simon), in "Analysis and Geometry", ed. by B. Fuchssteiner and W.A.J. Luxemburg, Wissenschaftsverlag, 87-113, 1992
  45. Generalized convex bodies and generalized envelopes, Contemp. Math., Vol. 140, ed. by E. Grinberg, (1992), 105-113
  46. On the problem of synthesis of offset dual shaped reflector antennas, Proceedings of the 1992 Antenna Applications Symposium, University of Illinois, Urbana, September, 1992, 1-23
  47. Evolution of nonparametric surfaces with speed depending on curvature, II. The mean curvature case, (with N. Uraltseva), Comm. on Pure and Apllied Math., Vol. 46, No.1(1993), 97-135
  48. On the energy conservation equation in the reflector mapping problem, (with E. Newman), Applied Mathematics Letters, Vol. 6, No. 1 (1993), 91-95.
  49. Evolution of nonparametric surfaces with speed depending on curvature, III. Some remarks on mean curvature and anisotropic flows, (with N. Uraltseva), in "Degenerate Diffusions", Ed. by W.-M. Ni.L.A. Peletier, and J.L. Vazques, IMA Volumes in Mathematics and its Applications, v.47, Springer-Verlag, 1993, pp.141-156
  50. Self-similar solutions and asymptotic behavior of flows of nonparametric surfaces driven by Gauss or mean curvature, Symposia in Pure Mathematics, Vol. 54, Part 1, American Math. Soc., 1993, pp. 389-402
  51. A formula for computing illumination intensity in a mirror optical system (with E. Newman and L. Prussner), Journal of the Optical Society of America A, Vol. 10, No. 9 (1993), pp. 1895-1901
  52. The ray tracing and energy conservation equations for mirror systems with two reflecting surfaces, Computers and Mathematics with Applications, Vol. 26, No. 7 (1993), 9-18
  53. Spacelike hypersurfaces with constant mean curvature in Lorentz space (with J.C.M. Barbosa), Mathematica Contemporanea, v.4 (1993), 27-44
  54. Stable spacelike hypersurfaces with constant mean curvature in Lorentz space (with J.C.M. Barbosa), Geometry and Global Analysis, Tohoku Univ., Sendai (1993), 161-164
  55. A new technique for synthesis of dual reflector systems (with L. Prussner), 10th Annual Review of Progress in Applied Electromagnetics, Monterey, CA, March, 1994, 45-52.
  56. Differential-geometric methods in design of reflector antennas (with E. Newman), Symposia Mathematica, Vol. 35 (1994), 205-223
  57. On analysis of reflector/refractor systems (with E. Newman), Proceedings of the IMACS Conference, v.2(1994), 854-857
  58. A comparison of synthesis techniques for reflector antennas (with D. Bodnar), Proceedings of the IMACS Conference, v.2(1994), 588-591
  59. New analytical and computational techniques for redirecting and reshaping energy flows, Proceedings of the IMACS Conference, v.2(1994), 854-857
  60. Optical design of a two-mirror asymmetrical reshaping system and its application in superbroadband color center lasers (with L.D. Prussner, D. Shealy, and S. Mirov), SPIE Proceedings, v.2263 (1994), 10-18
  61. On the regularity of the solution to a generalization of the Minkowski problem (with E. Lutwak), Journal of Differential Geometry, v.41, No.1(1995), 227-246
  62. Long time behavior of flows moving by mean curvature (with N. Uraltseva), American Math. Soc. Transl., (2), v. 164, (1995), 163-170
  63. A rigorous technique for synthesis of offset three-reflector systems, Proceedings of the 1995 Antenna Applications Symposium, Allerton Park, Monticello, Illinois, (1995), 1-12
  64. Determining the intensities produced by reflected and refracted wave fronts in geometrical optics, (with E. Newman), Journal of the Optical Society of America, A , v.12, no.4 (1995), 784-793
  65. The reflector problem for closed surfaces, Lecture Notes in Pure and Applied Mathematics, Partial Differential Equations and Applications, Marcel Dekker, Inc., vol.177 (1996), 265-270
  66. Determination of reflector surfaces from near-field scattering data, (with S. Kochengin), J. of Inverse Problems, vol. 13 (1997), 363-373
  67. Long time behavior of flows moving by mean curvature, II. (with N. Uraltseva), Topological Methods in Nonlinear Analysis, Vol.9, No.1 (1997), 17-28
  68. Determination of reflector surfaces from near-field scattering data II. Numerical Solution (with S. Kochengin), Numerische Mathematik, Vol. 79 (1998), 553-568
  69. On the design of reflectors with prespecified distribution of virtual sources and intensities (with S. Kochengin and O. von Tempski), Inverse Problems, vol. 14 (1998), 1-18
  70. On the numerical solution of the problem of reflector design with given far-field scattering data (with L. Caffarelli and S. Kochengin), Contemporary Mathematics, Vol.226(1999), 13-32.
  71. Polar hypersurfaces in spheres (with F. Brito, H.L. Liu, U. Simon, C.P. Wang), Geometry and Topology of Submanifolds, v. IX (1999), 33-47
  72. Electromagnetic scattering, 1D/2D obstacle problem, and discrete geodesics, Proceedings of the 16-th European Workshop on Computational Geometry, EURO-CG-2000, March 2000, pp. 66-69
  73. On the geometry of convex reflectors, Banach Center Publications, v. 57(2002), 155-169.
  74. A priori estimates for starshaped compact hypersurfaces with prescribed m-th curvature function in space forms (with J.L. Barbosa and J.H. Lira), Nonlinear Problems in Mathematical Physics and Related Topics, v. I(2002), 35-52. [This paper also appeared in the Russian translation of the volume, Nelinejnye Zadachi Matematicheskoy Phisiki i Smejnye Voprosy, tom 1(2002), 33-48.]
  75. Starshaped compact hypersurfaces with prescribed m-th mean curvature in elliptic space (with Y.Y. Li), Journal of Partial Diff. Eqs, 15(2002), 68-80.
  76. Mathematical aspects of design of beam shaping surfaces in geometrical optics, in Trends in Nonlinear Analysis, ed. by M. Kirkilionis, S. Kromker, R. Rannacher, F. Tomi, Springer-Verlag, 2003, pp. 193-224.
  77. Computational algorithms for constructing reflectors (with S. Kochengin), Computing and Visualization in Science, 6:15-21(2003).
  78. Optical design of single reflector systems and the Monge-Kantorovich mass transfer problem (with T. Glimm), J. of Math. Sciences, Vol. 117, No. 3(2003), 4096-4108.
  79. Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle (with T. Glimm), Indiana Univ.  Math. J., Vol. 53, No. 5(2004), 1255-1277.
  80. Geometric and variational methods in optical design of reflecting surfaces with prescribed irradiance properties, Proceedings of SPIE, Nonimaging Optics and Efficient Ellimination Systems, ed. by R. Winston and R.J. Koshel, v. 5942(2005), 594207-1 - 594207-12.
  81. Optical design of two-mirror beam-shaping systems. Convex and non-convex solutions for symmetric and non-symmetric data, Proceedings of SPIE, Laser Beam Shaping VI - Optical Engineering and Instrumentation, ed. by F. M. Dickey and D. L. Shealy, v. 5876(2005), 203-214.
  82. A variational solution of A.D. Aleksandrov problem of existence of a noncompact complete convex polytope with prescribed integral Gauss curvature, Banach Center Publications, v. 69(2005), 81-90.
  83. A rigorous method for synthesis of offset shaped reflector antennas, Computing Letters, vol. 2, No. 1-2(2006), 29-49.
  84. Freeform optical systems with prescribed irradiance properties in the near-field, Proceedings of SPIE, International Optical Design Conference 2006, ed. by G.G. Gregory, J.M. Howard and R.J. Koshel, v. 6342(2006), 634211-1 - 634211-12.
  85. Existence of optimal maps in the reflector-type problems (with W. Gangbo), ESAIM. Control, Optimization and Calculus of Variations, Vol. 13, No.1(2007), 93-106.
  86. Uniqueness of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space (with J.L. Barbosa and J.H. Lira), Illinois J. of Mathematics, Vol. 51, No.2 (2007), 571-582.
  87. Embedding $S^n$ into $R^{n+1}$ with prescribed integral Gauss curvature and optimal mass transport on $S^n$, Advances in Mathematics, Vol. 213(2007), 600-620.
  88. Optical design of freeform two-mirror beam-shaping systems, J. of Optical Society of America A, 24(12) 2007, 3741-3752.
  89. On the geometry of convex reflectors, II. Polar reflectors, Results in Math., no. 52 (2008), 359-367.
  90. Weak solutions of one inverse problem in geometric optics (with L. Caffarelli), J. of Math. Sciences, Vol. 154(1)(2008), 37-46, 2008.
  91. On design of freeform refractive beam-shapers, sensitivity to figure error and convexity of lenses, J. of Optical Society of America A, 25(12)(2008), 3067-3076.
  92. A Minkowski-style theorem for focal functions of compact convex reflectors, Transactions of the AMS, 360 (2008), no. 2, 563-574.
  93. Kummer configurations and $S_m$-reflector problems:Hypersurfaces in ${\mathbb{R}^{n+1}}$ with given mean intensity, Results in Math., 56(2009), 519-535.
  94. Sensitivity to figure error of a freeform refractive beam shaper, Proceedings of SPIE, Nonimaging Optics:VI Conference, ed. by R. Winston and J. Gordon, vol. 7423(2009), 04-1--04-10.
  95. A characterization of revolution quadrics by a system of partial differential equations, Proceedings of the AMS, 138(2010), no. 11, 4075-4080.
  96. Designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport, Archive for Rational Mechanics and Analysis, 201(2011), 1013-1045.
  97. An optimal mass transport approach to the near-field reflector problem in optical design (with T. Graf), Inverse Problems, 28(2012), 1-15.
  98. Ray mapping and illumination control (with J. Rubinstein and G. Wolansky),Journal of Photonics for Energy, 3(1)(2013), 035599-1--7.
  99. Differential equations for design of a freeform single lens with prescribed irradiance properties. Optical Engineering, 53(3)(2014), 031302-1--10.
  100. Controlling light with freeform optics: recent progress in computational methods for optical design of freeform lenses with prescribed irradiance properties (with B. Cherkasskiy), Proc. SPIE 9 191(2014), 919105-1--919105-7
  101. Controlling light with freeform optics, SPIE Newsroom. DOI:10.1117/2.1201009.001234. Published Online: September 21, 2014. http://spie.org/x110544.xml, 4 pages.
  102. On the use of the supporting quadric method in the problem of the light field eikonal calculation, (with L. Doskolovich, M. Moiseev, E. Bezus), Optics Express, 23(2015), 19605-19617.
  103. Diffraction effects in freeform optics (with M. Ricketts and R. Winston), Proc. SPIE 9572(2015), doi:10.1117/12.2191895.
  104. Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light, (with J. Rubinstein and G. Wolansky), Advances in Appl. Math., 62(2015), 160-183.