\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Two problems related to prescribed curvature measures

Abstract Related Papers Cited by
  • Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. Moreover, we promote Ivochkina's $C^2$ estimates for prescribed curvature equation in [12,13].
    Mathematics Subject Classification: Primary: 35J60, 35J66; Secondary: 53C21.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces, in "Current Topics in Partial Differential Equations" (Kinokuniya, Tokyo.), (1986), 1-26.

    [2]

    L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second-order elliptic equations. V. The Dirichlet problem for Weingarten hypersurfaces, Comm. Pure Appl. Math., 41 (1988), 47-70.doi: 10.1002/cpa.3160410105.

    [3]

    C. Gerhardt., "Curvature Problems," Series in Geometry and Topology, 39. International Press, Somerville, MA, 2006.

    [4]

    P. Guan, Topics Geometric fully nonlinear equations, Lecture Notes, 147-page Manuscript (2004). Available from: http://www.math.mcgill.ca/guan/zheda0508.pdf.

    [5]

    P. Guan, Private notes.

    [6]

    B. Guan and P. Guan, Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673.doi: 10.2307/3597202.

    [7]

    P. Guan and Y. Li, $C^{1,1}$ estimates for solutions of a problem of Alexandrov, Comm. Pure Appl. Math., 50 (1997), 789-811.doi: 10.1002/(SICI)1097-0312(199708)50:8<789::AID-CPA4>3.3.CO;2-B.

    [8]

    P. Guan and Y. Li, unpublished notes, 1995.

    [9]

    P. Guan, J. Li and Y. Li, Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942.

    [10]

    P. Guan, C. S. Lin and X. N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not. IMRN, 11 (2009), 1947-1975.doi: 10.1093/imrn/rnp007.

    [11]

    D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second Edition, Revised Third Printing, Springer-Verlag, 1998.

    [12]

    N. M. Ivochkina, Solution of the Dirichlet problem for curvature equations of order m, Mathematics of the USSR-Sbornik, 67 (1990), 317-339.

    [13]

    N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 192-217.

    [14]

    N. V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc., 347 (1995), 857-895.doi: 10.2307/2154876.

    [15]

    M. Lin and N. S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Austral. Math. Soc., 50 (1994), 317-326.

    [16]

    V. I. Oliker, Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature, " Sem. Inst. Mate. Appl. Giovanni Sansone," Univ. Studi Firenze, (1983).

    [17]

    A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces," translated from the Russian by Israel Program for Scientific Translations, Amer. Math. Soc., Providence, RI, 1973.

    [18]

    R. Schneider, "Convex Bodies: the Brunn-Minkowski Theory," Encyclopedia of Mathematics and its Applications, 44, Cambridge Univ. Press, Cambridge, 1993.doi: 10.1017/CBO9780511526282.

    [19]

    W. Sheng, J. Urbas and X. J. Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J., 123 (2004), 235-264.doi: 10.1215/S0012-7094-04-12321-8.

    [20]

    K. Takimoto, Solution to the boundary blowup problem for $k$-curvature equation, Calc. Var. Partial Differential Equations, 26 (2006), 357-377.doi: 10.1007/s00526-006-0011-7.

    [21]

    N. S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179.doi: 10.1007/BF00375406.

    [22]

    J. Urbas, An interior curvature bound for hypersurfaces of prescribed $k$-th mean curvature, J. Reine Angew. Math., 519 (2000), 41-57.doi: 10.1515/crll.2000.016.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(125) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return