# American Institute of Mathematical Sciences

May  2013, 33(5): 1975-1986. doi: 10.3934/dcds.2013.33.1975

## Two problems related to prescribed curvature measures

 1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China, China

Received  March 2012 Revised  September 2012 Published  December 2012

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].
Citation: Yong Huang, Lu Xu. Two problems related to prescribed curvature measures. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1975-1986. doi: 10.3934/dcds.2013.33.1975
##### References:
 [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3.3.CO;2-B. [8] P. Guan and Y. Li, unpublished notes, 1995.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.

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##### References:
 [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.
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