Two problems related to prescribed curvature measures

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

Two problems related to prescribed curvature measures

Yong Huang ^1, and Lu Xu ^1,

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

Received March 2012 Revised September 2012 Published December 2012

Abstract
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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].

Keywords: curvature equations., curvature measure, Hypersurfaces.

Mathematics Subject Classification: Primary: 35J60, 35J66; Secondary: 53C2.

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[8]	P. Guan and Y. Li, unpublished notes, 1995.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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