American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems - A, 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, (1986), 1.   Google Scholar

[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.  doi: 10.1002/cpa.3160410105.  Google Scholar

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P. Guan, Topics Geometric fully nonlinear equations,, Lecture Notes, (2004).   Google Scholar

[5]

P. Guan, Private, notes., ().   Google Scholar

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

[7]

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

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P. Guan and Y. Li, unpublished, notes, (1995).   Google Scholar

[9]

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

[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.  doi: 10.1093/imrn/rnp007.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second Edition, (1998).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

V. I. Oliker, Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature,, V, (1983).   Google Scholar

[17]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces,", translated from the Russian by Israel Program for Scientific Translations, (1973).   Google Scholar

[18]

R. Schneider, "Convex Bodies: the Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

show all references

References:
[1]

L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces,, in, (1986), 1.   Google Scholar

[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.  doi: 10.1002/cpa.3160410105.  Google Scholar

[3]

C. Gerhardt., "Curvature Problems,", Series in Geometry and Topology, 39 (2006).   Google Scholar

[4]

P. Guan, Topics Geometric fully nonlinear equations,, Lecture Notes, (2004).   Google Scholar

[5]

P. Guan, Private, notes., ().   Google Scholar

[6]

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

[7]

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

[8]

P. Guan and Y. Li, unpublished, notes, (1995).   Google Scholar

[9]

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

[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.  doi: 10.1093/imrn/rnp007.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second Edition, (1998).   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

V. I. Oliker, Existence and uniqueness of convex hypersurfaces with prescribed Gaussian curvature in spaces of constant curvature,, V, (1983).   Google Scholar

[17]

A. V. Pogorelov, "Extrinsic Geometry of Convex Surfaces,", translated from the Russian by Israel Program for Scientific Translations, (1973).   Google Scholar

[18]

R. Schneider, "Convex Bodies: the Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (1993).  doi: 10.1017/CBO9780511526282.  Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

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