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Two problems related to prescribed curvature measures
1. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China, China |
References:
[1] |
L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations. IV. Starshaped compact Weingarten hypersurfaces,, in, (1986), 1.
|
[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. |
[3] |
C. Gerhardt., "Curvature Problems,", Series in Geometry and Topology, 39 (2006).
|
[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. |
[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. |
[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P. Guan and Y. Li, unpublished, notes, (1995).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. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second Edition, (1998).
|
[12] |
N. M. Ivochkina, Solution of the Dirichlet problem for curvature equations of order m,, Mathematics of the USSR-Sbornik, 67 (1990), 317.
|
[13] |
N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$,, Leningrad Math. J., 2 (1991), 192.
|
[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. |
[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).
|
[18] |
R. Schneider, "Convex Bodies: the Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (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.
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.
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.
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.
doi: 10.1515/crll.2000.016. |
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.
|
[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. |
[3] |
C. Gerhardt., "Curvature Problems,", Series in Geometry and Topology, 39 (2006).
|
[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. |
[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. |
[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. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second Edition, (1998).
|
[12] |
N. M. Ivochkina, Solution of the Dirichlet problem for curvature equations of order m,, Mathematics of the USSR-Sbornik, 67 (1990), 317.
|
[13] |
N. M. Ivochkina, The Dirichlet problem for the equations of curvature of order $m$,, Leningrad Math. J., 2 (1991), 192.
|
[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. |
[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).
|
[18] |
R. Schneider, "Convex Bodies: the Brunn-Minkowski Theory,", Encyclopedia of Mathematics and its Applications, 44 (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.
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.
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.
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.
doi: 10.1515/crll.2000.016. |
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