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On the decay in $ W^{1,\infty} $ for the 1D semilinear damped wave equation on a bounded domain
A class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China |
Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $ \mathbb{H}^{n+1} $. Under some sufficient conditions, we prove the a priori estimates for solutions to the Monge-Ampère type equation $ \det(\kappa-\mathbf{1}) = f(X, \nu(X)) $. Moreover, we obtain an existence result for the compact horo-convex hypersurface $ M $ satisfying the above equation with various assumptions.
References:
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S. B. Alexander and R. J. Currier,
Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions, J. London Math. Soc., 41 (1990), 347-360.
doi: 10.1112/jlms/s2-41.2.347. |
[2] |
I. Ja. Bakel'man and B. E. Kantor,
Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature, Geometry and Topology, 1 (1974), 3-10.
|
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, (1986), 1–26. |
[5] |
F. J. de Andrade, J. L. M. Barbosa and J. H. S. de Lira,
Closed Weingarten hypersurfaces in warped product manifolds, Indiana Univ. Math. J., 58 (2009), 1691-1718.
doi: 10.1512/iumj.2009.58.3631. |
[6] |
J. M. Espinar, J. A. Gálvez and P. Mira,
Hypersurfaces in $\mathbb{H}^{n+1}$ and conformally invariant equations: The generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11 (2009), 903-939.
doi: 10.4171/JEMS/170. |
[7] |
L. C. Evans,
Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363.
doi: 10.1002/cpa.3160350303. |
[8] |
W. J. Firey,
Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21.
doi: 10.1112/S0025579300002321. |
[9] |
B. Guan and P. Guan,
Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673.
doi: 10.2307/3597202. |
[10] |
P. Guan, J. Li and Y. Li,
Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942.
doi: 10.1215/00127094-1645550. |
[11] |
P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not., (2009), 1947–1975.
doi: 10.1093/imrn/rnp007. |
[12] |
P. Guan, C. Ren and Z. Wang,
Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325.
doi: 10.1002/cpa.21528. |
[13] |
Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643. |
[14] |
N. M. Ivochkina,
The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.
|
[15] |
Q. Jin and Y. Li,
Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space, Discrete Contin. Dyn. Syst., 15 (2006), 367-377.
doi: 10.3934/dcds.2006.15.367. |
[16] |
Y. Li,
Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578.
doi: 10.1080/03605308908820666. |
[17] |
Y. Li and V. I. Oliker,
Starshaped compact hypersurfaces with prescribed $m$-th mean curvature in elliptic space, J. Partial Differential Equations, 15 (2002), 68-80.
|
[18] |
V. I. Oliker,
Hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampére type, Comm. Partial Differential Equations, 9 (1984), 807-838.
doi: 10.1080/03605308408820348. |
[19] |
V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, Recent Developments in Geometry (Los Angeles, CA, 1987), Contemp. Math., Amer. Math. Soc., Providence, RI, 101 (1989), 107–123.
doi: 10.1090/conm/101/1034975. |
[20] |
C. Ren and Z. Wang,
On the curvature estimates for Hessian equations, Amer. J. Math., 141 (2019), 1281-1315.
doi: 10.1353/ajm.2019.0033. |
[21] |
C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702. |
[22] |
C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234. |
[23] |
J. Spruck and L. Xiao,
A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form, Rev. Mat. Iberoam., 33 (2017), 547-554.
doi: 10.4171/RMI/948. |
show all references
References:
[1] |
S. B. Alexander and R. J. Currier,
Nonnegatively curved hypersurfaces of hyperbolic space and subharmonic functions, J. London Math. Soc., 41 (1990), 347-360.
doi: 10.1112/jlms/s2-41.2.347. |
[2] |
I. Ja. Bakel'man and B. E. Kantor,
Existence of a hypersurface homeomorphic to the sphere in Euclidean space with a given mean curvature, Geometry and Topology, 1 (1974), 3-10.
|
[3] |
L. Caffarelli, L. Nirenberg and J. Spruck,
The Dirichlet problem for nonlinear second order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math., 37 (1984), 369-402.
doi: 10.1002/cpa.3160370306. |
[4] |
L. Caffarelli, L. Nirenberg and J. Spruck, Nonlinear second order elliptic equations, IV. Starshaped compact Weingarten hypersurfaces, Current Topics in Partial Differential Equations, (1986), 1–26. |
[5] |
F. J. de Andrade, J. L. M. Barbosa and J. H. S. de Lira,
Closed Weingarten hypersurfaces in warped product manifolds, Indiana Univ. Math. J., 58 (2009), 1691-1718.
doi: 10.1512/iumj.2009.58.3631. |
[6] |
J. M. Espinar, J. A. Gálvez and P. Mira,
Hypersurfaces in $\mathbb{H}^{n+1}$ and conformally invariant equations: The generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11 (2009), 903-939.
doi: 10.4171/JEMS/170. |
[7] |
L. C. Evans,
Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math., 35 (1982), 333-363.
doi: 10.1002/cpa.3160350303. |
[8] |
W. J. Firey,
Christoffel's problem for general convex bodies, Mathematika, 15 (1968), 7-21.
doi: 10.1112/S0025579300002321. |
[9] |
B. Guan and P. Guan,
Convex hypersurfaces of prescribed curvatures, Ann. of Math., 156 (2002), 655-673.
doi: 10.2307/3597202. |
[10] |
P. Guan, J. Li and Y. Li,
Hypersurfaces of prescribed curvature measure, Duke Math. J., 161 (2012), 1927-1942.
doi: 10.1215/00127094-1645550. |
[11] |
P. Guan, C. Lin and X.-N. Ma, The existence of convex body with prescribed curvature measures, Int. Math. Res. Not., (2009), 1947–1975.
doi: 10.1093/imrn/rnp007. |
[12] |
P. Guan, C. Ren and Z. Wang,
Global $C^2$-estimates for convex solutions of curvature equations, Comm. Pure Appl. Math., 68 (2015), 1287-1325.
doi: 10.1002/cpa.21528. |
[13] |
Y. Hu, H. Li and Y. Wei, Locally constrained curvature flows and geometric inequalities in hyperbolic space, preprint, arXiv: 2002.10643. |
[14] |
N. M. Ivochkina,
The Dirichlet problem for the equations of curvature of order $m$, Leningrad Math. J., 2 (1991), 631-654.
|
[15] |
Q. Jin and Y. Li,
Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space, Discrete Contin. Dyn. Syst., 15 (2006), 367-377.
doi: 10.3934/dcds.2006.15.367. |
[16] |
Y. Li,
Degree theory for second order nonlinear elliptic operators and its applications, Comm. Partial Differential Equations, 14 (1989), 1541-1578.
doi: 10.1080/03605308908820666. |
[17] |
Y. Li and V. I. Oliker,
Starshaped compact hypersurfaces with prescribed $m$-th mean curvature in elliptic space, J. Partial Differential Equations, 15 (2002), 68-80.
|
[18] |
V. I. Oliker,
Hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampére type, Comm. Partial Differential Equations, 9 (1984), 807-838.
doi: 10.1080/03605308408820348. |
[19] |
V. I. Oliker, The Gauss curvature and Minkowski problems in space forms, Recent Developments in Geometry (Los Angeles, CA, 1987), Contemp. Math., Amer. Math. Soc., Providence, RI, 101 (1989), 107–123.
doi: 10.1090/conm/101/1034975. |
[20] |
C. Ren and Z. Wang,
On the curvature estimates for Hessian equations, Amer. J. Math., 141 (2019), 1281-1315.
doi: 10.1353/ajm.2019.0033. |
[21] |
C. Ren and Z. Wang, The global curvature estimates for the $n-2$ Hessian equation, preprint, arXiv: 2002.08702. |
[22] |
C. Ren and Z. Wang, Notes on the curvature estimates for Hessian equations, preprint, arXiv: 2003.14234. |
[23] |
J. Spruck and L. Xiao,
A note on starshaped compact hypersurfaces with prescribed scalar curvature in space form, Rev. Mat. Iberoam., 33 (2017), 547-554.
doi: 10.4171/RMI/948. |
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