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Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China |
2. | Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain |
3. | Departamento de Matemática Aplicada, & Research Unit Modeling Nature (MNat), Universidad de Granada, 18071 Granada, Spain |
$ \begin{eqnarray} \text{div} \left(\frac{a\nabla u}{\sqrt{1-a^2\vert \nabla u\vert^2}}\right)+\frac{g(\nabla u, \nabla a)}{\sqrt{1-a^2\vert \nabla u\vert^2}} = \lambda NH \end{eqnarray} $ |
$ 0 $ |
$ H $ |
$ 0 $ |
References:
[1] |
J. A. Aledo, A. Romero and R. M. Rubio, The existence and uniqueness of standard static splitting, Class. Quantum Grav., 32 (2015), 105004, 9 pp.
doi: 10.1088/0264-9381/32/10/105004. |
[2] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.
doi: 10.1007/BF01211061. |
[3] |
C. Bereanu, P. Jebelean and P. J. Torres,
Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287.
doi: 10.1016/j.jfa.2012.10.010. |
[4] |
C. Bereanu, P. Jebelean and P. J. Torres,
Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659.
doi: 10.1016/j.jfa.2013.04.006. |
[5] |
C. Bereanu, P. Jebelean and J. Mawhin,
The Dirichlet problem with mean curvature operator in Minkowski spaceÔĢöa variational approach, Adv. Nonlinear Stud., 14 (2014), 315-326.
doi: 10.1515/ans-2014-0204. |
[6] |
E. Calabi,
Examples of Bernstein problems for some nonlinear equations, Global Analysis, Amer. Math. Soc., Providence, R.I., 15 (1970), 223-230.
|
[7] |
S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 1998.
![]() ![]() |
[8] |
S.-Y. Cheng and S.-T. Yau,
Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104 (1976), 407-419.
doi: 10.2307/1970963. |
[9] |
C. Corsato, F. Obersnel, P. Omari and S. Rivetti,
Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227-239.
doi: 10.1016/j.jmaa.2013.04.003. |
[10] |
G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations, 55 (2016), Art. 72, 17 pp.
doi: 10.1007/s00526-016-1012-9. |
[11] |
G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 97, 28 pp.
doi: 10.1007/s00526-016-1029-0. |
[12] |
G. W. Dai,
Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.
doi: 10.3934/dcds.2016034. |
[13] |
G. W. Dai, Global bifurcation for problem with mean curvature operator on general domain, Nonlinear Differential Equations Appl., 24 (2017), Art. 30, 10 pp.
doi: 10.1007/s00030-017-0454-x. |
[14] |
G. W. Dai,
Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J., 67 (2018), 2103-2121.
doi: 10.1512/iumj.2018.67.7546. |
[15] |
G. W. Dai, A. Romero and P. J. Torres,
Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann-Lemaître-Robertson-Walker spacetimes, J. Differential Equations, 264 (2018), 7242-7269.
doi: 10.1016/j.jde.2018.02.014. |
[16] |
M. Dajczer, Submanifolds and Isometric Immersions, Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX, 1990. |
[17] |
E. N. Dancer,
On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.
doi: 10.1512/iumj.1974.23.23087. |
[18] |
E. N. Dancer,
Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.
doi: 10.1112/S002460930200108X. |
[19] |
D. Fuente, A. Romero and P. J. Torres, Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner-Nordström spacetimes, Class. Quantum Grav., 32 (2015), 035018, 17 pp. Corrigendum: Class. Quantum Grav., 35 (2018), 059501, 2 pp.
doi: 10.1088/1361-6382/aaa5c9. |
[20] |
E. L. Ince, Ordinary Differential Equation, Dover Publication, New York, 1944. |
[21] |
J. L. Kazdan, Applications of Partial Differential Equations to Problems in Geometry, Grad. Texts in Math., Springer, 2004. |
[22] |
B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.
![]() ![]() |
[23] |
R. Osserman,
The minimal surface equation, Seminar on Nonlinear Partial Differential Equations, Math. Sci. Res. Inst. Publ., Springer, New York, 2 (1984), 237-259.
doi: 10.1007/978-1-4612-1110-5_13. |
[24] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
P. H. Rabinowitz,
On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.
doi: 10.1016/0022-0396(73)90061-2. |
[26] |
R. K. Sachs and H. H. Wu, General Relativity for Mathematicians, Graduate Texts in Mathematics, Vol. 48. Springer-Verlag, New York-Heidelberg, 1977. |
[27] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140. American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/gsm/140. |
[28] |
A. E. Treibergs,
Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39-56.
doi: 10.1007/BF01404755. |
[29] |
W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics. Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0601-9. |
[30] |
G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23. Princeton University Press, Princeton, N. J. 1958. |
show all references
References:
[1] |
J. A. Aledo, A. Romero and R. M. Rubio, The existence and uniqueness of standard static splitting, Class. Quantum Grav., 32 (2015), 105004, 9 pp.
doi: 10.1088/0264-9381/32/10/105004. |
[2] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.
doi: 10.1007/BF01211061. |
[3] |
C. Bereanu, P. Jebelean and P. J. Torres,
Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287.
doi: 10.1016/j.jfa.2012.10.010. |
[4] |
C. Bereanu, P. Jebelean and P. J. Torres,
Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659.
doi: 10.1016/j.jfa.2013.04.006. |
[5] |
C. Bereanu, P. Jebelean and J. Mawhin,
The Dirichlet problem with mean curvature operator in Minkowski spaceÔĢöa variational approach, Adv. Nonlinear Stud., 14 (2014), 315-326.
doi: 10.1515/ans-2014-0204. |
[6] |
E. Calabi,
Examples of Bernstein problems for some nonlinear equations, Global Analysis, Amer. Math. Soc., Providence, R.I., 15 (1970), 223-230.
|
[7] |
S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 1998.
![]() ![]() |
[8] |
S.-Y. Cheng and S.-T. Yau,
Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math., 104 (1976), 407-419.
doi: 10.2307/1970963. |
[9] |
C. Corsato, F. Obersnel, P. Omari and S. Rivetti,
Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227-239.
doi: 10.1016/j.jmaa.2013.04.003. |
[10] |
G. W. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations, 55 (2016), Art. 72, 17 pp.
doi: 10.1007/s00526-016-1012-9. |
[11] |
G. W. Dai, Two Whyburn type topological theorems and its applications to Monge-Ampère equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 97, 28 pp.
doi: 10.1007/s00526-016-1029-0. |
[12] |
G. W. Dai,
Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros, Discrete Contin. Dyn. Syst., 36 (2016), 5323-5345.
doi: 10.3934/dcds.2016034. |
[13] |
G. W. Dai, Global bifurcation for problem with mean curvature operator on general domain, Nonlinear Differential Equations Appl., 24 (2017), Art. 30, 10 pp.
doi: 10.1007/s00030-017-0454-x. |
[14] |
G. W. Dai,
Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J., 67 (2018), 2103-2121.
doi: 10.1512/iumj.2018.67.7546. |
[15] |
G. W. Dai, A. Romero and P. J. Torres,
Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann-Lemaître-Robertson-Walker spacetimes, J. Differential Equations, 264 (2018), 7242-7269.
doi: 10.1016/j.jde.2018.02.014. |
[16] |
M. Dajczer, Submanifolds and Isometric Immersions, Mathematics Lecture Series, 13. Publish or Perish, Inc., Houston, TX, 1990. |
[17] |
E. N. Dancer,
On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1973/74), 1069-1076.
doi: 10.1512/iumj.1974.23.23087. |
[18] |
E. N. Dancer,
Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.
doi: 10.1112/S002460930200108X. |
[19] |
D. Fuente, A. Romero and P. J. Torres, Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner-Nordström spacetimes, Class. Quantum Grav., 32 (2015), 035018, 17 pp. Corrigendum: Class. Quantum Grav., 35 (2018), 059501, 2 pp.
doi: 10.1088/1361-6382/aaa5c9. |
[20] |
E. L. Ince, Ordinary Differential Equation, Dover Publication, New York, 1944. |
[21] |
J. L. Kazdan, Applications of Partial Differential Equations to Problems in Geometry, Grad. Texts in Math., Springer, 2004. |
[22] |
B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.
![]() ![]() |
[23] |
R. Osserman,
The minimal surface equation, Seminar on Nonlinear Partial Differential Equations, Math. Sci. Res. Inst. Publ., Springer, New York, 2 (1984), 237-259.
doi: 10.1007/978-1-4612-1110-5_13. |
[24] |
P. H. Rabinowitz,
Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.
doi: 10.1016/0022-1236(71)90030-9. |
[25] |
P. H. Rabinowitz,
On bifurcation from infinity, J. Differential Equations, 14 (1973), 462-475.
doi: 10.1016/0022-0396(73)90061-2. |
[26] |
R. K. Sachs and H. H. Wu, General Relativity for Mathematicians, Graduate Texts in Mathematics, Vol. 48. Springer-Verlag, New York-Heidelberg, 1977. |
[27] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140. American Mathematical Society, Providence, RI, 2012.
doi: 10.1090/gsm/140. |
[28] |
A. E. Treibergs,
Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math., 66 (1982), 39-56.
doi: 10.1007/BF01404755. |
[29] |
W. Walter, Ordinary Differential Equations, Graduate Texts in Mathematics, 182. Readings in Mathematics. Springer-Verlag, New York, 1998.
doi: 10.1007/978-1-4612-0601-9. |
[30] |
G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23. Princeton University Press, Princeton, N. J. 1958. |

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