American Institute of Mathematical Sciences

Advanced
November  2020, 13(11): 3047-3071. doi: 10.3934/dcdss.2020118

Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes

Guowei Dai 1, , Alfonso Romero 2, and  Pedro J. Torres 3,,

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

* Corresponding author

Received  March 2019 Published  November 2020 Early access  October 2019

Fund Project: The first author is supported by NNSF of China (No. 11871129) and Xinghai Youqing funds from Dalian University of Technology, the second one by Spanish MINECO Grant with FEDER funds MTM2016-78807-C2-1-P and the third author by Spanish MINECO Grant with FEDER funds MTM2017-82348-C2-1-P

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime
$ \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} $
with
$ 0 $
-Dirichlet boundary condition on the unit ball. According to the behavior of
$ H $
 near
$ 0 $
, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs.
Keywords: Bifurcation, mean curvature operator, one-sign solution, entire spacelike graph.
Mathematics Subject Classification: 35B32, 53A10.
Citation: Guowei Dai, Alfonso Romero, Pedro J. Torres. Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3047-3071. doi: 10.3934/dcdss.2020118
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.

Figure 1.  Bifurcation diagrams of Theorem 1.1
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034
[2]

Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839
[3]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[4]

Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[5]

Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
[6]

Zixiao Liu, Jiguang Bao. Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022081
[7]

Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225
[8]

Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823
[9]

Alberto Farina, Miguel Angel Navarro. Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1233-1256. doi: 10.3934/dcds.2020076
[10]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
[11]

Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
[12]

Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773
[13]

Vittorio Martino. On the characteristic curvature operator. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
[14]

Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142
[15]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[16]

Xumin Jiang. Isometric embedding with nonnegative Gauss curvature under the graph setting. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3463-3477. doi: 10.3934/dcds.2019143
[17]

G. Kamberov. Prescribing mean curvature: existence and uniqueness problems. Electronic Research Announcements, 1998, 4: 4-11.

[18]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[19]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[20]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (257)
  • HTML views (372)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy