American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 159-169. doi: 10.3934/proc.2013.2013.159

On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space

Chiara Corsato 1, , Franco Obersnel 1, , Pierpaolo Omari 1, and  Sabrina Rivetti 1,

1. 

Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy, Italy

Received  September 2012 Revised  March 2013 Published  November 2013

We develop a lower and upper solution method for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space \begin{equation*} \begin{cases} -{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u) & \hbox{ in } \Omega, \\ u=0& \hbox{ on } \partial \Omega. \end{cases} \end{equation*} Here $\Omega$ is a bounded regular domain in $\mathbb {R}^N$ and the function $f$ satisfies the Carathéodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.
Keywords: lower and upper solutions, Mean curvature, partial differential equation, quasilinear, Dirichlet condition, existence, elliptic, Minkowski space, multiplicity..
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J62, 35J75, 35J93, 35A01, 47H0.
Citation: 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
References:
[1]

R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature,, Comm. Math. Phys. \textbf{87} (1982/83), 87 (): 131. Google Scholar

[2]

C. Bereanu, P. Jebelean, and P. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space,, J. Funct. Anal. \textbf{264} (2013), 264 (2013), 270. Google Scholar

[3]

C. Bereanu, P. Jebelean, and P. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space,, J. Funct. Anal. \textbf{265} (2013), 265 (2013), 644. Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations \textbf{23} (2010), 23 (2010), 801. Google Scholar

[5]

I. Coelho, C. Corsato, F. Obersnel, and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation,, Adv. Nonlinear Stud. \textbf{12} (2012), 12 (2012), 621. Google Scholar

[6]

I. Coelho, C. Corsato, and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball,, Topol. Methods Nonlinear Anal. (2013), (2013). Google Scholar

[7]

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. \textbf{405} (2013) 227-239., 405 (2013), 227. Google Scholar

[8]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds,, Comm. Math. Phys. \textbf{89} (1983), 89 (1983), 523. Google Scholar

[9]

J. Mawhin, Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator,, Milan J. Math. \textbf{79} (2011), 79 (2011), 95. Google Scholar

[10]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential,, Comm. Partial Differential Equations {\bf 21} (1996), 21 (1996), 721. Google Scholar

show all references

References:
[1]

R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature,, Comm. Math. Phys. \textbf{87} (1982/83), 87 (): 131. Google Scholar

[2]

C. Bereanu, P. Jebelean, and P. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space,, J. Funct. Anal. \textbf{264} (2013), 264 (2013), 270. Google Scholar

[3]

C. Bereanu, P. Jebelean, and P. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space,, J. Funct. Anal. \textbf{265} (2013), 265 (2013), 644. Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations \textbf{23} (2010), 23 (2010), 801. Google Scholar

[5]

I. Coelho, C. Corsato, F. Obersnel, and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation,, Adv. Nonlinear Stud. \textbf{12} (2012), 12 (2012), 621. Google Scholar

[6]

I. Coelho, C. Corsato, and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball,, Topol. Methods Nonlinear Anal. (2013), (2013). Google Scholar

[7]

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. \textbf{405} (2013) 227-239., 405 (2013), 227. Google Scholar

[8]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds,, Comm. Math. Phys. \textbf{89} (1983), 89 (1983), 523. Google Scholar

[9]

J. Mawhin, Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator,, Milan J. Math. \textbf{79} (2011), 79 (2011), 95. Google Scholar

[10]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential,, Comm. Partial Differential Equations {\bf 21} (1996), 21 (1996), 721. Google Scholar

[1]

Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271
[2]

Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963
[3]

Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006
[4]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9
[5]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[6]

Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155
[7]

Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
[8]

Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061
[9]

Roberta Filippucci, Chiara Lini. Existence of solutions for quasilinear Dirichlet problems with gradient terms. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 267-286. doi: 10.3934/dcdss.2019019
[10]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[11]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315
[12]

Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169
[13]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[14]

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 & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
[15]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
[16]

Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006
[17]

Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112
[18]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[19]

Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1691-1706. doi: 10.3934/dcds.2017070
[20]

Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036

 Impact Factor: 

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences