American Institute of Mathematical Sciences

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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. 264 (2013), 270-287.  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. 265 (2013), 644-659.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), 801-810.  Google Scholar

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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. 12 (2012), 621-638.  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), in press. Available at: http://www.dmi.units.it/pubblicazioni/Quaderni_Matematici/624_2012.pdf 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. 405 (2013) 227-239.  Google Scholar

[8]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds, Comm. Math. Phys. 89 (1983), 523-553.  Google Scholar

[9]

J. Mawhin, Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator, Milan J. Math. 79 (2011), 95-112.  Google Scholar

[10]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations 21 (1996), 721-733.  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. 264 (2013), 270-287.  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. 265 (2013), 644-659.  Google Scholar

[4]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23 (2010), 801-810.  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. 12 (2012), 621-638.  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), in press. Available at: http://www.dmi.units.it/pubblicazioni/Quaderni_Matematici/624_2012.pdf 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. 405 (2013) 227-239.  Google Scholar

[8]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds, Comm. Math. Phys. 89 (1983), 523-553.  Google Scholar

[9]

J. Mawhin, Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator, Milan J. Math. 79 (2011), 95-112.  Google Scholar

[10]

P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. Partial Differential Equations 21 (1996), 721-733.  Google Scholar

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