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

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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

Abstract
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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.
[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.
[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.
[4]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations \textbf{23} (2010), 23 (2010), 801.
[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.
[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).
[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.
[8]	C. Gerhardt, $H$-surfaces in Lorentzian manifolds,, Comm. Math. Phys. \textbf{89} (1983), 89 (1983), 523.
[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.
[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.

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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.
[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.
[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.
[4]	H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations \textbf{23} (2010), 23 (2010), 801.
[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.
[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).
[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.
[8]	C. Gerhardt, $H$-surfaces in Lorentzian manifolds,, Comm. Math. Phys. \textbf{89} (1983), 89 (1983), 523.
[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.
[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.

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