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

Pages: 159 - 169, Issue special, November 2013

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Chiara Corsato - Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy (email)
Franco Obersnel - Università degli Studi di Trieste, Dipartimento di Matematica e Geoscienze - Sezione di Matematica e Informatica, Via A. Valerio 12/1, 34127 Trieste, Italy (email)
Pierpaolo Omari - Università degli Studi di Trieste, Dipartimento di Matematica e Geoscienze - Sezione di Matematica e Informatica, Via A. Valerio 12/1, 34127 Trieste, Italy (email)
Sabrina Rivetti - Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy (email)

Abstract: 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:  Mean curvature, Minkowski space, partial differential equation, quasilinear, elliptic, Dirichlet condition, lower and upper solutions, existence, multiplicity.
Mathematics Subject Classification:  Primary: 35J25; Secondary: 35J62, 35J75, 35J93, 35A01, 47H07.

Received: September 2012;      Revised: March 2013;      Published: November 2013.

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