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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

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Volume 24, Issue 1

Issues

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

Chiara Corsato
  • Dipartimento di Scienze Economiche, Aziendali, Matematiche e Statistiche, Università degli Studi di Trieste, Piazzale Europa 1, 34127 Trieste, Italy
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/ Franco Obersnel
  • Dipartimento di Matematica e Geoscienze, Sezione di Matematica e Informatica, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy
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/ Pierpaolo Omari
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  • Dipartimento di Matematica e Geoscienze, Sezione di Matematica e Informatica, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy
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Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/gmj-2016-0078

Abstract

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space

{-div(u1-|u|2)=f(x,u,u)in Ω,u=0on Ω.

The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

Keywords: Mean curvature; Lorentz–Minkowski space; partial differential equation; quasilinear elliptic problem; Dirichlet condition; lower and upper solutions; existence; multiplicity; order stability; orderinstability

MSC 2010: 35J25; 35J62; 35J75; 35J93; 35B35; 47H07

Dedicated to Professor Ivan Tarielovich Kiguradze

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About the article

Received: 2016-09-23

Accepted: 2016-11-21

Published Online: 2017-01-12

Published in Print: 2017-03-01


This paper was written under the auspices of INdAM-GNAMPA. The second and the third named authors have also been supported by the University of Trieste, in the frame of the 2015 FRA project “Differential Equations: Qualitative and Computational Theory”.


Citation Information: Georgian Mathematical Journal, Volume 24, Issue 1, Pages 113–134, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2016-0078.

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