Show Summary Details
More options …

# 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

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2017: 0.23

Online
ISSN
1572-9176
See all formats and pricing
More options …
Volume 24, Issue 1

# 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
• Email
• Other articles by this author:
/ 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
• Email
• Other articles by this author:
/ Pierpaolo Omari
• Corresponding author
• Dipartimento di Matematica e Geoscienze, Sezione di Matematica e Informatica, Università degli Studi di Trieste, Via A. Valerio 12/1, 34127 Trieste, Italy
• Email
• Other articles by this author:
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

$\left\{\begin{array}{cccc}\hfill -div\left(\frac{\nabla u}{\sqrt{1-{|\nabla u|}^{2}}}\right)& =f\left(x,u,\nabla u\right)\hfill & & \hfill \text{in}\mathrm{\Omega },\\ \hfill u& =0\hfill & & \hfill \text{on}\partial \mathrm{\Omega }.\end{array}$

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.

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

Dedicated to Professor Ivan Tarielovich Kiguradze

## References

• [1]

Amann H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. Google Scholar

• [2]

Bartnik R., Maximal surfaces and general relativity, Miniconference on Geometry and Partial Differential Equations (Canberra 1986), Proc. Centre Math. Appl. Austral. Nat. Univ. 12, Australian National University, Canberra (1987), 24–49. Google Scholar

• [3]

Bartnik R. and Simon L., Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), no. 1, 131–152. Google Scholar

• [4]

Bereanu C., de la Fuente D., Romero A. and Torres P. J., Existence and multiplicity of entire radial spacelike graphs with prescribed mean curvature function in certain Friedmann–Lemaître–Robertson–Walker spacetimes, Commun. Contemp. Math. (2016), 10.1142/S0219199716500061. Google Scholar

• [5]

Bereanu C., Jebelean P. and Mawhin J., The Dirichlet problem with mean curvature operator in Minkowski space – A variational approach, Adv. Nonlinear Stud. 14 (2014), no. 2, 315–326. Google Scholar

• [6]

Bereanu C., Jebelean P. and Torres P. J., Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), no. 4, 644–659. Google Scholar

• [7]

Bereanu C., Jebelean P. and Torres P. J., Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal. 264 (2013), no. 1, 270–287. Google Scholar

• [8]

Bergner M., The Dirichlet problem for graphs of prescribed anisotropic mean curvature in ${ℝ}^{n+1}$, Analysis (Munich) 28 (2008), no. 2, 149–166. Google Scholar

• [9]

Bergner M., On the Dirichlet problem for the prescribed mean curvature equation over general domains, Differential Geom. Appl. 27 (2009), no. 3, 335–343. Google Scholar

• [10]

Cheng S. Y. and Yau S. T., Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces, Ann. of Math. (2) 104 (1976), no. 3, 407–419. Google Scholar

• [11]

Coelho I., Corsato C., Obersnel F. and Omari P., Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud. 12 (2012), no. 3, 621–638. Google Scholar

• [12]

Coelho I., Corsato C. and Rivetti S., Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 23–39. Google Scholar

• [13]

Corsato C., De Coster C. and Omari P., The Dirichlet problem for a prescribed anisotropic mean curvature equation: Existence, uniqueness and regularity of solutions, J. Differential Equations 260 (2016), no. 5, 4572–4618. Google Scholar

• [14]

Corsato C., Obersnel F., Omari P. and Rivetti S., On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, Discrete Contin. Dyn. Syst. Suppl. 2013 (2013), 159–169. Google Scholar

• [15]

Corsato C., Obersnel F., Omari P. and Rivetti S., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl. 405 (2013), no. 1, 227–239. Google Scholar

• [16]

De Coster C., Obersnel F. and Omari P., A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness, Handbook of Differential Equations: Ordinary Differential Equations. Vol. III, Elsevier, Amsterdam (2006), 203–339. Google Scholar

• [17]

De Coster C. and Omari P., Unstable periodic solutions of a parabolic problem in the presence of non-well-ordered lower and upper solutions, J. Funct. Anal. 175 (2000), no. 1, 52–88. Google Scholar

• [18]

De Coster C. and Omari P., Stability and Instability in Periodic Parabolic Problems via Lower and Upper Solutions, Quad. Mat. (II Ser.) 539, Università di Trieste, Trieste, 2003; http://www.dmi.units.it/~omari/DCOARB/DCO.pdf.

• [19]

de la Fuente D., Romero A. and Torres P. J., Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzchild and Reissner–Nordström spacetimes, Classical and Quantum Gravity 32 (2015), no. 3, Article ID 035018. Google Scholar

• [20]

de la Fuente D., Romero A. and Torres P. J., Radial solutions of the Dirichlet problem for the prescribed mean curvature equation in a Robertson–Walker spacetime, Adv. Nonlinear Stud. 15 (2015), no. 1, 171–181. Google Scholar

• [21]

Dupaigne L., Stable Solutions of Elliptic Partial Differential Equations, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, Chapman & Hall/CRC, Boca Raton, 2011. Google Scholar

• [22]

Gerhardt C., H-surfaces in Lorentzian manifolds, Comm. Math. Phys. 89 (1983), no. 4, 523–553. Google Scholar

• [23]

Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 Edition, Classics Math., Springer, Berlin, 2001. Google Scholar

• [24]

Hess P., Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser. 247, Longman Scientific & Technical, Harlow, 1991. Google Scholar

• [25]

Lieberman G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), no. 11, 1203–1219. Google Scholar

• [26]

López R., Constant Mean Curvature Surfaces with Boundary, Springer Monogr. Math., Springer, Heidelberg, 2013. Google Scholar

• [27]

Marquardt T., Remark on the anisotropic prescribed mean curvature equation on arbitrary domains, Math. Z. 264 (2010), no. 3, 507–511. Google Scholar

• [28]

Mawhin J., Radial solutions of Neumann problem for periodic perturbations of the mean extrinsic curvature operator, Milan J. Math. 79 (2011), no. 1, 95–112. Google Scholar

• [29]

Serrin J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. Lond. Ser. A 264 (1969), 413–496. Google Scholar

• [30]

Treibergs A. E., Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39–56. Google Scholar

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,

Export Citation