Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T14:05:24.322Z Has data issue: false hasContentIssue false
  • English
  • Français

The spectrum of the mean curvature operator

Part of: Elliptic equations and systems Existence theories Linear function spaces and their duals Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  01 April 2020

Mihai Mihăilescu
Mihai Mihăilescu*
Affiliation:
Department of Mathematics, University of Craiova, 200585Craiova, Romania The Research Institute of the University of Bucharest, University of Bucharest, 050663Bucharest, Romania (mmihailes@yahoo.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the spectrum of the relativistic mean curvature operator on a bounded domain Ω ⊂ ℝN (N ⩾ 1) having smooth boundary, subject to the homogeneous Dirichlet boundary condition, is exactly the interval (λ1(2), ∞), where λ1(2) stands for the principal frequency of the Laplace operator in Ω.

Keywords

Eigenvalue problemsΓ-convergenceMean curvature operatorp-LaplacianVariational inequalities

MSC classification

Primary: 35J92: Quasilinear elliptic equations with $p$-Laplacian
Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 49J40: Variational methods including variational inequalities 49J45: Methods involving semicontinuity and convergence; relaxation
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 451 - 463
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Arcoya, D., Bereanu, C. and Torres, P. J.. Critical point theory for the Lorentz force equation. Arch. Rational Mech. Anal. 232 (2019), 1685–1724.CrossRefGoogle Scholar
2Bartnik, R. and Simon, L.. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Commun. Math. Phys. 87 (1982–83), 131–152.CrossRefGoogle Scholar
3Bereanu, C., Jebelean, P. and Mawhin, J.. The Dirichlet problem with mean curvature operator in Minkowski space – a variational approach. Adv. Nonlinear Stud. 14 (2014), 315–326, Corrigendum. Adv. Nonlinear Stud. 16 (2016), 173–174.CrossRefGoogle Scholar
4Bocea, M. and Mihăilescu, M.. Eigenvalue problems in Orlicz-Sobolev spaces for rapidly growing operators in divergence form. J. Differ. Equ. 256 (2014), 640–657.CrossRefGoogle Scholar
5Bonheure, D., Colasuonno, F. and Földes, J.. On the Born-Infeld equation for electrostatic fields with a superposition of point charges. Annali di Matematica Pura ed Applicata 198 (2019), 749–772.CrossRefGoogle Scholar
6Bonheure, D., D'Avenia, P. and Pomponio, A.. On the electrostatic Born-Infeld equation with extended charges. Commun. Math. Phys. 346 (2016), 877–906.CrossRefGoogle Scholar
7Brezis, H. and Mawhin, J.. Periodic solutions of the forced relativistic pendulum. Differ. Integral Equ. 23 (2010), 801–810.Google Scholar
8Cheng, S.-Y. and Yau, S.-T.. Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces. Ann. Math. 104 (1976), 407–419.CrossRefGoogle Scholar
9Coelho, 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), 23–39.CrossRefGoogle Scholar
10Corsato, C., Obersnel, F. and Omari, P.. The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space. Georgian Math. J. 24 (2017), 113–134.CrossRefGoogle Scholar
11Corsato, 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), 227–239.CrossRefGoogle Scholar
12Dacorogna, B.. Direct methods in the calculus of variations. In Applied mathematical sciences, vol. 78 (Berlin: Springer-Verlag; 1989).Google Scholar
13Dal Maso, G.. An introduction to Γ-convergence. Progress in nonlinear differential equations applications, vol. 8. (Boston, MA: Birkäuser; 1993).Google Scholar
14De Giorgi, E.. Sulla convergenza di alcune succesioni di integrali del tipo dell'area. Rend. Mat. 8 (1975), 277–294.Google Scholar
15De Giorgi, E. and Franzoni, T.. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842–850.Google Scholar
16Flaherty, F.. The boundary value problem for maximal hypersurfaces. Proc. Natl. Acad. Sci. USA 76 (1979), 4765–4767.CrossRefGoogle ScholarPubMed
17Jost, J. and Li-Jost, X.. Calculus of variations (Cambridge University Press, New York; 2008).Google Scholar
18Kiessling, M. K.-H.. On the quasilinear elliptic PDE $-\nabla \cdot (\nabla u/\sqrt 1-|\nabla u|^2)=4\pi \sum _ka_k\delta _{n_{k}}$ in physics and geometry. Commun. Math. Phys. 314 (2012), 509–523.CrossRefGoogle Scholar
19Lê, A.. Eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64 (2006), 1057–1099.CrossRefGoogle Scholar
20Struwe, M.. Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems (Heidelberg: Springer, 1996).CrossRefGoogle Scholar
21Szulkin, A.. Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 77–109.CrossRefGoogle Scholar
22Trudinger, N. S.. On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math. 20 (1967), 721–747.CrossRefGoogle Scholar