Generalized Spectrum of Steklov-Robin Type Problem for Elliptic Systems

Alzaki Fadlallah, Kwadwo Antwi-Fordjour, Marius N. Nkashama

Department of Mathematics, University of Alabama at Birmingham, Birmingham, USA.

DOI: 10.4236/am.2015.62039                

Abstract

We will study the generalized Steklov-Robin eigenproblem (with possibly matrix weights) in which the spectral parameter is both in the system and on the boundary. The weights may be singular on subsets of positive measure. We prove the existence of an increasing unbounded sequence of eigenvalues. The method of proof makes use of variational arguments.

Keywords

Steklov-Robin, Variational Arguments, Matrix Weights

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Auchmuty, G. (2012) Bases and Comparison Results for Linear Elliptic Eigenproblems. Journal of Mathematical Analysis and Applications, 390, 394-406. http://dx.doi.org/10.1016/j.jmaa.2012.01.051
[2]	Mavinga, N. (2012) Generalized Eigenproblem and Nonlinear Elliptic Equations with Nonlinear Boundary Conditions. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142, 137-153.
[3]	de Godoi, J.D.B., Miyagaki, O.H. and Rodrigues, R.S. (2013) Steklov-Neumann Eigenvalue Problems for a Class of Elliptic System. SP-Brazil.
[4]	Steklov, L.M.W. (1902) Sur les problèmes fundamentaux de la physique mathématique. Annali Scuola Norm. Sup. Pisa, 19, 455-490.
[5]	Küfner, A., John, O. and Fucik, S. (1977) Function Spaces. Noordhoff International Publishing, Leyden.
[6]	Necas, J. (1967) Les Mèthodes directes en thèorie des equations elliptiques. Masson, Paris.
[7]	Grisvard, P. (1985) Elliptic Problems in Nonsmooth Domains. Pitman, Boston.
[8]	Fadlallah, A. (2015) Elliptic Equations and Systems with Nonlinear Boundary Conditions. Ph.D. Dissertation, University of Alabama at Birmingham (UAB), Birmingham.