scholarly journals Eigenvalues of the Laplacian for the third boundary value problem

1987 ◽  
Vol 29 (1) ◽  
pp. 79-87 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Function ◽  
Boundary Value ◽  
Two Dimensional ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions ◽  
The Third ◽  
Eigenvalues Of The Laplacian ◽  
Third Boundary Value Problem

AbstractThe spectral function , where are the eigenvalues of the two-dimensional Laplacian, is studied for a variety of domains. The dependence of θ(t) on the connectivity of a domain and the impedance boundary conditions is analysed. Particular attention is given to a doubly-connected region together with the impedance boundary conditions on its boundaries.

scholarly journals Some asymptotic spectral formulae for the eigenvalues of the Laplacian

1988 ◽  
Vol 30 (2) ◽  
pp. 220-229 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Positive Function ◽  
The Third ◽  
Asymptotic Formulae ◽  
Eigenvalues Of The Laplacian ◽  
Third Boundary Value Problem

AbstractIn this paper we shall derive some asymptotic formulae for spectra of the third boundary value problem in Rn, n = 2 or 3, linked with variation of a positive function entering the boundary conditions. Further results may be obtained.

scholarly journals Hearing the shape of membranes: further results

1990 ◽  
Vol 13 (3) ◽  
pp. 591-598
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Conditions ◽  
Spectral Function ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions ◽  
Eigenvalues Of The Laplacian ◽  
Spherical Domains

The spectral functionθ(t)=∑m=1∞exp(−tλm),t>0where{λm}m=1∞are the eigenvalues of the Laplacian inRn,n=2or3, is studied for a variety of domains. Particular attention is given to circular and spherical domains with the impedance boundary conditions∂u∂r+γju=0onΓj(orSj),j=1,…,JwhereΓjandSj,j=1,…,Jare parts of the boundaries of these domains respectively, whileγj,j=1,…,Jare positive constants.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

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