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.

scholarly journals AN INVERSE PROBLEM FOR A GENERAL DOUBLY-CONNECTED BOUNDED DOMAIN: AN EXTENSION TO HIGHER DIMENSIONS

1997 ◽  
Vol 28 (4) ◽  
pp. 277-295
Author(s):  
E. M. E. ZAYED
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Spectral Function ◽  
Bounded Domain ◽  
Higher Dimensions ◽  
Piecewise Smooth ◽  
Bounding Surface ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions ◽  
Negative Laplacian

The spectral function $\Theta(t)=\sum_{\nu=1}^\infty \exp(-t\lambda_\nu)$, where $\{\lambda_\nu\}_{\nu=1}^\infty$ are the eigenvalues of the negative Laplacian $-\nabla^2=-\sum_{i=1}^3(\frac{\partial}{\partial x_i})^2$ in the $(x^1, x^2, x^3)$-space, is studied for an arbitrary doubly connected bounded domain $\Omega$ in $R^3$ together with its smooth inner bounding surface $\tilde S_1$ and its smooth outer bounding surface $\tilde S_2$, where piecewise smooth impedance boundary conditions on the parts $S_1^*$, $S_2^*$ of $\tilde S_1$ and $S_3^*$, $S_4^*$ of $\tilde S_2$ are considered, such that $\tilde S_1=S_1^*\cup S_2^*$ and $\tilde S_2=S_3^*\cup S_4^*$.

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 Heat asymptotics with spectral boundary conditions. II

2003 ◽  
Vol 133 (2) ◽  
pp. 333-361 ◽  
Author(s):  
Peter Gilkey ◽  
Klaus Kirsten
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Dirac Type ◽  
Heat Trace ◽  
Operator Of Dirac Type ◽  
Laplace Type

Let P be an operator of Dirac type on a compact Riemannian manifold with smooth boundary. We impose spectral boundary conditions and study the asymptotics of the heat trace of the associated operator of Laplace type.

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.

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