On operators generated by elliptic boundary problems with a spectral parameter in boundary conditions

1995 ◽  
Vol 23 (2) ◽  
pp. 205-231 ◽  
Author(s):  
Alexander Kozhevnikov ◽  
Sasun Yakubov
Keyword(s):  
Boundary Conditions ◽  
Spectral Parameter ◽  
Elliptic Boundary ◽  
Boundary Problems

Eigenvalue asymptotics for an elliptic boundary problem

2007 ◽  
Vol 137 (2) ◽  
pp. 281-302 ◽  
Author(s):  
M. Faierman ◽  
M. Möller
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Weight Function ◽  
Boundary Problem ◽  
Spectral Parameter ◽  
Elliptic Boundary ◽  
Bounded Region ◽  
Eigenvalue Asymptotics ◽  
Elliptic Boundary Problem ◽  
Parameter Condition

We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

BIORTHOGONAL SPLINE WAVELETS ON THE INTERVAL FOR THE RESOLUTION OF BOUNDARY PROBLEMS

1996 ◽  
Vol 06 (06) ◽  
pp. 749-791 ◽  
Author(s):  
ROLAND MASSON
Keyword(s):  
Wavelet Transform ◽  
Boundary Conditions ◽  
Elliptic Boundary ◽  
Stability And Convergence ◽  
Spline Functions ◽  
Wavelet Basis ◽  
Boundary Problems ◽  
Spline Wavelets ◽  
Points Of View ◽  
Wavelets On The Interval

We construct biorthogonal spline wavelets on the interval for different spline orders. We then discuss the accuracy of the construction and test the wavelet transform. Building the corresponding wavelet basis for [Formula: see text] boundary conditions we use it for the resolution of elliptic boundary problems both with pure Galerkin and Petrov-Galerkin schemes. The results are compared from stability and convergence points of view. The main conclusion is that using the biorthogonal counterpart of spline functions as trial spaces for low order splines provides much more advantages than using spline functions themselves as trial spaces for high order splines.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

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