scholarly journals On fractional powers of a class of elliptic differential operators with feedback boundary conditions

1990 ◽  
Vol 66 (9) ◽  
pp. 287-290
Author(s):  
Takao Nambu
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Fractional Powers ◽  
Elliptic Differential Operators

Heat Kernel Expansions in the Case of Conic Singularities

2003 ◽  
Vol 18 (12) ◽  
pp. 2197-2203 ◽  
Author(s):  
R. Seeley
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Zeta Function ◽  
Differential Operators ◽  
Smooth Boundary ◽  
General Nature ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
The Third ◽  
Elliptic Differential Operators

For positive elliptic differential operators Δ, the asymptotic expansion of the heat trace tr(e-tΔ) and its related zeta function ζ(s, Δ) = tr(Δ-s) have numerous applications in geometry and physics. This article discusses the general nature of the boundary conditions that must be considered when there is a singular stratum, and presents three examples in which a choice of boundary conditions at the singularity must be made. The first example concerns the signature operator on a manifold with a singular stratum of conic type. The second concerns the "Zaremba problem" for a nonsingular manifold with smooth boundary, posing Dirichlet conditions on part of the boundary and Neumann conditions on the complement; the intersection of these two regions can be viewed as a singular stratum of conic type, and a boundary condition must be imposed along this stratum. The third example is a one-dimensional manifold where the operator at one end has a singularity like that in conic problems, and the choice of boundary conditions affects not just the residues at the poles of the zeta function, but also the very location of the poles

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

2021 ◽  
pp. 1-16
Author(s):  
Alexander Dabrowski
Keyword(s):  
General Type ◽  
Differential Operators ◽  
Laplacian Operator ◽  
Variational Characterization ◽  
Repeated Eigenvalues ◽  
Direct Extension ◽  
Asymptotic Formulae ◽  
Elliptic Differential Operators ◽  
Domain Perturbations ◽  
Boundary Deformation

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

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