Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |
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Page 52
... normalized eigenfunction for D n ' n = 0,1,2 .... If DXD , then n n and → almost n uniformly in E. Proof : Let X be the characteristic - 52 - Continuous Dependence b) Monotone Dependence c) Elementary Distortion a) Characteristic ...
... normalized eigenfunction for D n ' n = 0,1,2 .... If DXD , then n n and → almost n uniformly in E. Proof : Let X be the characteristic - 52 - Continuous Dependence b) Monotone Dependence c) Elementary Distortion a) Characteristic ...
Page 56
... characteristic convergence of Transfinite diameter also is not continuous with respect to convergence in the Caratheodory sense ( as the above example shows ) , but it is continuous with respect to convergence in the Fréchet sense [ 15 ...
... characteristic convergence of Transfinite diameter also is not continuous with respect to convergence in the Caratheodory sense ( as the above example shows ) , but it is continuous with respect to convergence in the Fréchet sense [ 15 ...
Page 67
... convergence theorem , the integral approaches zero with * Also , ป is bounded on D and ατ * du μ Əz √ TZ - ST 4 * ( 5 ) ≤ M on compact subsets of E. Finally , the characteristic convergence of D * to D and the fact that D is ...
... convergence theorem , the integral approaches zero with * Also , ป is bounded on D and ατ * du μ Əz √ TZ - ST 4 * ( 5 ) ≤ M on compact subsets of E. Finally , the characteristic convergence of D * to D and the fact that D is ...
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analytic annulus applying Green's B₂ Borel measures boundary component bounded convergence bounded sets characteristic convergence compact operator compact set compact subsets conclude connected admissible domains constant continuous extension continuous with respect convergence theorem D₁ D₂ defined definition denote dielectric dielectric Green's function disc of radius domain of given double-layer potentials eigen eigenfunction eigenfunctions associated eigenvalue and associated extremal problem feL D finite follows given transfinite diameter Green's function Green's identity harmonic Hence Hölder continuous implies kernel log|z monotonicity Moreover neighborhood non-negative normalized eigenfunction o(p˛ pointwise and boundedly proof satisfies Schwarz inequality simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.4 transfinite diameter uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity ατ πλ эко