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The Dielectric Operators
Negative Eigenvalues and Associated
k Extremal Problems
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analytic annulus associated non-negative normalized Borel measures boundary component bounded sets characteristic convergence compact operator compact set compact subsets conclude conformal mapping connected admissible domains constant continuous extension continuous with respect cp(z D X D defined definition denote dielectric Green's function differential equation disc of radius domain of given dominated convergence theorem double-layer potentials eigen eigenfunctions associated eigenvalue and associated extremal problem follows formula gg(z given transfinite diameter Green's identity applied harmonic Hence Hilbert space implies integral equation kernel Lebesgue measure log r2 log|z Moreover neighborhood non-negative normalized eigenfunction operators G pointwise and boundedly satisfies Schiffer Schwarz inequality sequence simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.3 uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity variational vergence weak derivative zero