Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |
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analytic annulus applying Green's associated non-negative normalized Borel measures bounded convergence bounded sets characteristic convergence compact operator compact set compact subsets conclude constant continuous extension continuous with respect convergence theorem D₁ D₂ defined definition denote 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 identity harmonic Hence Hölder continuous implies kernel L¹(D log|z monotonicity Moreover neighborhood pointwise and boundedly proof r₂ satisfies Schiffer Schwarz inequality simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.4 transfinite diameter u₂ uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity variational ατ πλ эко