Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |
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Page 17
... bounded , equi - continuous sets of bounded sets of c ( K ) for each 2 a € ( 0,1 ) and each compact set K of the finite plane . αε From the Ascoli theorem it follows that L is a compact operator from L2 ( D ) → c ( K ) when c ( K ) is ...
... bounded , equi - continuous sets of bounded sets of c ( K ) for each 2 a € ( 0,1 ) and each compact set K of the finite plane . αε From the Ascoli theorem it follows that L is a compact operator from L2 ( D ) → c ( K ) when c ( K ) is ...
Page 20
... bounded sets of L2 ( D ) into locally equibounded , locally equicontinuous sets of functions on D. Hence , the conditions of Ascoli's theorem are again satisfied , and we conclude that He maps bounded sets of L2 ( D ) into bounded sets - - ...
... bounded sets of L2 ( D ) into locally equibounded , locally equicontinuous sets of functions on D. Hence , the conditions of Ascoli's theorem are again satisfied , and we conclude that He maps bounded sets of L2 ( D ) into bounded sets - - ...
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John Lewis Troutman. bounded sets of L2 ( D ) into bounded sets of L2 ( D ) which are harmonic and compact in the sense of almost uniform convergence . Moreover , since h2 € L2 ( D x D ) , it follows that 2 | x , f ( z ) | 2 ≤ = { h2 ...
John Lewis Troutman. bounded sets of L2 ( D ) into bounded sets of L2 ( D ) which are harmonic and compact in the sense of almost uniform convergence . Moreover , since h2 € L2 ( D x D ) , it follows that 2 | x , f ( z ) | 2 ≤ = { h2 ...
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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(p² 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