Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |
From inside the book
Results 1-3 of 5
Page 40
... vanish identically on the exterior boundary component ( that component of the boundary adjacent to the unbounded component D of D ) , since by the maximum principle , Ф would vanish identically in D and so would not be an eigenfunction ...
... vanish identically on the exterior boundary component ( that component of the boundary adjacent to the unbounded component D of D ) , since by the maximum principle , Ф would vanish identically in D and so would not be an eigenfunction ...
Page 42
... vanish identically on · 6 . D - neighborhood adjacent ป cannot vanish identically in a to any boundary arc of dà ̧ . Note that these results were obtained through a fortunate series of interactions between disciplines of analysis which ...
... vanish identically on · 6 . D - neighborhood adjacent ป cannot vanish identically in a to any boundary arc of dà ̧ . Note that these results were obtained through a fortunate series of interactions between disciplines of analysis which ...
Page 83
... identity applied to D and the differential equation ( 2.11 ) for in this region , we have s быс asy = -S R2yat & = + 5 D ad +12 ψάτι < 0 . Therefore ang cannot vanish identically on OD and so from ( 2.425 ) and the continuity of an ( 5 ...
... identity applied to D and the differential equation ( 2.11 ) for in this region , we have s быс asy = -S R2yat & = + 5 D ad +12 ψάτι < 0 . Therefore ang cannot vanish identically on OD and so from ( 2.425 ) and the continuity of an ( 5 ...
Other editions - View all
Common terms and phrases
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