## Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |

### From inside the book

Results 1-3 of 6

Page 17

0l%IWI2, where is any positive constant dependent on K. Hence L maps

a e (0,l) and each compact set K of the finite plane. * From the Ascoli theorem it

follows that L is a compact operator from L (D) -» C (K) when C (K) is supplied

with the topology of uniform convergence . Note also that for z e D, (Lf)(z) is

harmonic and is at most logarithmically infinite at infinity. */ For a e (0,1], n = 0,1,2.

.., C (S) is the set ...

0l%IWI2, where is any positive constant dependent on K. Hence L maps

**bounded sets**of L (D) into equi -bounded, equi-continuous sets of Ca(K) for eacha e (0,l) and each compact set K of the finite plane. * From the Ascoli theorem it

follows that L is a compact operator from L (D) -» C (K) when C (K) is supplied

with the topology of uniform convergence . Note also that for z e D, (Lf)(z) is

harmonic and is at most logarithmically infinite at infinity. */ For a e (0,1], n = 0,1,2.

.., C (S) is the set ...

Page 20

Hef)(z)| <^||f||2 / |h (2,C)|2dTc (1.28) < - J^||f ||2 for z e K , where is a constant for

each compact set K. Similarly for z,zq e K : (1.29) |(H.f)(z)-(H_f)(zJ| <§ ||f||, / |he(«,

S)-he(«0,t)|ZdT5 and the integral is arbitrarily small providing lz_z0l ^s sufficiently

small in view of the uniform continuity of h on K x D. Thus H£ maps

of L (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 ...

Hef)(z)| <^||f||2 / |h (2,C)|2dTc (1.28) < - J^||f ||2 for z e K , where is a constant for

each compact set K. Similarly for z,zq e K : (1.29) |(H.f)(z)-(H_f)(zJ| <§ ||f||, / |he(«,

S)-he(«0,t)|ZdT5 and the integral is arbitrarily small providing lz_z0l ^s sufficiently

small in view of the uniform continuity of h on K x D. Thus H£ maps

**bounded sets**of L (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 ...

Page 21

compact in the sense of almost uniform convergence. 2 Moreover, since h £ L (

D X D), it follows that and we may apply the Lebesgue dominated convergence

theorem to any almost uniformly convergent subsequence of the images of the

Thus H is 2 a compact operator on L (D). From (1.21) we see that G as a linear

combination of ...

**bounded sets**of L (D) into**bounded sets**of L (D) which are harmonic andcompact in the sense of almost uniform convergence. 2 Moreover, since h £ L (

D X D), it follows that and we may apply the Lebesgue dominated convergence

theorem to any almost uniformly convergent subsequence of the images of the

**bounded sets**of 2 2 L (D) and conclude that the convergence is also in L (D).Thus H is 2 a compact operator on L (D). From (1.21) we see that G as a linear

combination of ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Other editions - View all

### Common terms and phrases

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 D X D defined definition denote dielectric Green's function differential equation disc of radius Doctor of Philosophy domain of given dominated convergence theorem double-layer potentials dz dz eigen eigenfunctions associated eigenvalue and associated extremal problem follows given transfinite diameter Green's identity applied harmonic Hence Hilbert space implies interior minimum kernel Lebesgue measure log r2 log|z Moreover neighborhood non-negative normalized eigenfunction obtain operators G pointwise and boundedly satisfies Schwarz inequality sequence simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.4 uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity variational vergence weak derivative zero