By Lawrence Zalcman

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W n _< n4-n of Melnikov's is a peak point for dr = = . R(X) . We have ~o Clearly, value by = o ~(r~~ ( - ½ 7)=-~. 3 follows Although sufficient necessary. 2. 2 and 7 . 3 are t h e y are far f r o m A n = [Iz-3 -2-n-21 < n-12-n] and oO set X = A \ ~ that, but for An One s h o u l d continuous . problem every discussed Let D u 6 a regular D x 6 8D if CR(~X ) . point the r e l a t i o n A u u point as z -+ x [105] showed for the D i r i c h l e t u problem of h a r m o n i c be a real we can associate• (continuous is a r e g u l a r Wiener of the theory To e a c h s u c h ~(z) -~ u(x) .

R(X) ~ =~4n Theorem 7 •3. c x n {Iz-xl x Proof. = = 4n mn = = then (2 n ~/'Wnn)2 _<~ (~)(2n~ r] x E X (We h a v e . 12). Let < theorem. mn = ~ and ~n If Now apply Melnikov' s theorem. w n _< n4-n of Melnikov's is a peak point for dr = = . R(X) . We have ~o Clearly, value by = o ~(r~~ ( - ½ 7)=-~. 3 follows Although sufficient necessary. 2. 2 and 7 . 3 are t h e y are far f r o m A n = [Iz-3 -2-n-21 < n-12-n] and oO set X = A \ ~ that, but for An One s h o u l d continuous . problem every discussed Let D u 6 a regular D x 6 8D if CR(~X ) .

1 f l l . - o depend on G . (K~]). ~s -456. MELNIKOV' S THEOPd~4 We are now p r e p a r e d proceeding with the proof, set some notation. and £n = [Izl For z 6 C from z = 2 -n] of • If p(z,S) S . 1 to prove Melnikov's Finally, = inf Before it will be convenient let S c ¢ theorem. A n = [2 -n-1 ~ we shall write to Izl ~ 2 -n] CS = 2 \ [Iz-x I :x E S] , the distance d(S) = sup [Iz-gl :z,C 6 S] is S . (Melnikov). Let y(CX fl {2 -n-1 _< Iz-xl _< 2 - n ] ) . for R(X) x E X and let Then x Yn = is a peak point if and only if @O 2n Yn = ~ n=0 Proof.