By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)
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Extra resources for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982
That u(x+lllxle )' s we have [ ~ Then, if ~(x+lllxle) lu(xl-u(x+lllxle) III xl s] p luk(xl-uk(x+lllxle s ) [lllxl s =[ ~ k~l[ u(x) > min ui(x+lllxle) =u(x+lllxle s )' 1~~q s lu(x)-uk(x+lllxle s )I p III xl ] q ]p = l~(x)-~(x+lllxles) lllxl I p ] 52 Petru Caraman and, as above, in the hypothesis (20), we obtain (19) in this case as well. Finally, since uE21, '1t Hence, since each we deduce by (19) that is an arbi trary function of Uk' q k Z cap (E ,E ,D) , 1 k=l p 0 as desired. The same argument still holds for (iii'), max ( u 1 (x) , ••• , u (i') q but with u(x) (x) ] • is trivial since u =0 is an admissible function; arguing as for (ii), we also establish (ii').
We had to introduce the condition supposed to be continuous in D U Eo U E1 d(E ,E ) >0 since u is o 1 and to have boundary values o on Eo and on E1 and then, if Eo n E1 n D 'f~, E~ n E1 'f ~ or Eo n E1 'f~, at the points of such a set, u has to be at the same time equal to 0 and to 1. Arguing as in the preceding lemma, we obtain COROLLARY 2. --From each of the above definitions for the p-capacity, we obtain the corresponding definition for the conformal capacity if we take and suppose that D p =n is contained in a fixed ball.
Union of domains of the form o (k = 1,2, ••• ), then 40 Petru Caraman Rem a r k . From this Lemma 3 it follows that it does not matter if two different components of D have common boundary points and if some of these common boundary points belong to Eo or to E . l An important role in the generalization of Ziemer's relation (1) is played by PROPOSITION 3. pPdm, (r) OD where F (r) is the class of admissible functions p E F (r) Rn, continuous in D and 0 in CD (see our note ). bounded in Arguing as in the preceding proposition, we have also LEMMA 4.