Weighted Sobolev spaces. by Kufner, Alois

By Kufner, Alois

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Example text

The function the domain v. (0,x) . From the inclusion a as well, namely, in v. € (U (after substituting v iX € the hypersurface by x ^(y) r. now is the distance of the point We find y from which results by translating the hypersurface in the direction of the Let now i ; d M' e ) z = y-xh ) that ^ ^ ^ i ^ - B . X ) ; c£, e ) , where the function r. ;d_^,e) . Consequently, as has been an arbitrary derivative (of an order tion wnich yields J(A) = | | g - g J | P H c < n P M' that is, < 0 < y < X .

9) - can be chosen in such a way that they fulfil the following conditions: (a) The point (b) (y' y 1 N ) belongs to one and only one of the sets r. r. ,m x is distance from the sets U. for j = positive. 10 (ii). 11 (ii) are fulfilled. 31) dM(x) = |x - x Q | . Let us now briefly deal with the distance just mentioned. 12. LEMMA. 33) a ( 'IN ^ * I N ^ i y i > • Then there exist the following -positive constants inequalities c. 35) c2(|y I + a i < y p - yi N ) i lyl i ly + a x (yp - y 1 N The proof is left to the reader.

A € £ ° . It is evident that the and the corresponding sets B. , U. r. 9) - can be chosen in such a way that they fulfil the following conditions: (a) The point (b) (y' y 1 N ) belongs to one and only one of the sets r. r. ,m x is distance from the sets U. for j = positive. 10 (ii). 11 (ii) are fulfilled. 31) dM(x) = |x - x Q | . Let us now briefly deal with the distance just mentioned. 12. LEMMA. 33) a ( 'IN ^ * I N ^ i y i > • Then there exist the following -positive constants inequalities c.

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