By Yu. D. Burago; V. G. Maz'ya

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**Example text**

The proof for case b) can be carried out with· trivial changes. Note, further, that it suffices to examine the case u,;t E + cp. 1°. E R" \ wt E. ) We assume the contrary. ; and ~,. Then, Therefore, and, hence, ~i 1 ~, _.. 0 as ;. - oo . "(It;+~,<~,)\lt,C~)). We divide this inequality by v. 8). The assertion of item 1 o is proved. 2 °. Let Kt. ( \t 1) be a decreasing function possessing the properties of an averaging kernel. )=~n (I£ C~)f\E). Going over to spherical coordinates we obtain According to 1 o, when ~ ' Eo , !

Where -aE. t E . This 3D 11ULUYAIUATE POI'ENTIAL THEORY AND THE SOLUTION OF BOUNDARY VALUE PROBLEMS (\) 2:. (B) 4=. 'l1ae illlerior Neumann problem for any finite charge with zero total mass has a so- lution representable as a simple-layer potential. Such a solution is unique with accuracy up to a con- stant term. Proof. As has been shown in Lemma 21, it follows from Condition (B) that the Fredholm ra- dius of operator T is greater than unity and, hence, the Fredholm theorem is valid for the integral equations of the problems being considered.

If 'Q is the unit ball in R3 , then for any E. c Q the inequality is valid. )= V~ ~. LEMMA 9. Let Q be the unit ball in R". , ( Proof. s1\ Q) =V _it We restrict ourselves to the case V< 'II::i: . examine only the set E. which is the intersection of Q symmetrization of metric relative to E relative to ray As in the preceding lemma, it suffices to with a polyhedron. =oE'C\'aQ. Byvirtueofthe isoperimetric property of a ball, 11_ ( s') ~ Pn (E'). Elementary calculations show that among all the balls smallest value of the quantity Pn ( 8 ') tCompare with [11], Appendices A and C.

### Potential Theory and Function Theory for Irregular Regions by Yu. D. Burago; V. G. Maz'ya

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