By Yu. D. Burago; V. G. Maz'ya
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Additional resources for Potential Theory and Function Theory for Irregular Regions
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 , Appendices A and C.
Potential Theory and Function Theory for Irregular Regions by Yu. D. Burago; V. G. Maz'ya