Potential techniques for boundary value problems on C1-domains

Abstract

Recently BjSrn Dahlberg through a very careful study of the Poisson kernel of D resolved the Dirichlet problem in the case of C ~ domains for data in L~(~D), 1 <p < ~, and in the case of Lipschitz domains for data in L'(~D), 2~<p< ~. (See [3].)While Dahlberg's results did not cover the Noumann problem nor give the regularity mentioned above for the Dirichlet problem, there.remainsalong the lines of this work -th6 very open question of the use of thb double and single layer p0ten~ials in the case of Lipschitz domains.Throughout this work D will denote a bounded domain of R ~. Points of D will generally be denoted Jay the capital letters X and Y, and points on the boundary of D, ~D, will be representedoby 9 the capital letters P and Q: Definition.D ~ C ~ (or ~D ~ C ~) means tha t corresponding to each point Q eaD there is a system of coordinates of R" with origin Q and a sphere, B(Q, 6), with center Q and radius 6 >0, such thatlwith respect to this coordinate systemwhere q~EC~(R ~ 1), the space of functions in CI(R ~ -1) with compact support, and ~( 0)= (a~/az,)(0) =0, i = 1 ..... n-~.Remark.If D E C 1 and e > 0 is given we can find a finite number of spheres, (B(Qj, 6~.)}'j~ 1, B QjE~D, such that ~D~ U]-I (Qj, 61) and with and D N B(Qj, 6j) = {(x, t): t > ~j(x)} N B(Qj, 6j) ~j~C~ (R ~ 1), ~j(0) = ~ " =0, i=l ..... n-l, oxt maxlv~,lx)l < vCj(x) = (a% x \~xl (x),)).... a~i

Locations

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