Construction of boundary operators for the laplacian. 1. Using of simple layer potential

Abstract

1. Functional space Hr,A=. Let G be a bounded open C' -set [ I ] in R3 the boundary of which is 1-smooth surface r. We write G'= R3 \ G and introduce [2] Sobolev spaces W2(G), W2,0(G')=(ueD'(G'):u/r,DuEL2(G')}, where r is the distance of the point xE G' to the coordinates origin 0 and W2'2(f) Introduce the space H' = W2(G)x Wl0(G'). We write the elements of H' as u _ (u,iue), U E W (G), Ue E W2 ,0(G') and define [3] the linear continuous operators yru =UIr, yu = Ur, e : W (G) w2 (r), y,o: W2,0(G') -+ yIU = au /aiinr, yfu = au /aI|r , yl : W2 (G) -+ W,"/2(r), ye W21 (G') + W2 "2(r) where ii exterior normal to r, W2(r) be a dual space to W?'2(J).Wewrite y0 =y _sy = yY , E > e= const. Introduce the spaces H,' = {u E H' : yru = 01, (u,V)Hi = JVUVv, IIUIIHI = (U)t, GuG' r H