$L_p$-Theory of Boundary Integral Equations on a Contour with Inward Peak

Vladimir Maz'ya, A. Soloviev
1998 Zeitschrift für Analysis und ihre Anwendungen  
Boundary integral equations of the second kind in the logarithmic potential theory are studied under the assumption that the contour has an inward peak. For each equation we find a pair of function spaces such that the corresponding operator bijectively maps one of them onto another. and the Neumann problem Au =0 in ci au (1.2) ônr J in a bounded plane simply connected domain ci with inward peak z = 0 on the boundary 1'. Here and elsewhere we assume that the normal n is directed outwards. We
more » ... ted outwards. We look for a solution of the problem (1.1) in the form where Wa is the double layer potential (Wa)(z) = Ir a(q 1 ds q , V. Maz'ya: Let qJ(I') denote the space of restrictions to r \ {O} of real functions of the form P(Z) = Iot ) Rez k , where m = --+ fl . A norm of p is defined by El = It(k)I. The space 9)t"([') is defined as the direct sum of 9l(f) and q3(r). By T,6(r) we denote the space of functions on r \ {O} represented in the form = da 4', where 0 E 9)I([') and t,1' (za) = 0 for a fixed point zo E F \ {O}. We supply
doi:10.4171/zaa/843 fatcat:esrkre75wjal3eebaaaohyrr6q