Singular Integral Equations Boundary Problems Of Function Theory And Their Application To Mathematical Physics N I Muskhelishvili File

Title: Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics Author: N. I. Muskhelishvili (also spelled Muskhelishvili) Original Russian Publication: 1946 (frequently revised) English Translation: 1953 (P. Noordhoff, Groningen; later Dover reprints)

with given Hölder-continuous ( G(t) \neq 0 ) and ( g(t) ). The of the problem is Their boundary values on ( \Gamma ) satisfy

with ( a(t), b(t) ) Hölder continuous. The key is to set \textP.V. \int_\Gamma \frac\phi(t)t-t_0

defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy ( \Phi^-(z) ) outside.

[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(t)t-z , dt ]

[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ]

is bounded on Hölder spaces and ( L^p ) ((1<p<\infty)). Find a sectionally analytic function ( \Phi(z) ) (vanishing at infinity as ( O(1/z) ) for the “exterior” problem) satisfying on ( \Gamma ):