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 ):
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)
defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy is bounded on Hölder spaces and ( L^p
This is a foundational text in analytical methods for applied mathematics, elasticity, and potential theory. It systematically develops the theory of using the apparatus of boundary value problems of analytic functions (Riemann–Hilbert and Hilbert problems). Core Mathematical Content 1. Prerequisite: Cauchy-Type Integrals and the Plemelj–Sokhotski Formulas Let ( \Gamma ) be a smooth or piecewise-smooth closed contour in the complex plane (often the real axis or a circle). For a Hölder-continuous function ( \phi(t) ) on ( \Gamma ), the Cauchy-type integral
where P.V. denotes the Cauchy principal value. The singular integral operator It systematically develops the theory of using the
[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ]
[ \Phi^\pm(t_0) = \pm \frac12 \phi(t_0) + \frac12\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt, ] \textP.V. \int_\Gamma \frac\phi(t)t-t_0
[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(t)t-z , dt ]
[ (a(t) + b(t)) \Phi^+(t) - (a(t) - b(t)) \Phi^-(t) = f(t). ]