# Piecewise Smooth and Open Contours

Chapter
Part of the Frontiers in Mathematics book series (FM)

## Abstract

In this chapter, we study different problems connected with approximation methods for singular integral equations of the form
$$\begin{gathered} t) \equiv a(t)u(t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{u(s) ds}} {{s - t}}} + c(t)\overline {u(t)} + \frac{{d(t)}} {{\pi i}}\int_\Gamma {\frac{{\overline {u(s)} ds}} {{s - t}} + \frac{{e(t)}} {{\pi i}} \overline {\int_\Gamma {\frac{{u(s)ds}} {{s - t}}} } } \hfill \\ + \frac{{f(t)}} {{\pi i}}\int\limits_\Gamma { \overline {\frac{{\overline {u(s)} ds}} {{s - t}}} } + \int_\Gamma {k_1 (t,s)u(s)ds + } \int_\Gamma {k_2 (t,s)\overline {u(s)} ds = g(t), + t \in \Gamma ,} \hfill \\ \end{gathered}$$
(4.1)
where a, b, c, d, e, f, g, k1, and k2 are given functions, and Γ is either a simple open or closed piecewise smooth curve in the complex plane ℂ. A case of particular interest is the double layer potential equation
$$(Au)(t) \equiv a(t)u(t) + \frac{{b(t)}} {\pi }\int_\Gamma {u(s)\frac{d} {{dn_s }}\log |t - s|d\Gamma _s + (Tu)(t) = g(t)} , t \in \Gamma$$
(4.2)
where n s refers to the inner normal to Γ at s, and T stands for a compact operator.

## Keywords

Local Operator Toeplitz Operator Singular Integral Equation Corner Point Essential Spectrum

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## Copyright information

© Birkhäuser Verlag AG 2008