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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