Piecewise Smooth and Open Contours

Part of the Frontiers in Mathematics book series (FM)


In this chapter, we study different problems connected with approximation methods for singular integral equations of the form
$$ \begin{gathered} t) \equiv a(t)u(t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{u(s) ds}} {{s - t}}} + c(t)\overline {u(t)} + \frac{{d(t)}} {{\pi i}}\int_\Gamma {\frac{{\overline {u(s)} ds}} {{s - t}} + \frac{{e(t)}} {{\pi i}} \overline {\int_\Gamma {\frac{{u(s)ds}} {{s - t}}} } } \hfill \\ + \frac{{f(t)}} {{\pi i}}\int\limits_\Gamma { \overline {\frac{{\overline {u(s)} ds}} {{s - t}}} } + \int_\Gamma {k_1 (t,s)u(s)ds + } \int_\Gamma {k_2 (t,s)\overline {u(s)} ds = g(t), + t \in \Gamma ,} \hfill \\ \end{gathered} $$
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
$$ (Au)(t) \equiv a(t)u(t) + \frac{{b(t)}} {\pi }\int_\Gamma {u(s)\frac{d} {{dn_s }}\log |t - s|d\Gamma _s + (Tu)(t) = g(t)} , t \in \Gamma $$
where n s refers to the inner normal to Γ at s, and T stands for a compact operator.


Local Operator Toeplitz Operator Singular Integral Equation Corner Point Essential Spectrum 
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© Birkhäuser Verlag AG 2008

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