\(L_p\)-theory of boundary integral equations on a contour with outward peak
The authors consider the Dirichlet problem \(\Delta u=0\) in \(D\), \(u=f\) on \(\Gamma\setminus \{0\}\), where \(D\) is a plane simply connected bounded domain having boundary \(\Gamma\) with a peak \((0,0)\). This means that \(\Gamma\setminus \{0\}\) belongs to the class \(C^2\) and \(D\) is given near 0 by the inequalities \[ \kappa_-(x)< y<\kappa_+(x), \quad 0<x<\delta, \] where \(x^{-\mu-1} \kappa_\pm(x)\in C^2[0,\delta]\), \(\lim_{x\to+0} x^{-\mu-1} \kappa_\pm(x)= \alpha_\pm\) with \(\mu>0\) and \(\alpha_+> \alpha_-\). The Dirichlet problem is reduced to the integral equation \[ (\pi I-T)\sigma=-f \quad\text{on }\Gamma\setminus \{0\}, \tag{1} \] where \(T\sigma\) is the double layer potential. The Fredholm theory cannot be directly applied to (1) considered in the space of continuous functions. The authors introduce a pair of functional spaces such that the operator \(\pi I-T\) maps one of them onto another. These spaces are introduced as spaces of absolutely continuous functions on \(\Gamma\setminus \{0\}\) with special weights. \(\text{Ker}(\pi I-T)\) is described. The exterior Dirichlet and Neumann problems are also studied.
scientific article; zbMATH DE number 1213756