Integral equations for the Dirichlet and Neumann boundary value problems in a plane domain with a cusp on the boundary (Q1360823)

Let \(\Omega\) be a Jordan domain with a boundary \(\Gamma\). The curve \(\Gamma\) is piecewise-smooth with unique interior or exterior cusp at point \((0,0)\). In a vicinity of \((0,0)\) the curve \(\Gamma\) is a sum of two curves \[ \kappa_+(x)= {\textstyle \frac12} \kappa_+'' (0)x^2+\cdots, \qquad \kappa_-(x)= {\textstyle \frac12} \kappa_- ''(0) x^2+\cdots, \] \[ \kappa_-''(0)< \kappa_+''(0), \qquad \kappa_\pm(x)\in C^\infty[0,\delta]. \] Let \[ L_{p,\alpha}^k= \biggl\{\sigma: \int_\Gamma |\sigma^{(k)}(q)|^p q^\alpha ds(q)<\infty \biggr\}, \qquad k=0,1. \] The author studies the integral equations which appear under solution of the interior Dirichlet problem and exterior Neumann problem for the domain \(\Omega\) by simple and double layer potentials. He introduces relevant operators \(T_n\), \(n=1,\dots, 5\). The main result is a description of the ranges \(T_n(L_{p,p+\alpha}^0)\), \(n=1,3,5\), \(T_n(L_{p,p+\alpha}^1)\), \(n=2,4\).