Hypersingular integral equations of the first kind of diffraction problems for electromagnetic waves on nonclosed surfaces (Q1309091)

The paper consists of two parts. In the first one the authors study the mapping properties of the following hyper-singular integral operator \[ \int^ 1_{-1} {\varphi (t) dt \over (x-t)^ 2} + \int^ 1_{-1} k(x,t) \varphi (t)dt = f(t), \] in Hölder spaces of functions with weight, where \(k(x,t)\) has a weak singularity and the first integral is understood as the finite part by Hadamard. A projection method is suggested to solve such equations; as a trial finite-dimensional space they choose a set of piecewise-constants with exactly \(N-1\) jumps at fixed knots. Some estimations of convergence are demonstrated as well. In the second part a vector boundary value problem for Maxwell's equations is considered, which describes the diffraction of electromagnetic waves by a thin open surface \(S \subset \mathbb{R}^ 3\). For the corresponding boundary integral equation, which is hypersingular, some mapping properties in special Hölder spaces with weight are proved.