On the index and spectrum of integral operators of potential type along Radon curves

A study is made of how classical integral equations of mathematical physics are affected by nonregularity of the contour of integration. A criterion is obtained for a matrix integral equation with operator of potential type acting in $L_p$ $(1<p<\infty)$ to be Noetherian, and the index is computed. It is established that an integral equation corresponding to the interior Dirichlet problem for harmonic functions is Noetherian in $L_p$ for all $p$ except for a finite or countable number of values determined by the angles of the contour; the defect numbers, which depend on $p$ and the angles mentioned, are found. Analogous results are obtained for the system of integral equations of the planar theory of elasticity. The non-Noetherian spectrum of a matrix integral operator of potential type acting in a space of continuous vector-valued functions is described. This result is illustrated by an example of an operator in elasticity theory (for which, in particular, the Fredholm radius is found) and of the direct value of a double layer potential.