On the least positive eigenvalue of integral equations with equimeasurable kernels

Abstract

Rearrangements of sets of numbers and rearrangements of functions were defined and investigated in detail in the book of Hardy, Littlewood and Polya [4, Chapter X]. Using this notion, classes of nonhomogeneous strings, membranes, rods and plates with equimeasurable density were considered by P. R. Beesack and the author and the extrema of the principal frequency were found for these classes [1; 2; 7]. Here we deal with an analogous question for integral equations. For any equimeasurable class of non-negative and symmetric L2 kernels we find the maximum of the first eigenvalue. The proof is elementary and uses only the maximum property of the reciprocal of the first eigenvalue [3; 8; 9] and a simple geometric idea. After the proof of this theorem we add some remarks on the corresponding minimum and on the same problem for matrices.