Rearrangements of functions and lower bounds for eigenvalues of differential equations

摘要

We are concerned with eigenvalue problems of the formsubject to boundary conditionsin the first case andin the second case. The eigenvalues of these systems depend on the coefficientsand we denote them accordinglyand. We introduce some new kinds of rearrangements of functions and relate them to those given by B. Schwarz 19. Our main results are that if#x2266;0 thenand ifthen