Equimeasurable rearrangements of functions and fourth order boundary value problems (Q1915760)

The boundary value problem (1) \((p(x) y'')''+ \lambda y''+ F(x) y= 0\), \(x\in [0, l]\), (2) \(y(0)= y(l)= y''(0)= y''(l)= 0\), arises in the study of the buckling of a beam which is supported on an elastic foundation. The functions \(p(x)\), \(F(x)\) are the stiffness and elastic destructive force per unit length. The rearrangements of functions \(f_{\pm m}\) that are equimeasurable with \(f\) and periodic in \([0, l]\) with period equal to \({l\over n}\) and satisfy the symmetry requirement are introduced. Some theorems on these functions follow. The \(n\)th eigenvalue of the problem defined by (1) and (2) above is shown to have a lower bound expressible in the form (3) \(\lambda_n(p, F)\geq \lambda_n(\overline p_{-n}, F^*_{+ n}(\lambda, \xi))\). In the above \(\overline p_{- n}\) is the rearrangement which is equimeasurable with \(p(x)\), periodic in \([0, l]\) with period equal to \({l\over n}\) and increasing in the interval \([0, {l\over 2n}]\). The function \(F^*_{+ n}(x, \xi)\) equals \(\overline F_n(x) H^*_n(x, \xi)\) and \(H^*_n(x, \xi)\) is the periodic generalization of the step function with period \({l\over n}\).