On a family of extremum problems and the properties of an integral

A. P. Buslaev ; V. A. Kondrat'ev ; A. I. Nazarov

Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 830-838

Résumé

The following extremum problem is studied: $$ \int _0^1\bigl(y''(t)\bigr)^p\,dt\bigg/ \int _0^1\bigl(y'(t)\bigr)^q\,dt \to\min $$ over all $y$, with $y(0)=y(1)=0$ and $y'(0)=y'(1)=0$, which leads to the integral $$ \int_{\mathbb R}\bigl(\max(0,1+\mu x-|x|^q)\bigr)^{1/p'}\,dx $$ and yields exact estimates for the eigenvalues of differential operators in the generalized Lagrange problem on the stability of a column.