Geodesic
On a family of extremum problems and the properties of an integral
Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 830-838.

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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. 
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A. P. Buslaev; V. A. Kondrat'ev; A. I. Nazarov. On a family of extremum problems and the properties of an integral. Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 830-838. http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a3/

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