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Estimates for the first eigenvalue in some Sturm-Liouville problems. (English. Russian original) Zbl 0883.34027
Russ. Math. Surv. 51, No. 3, 439-508 (1996); translation from Usp. Mat. Nauk 51, No. 3, 73-144 (1996).
A famous problem of Lagrange is to find the strongest column which is the body of revolution of a plane curve around some axis in the plane of the curve, where, for a column of unit length and volume, strength means the resistance to compression along the axis of revolution. It leads to an extremal problem for the critical value of the load $$\lambda$$ along the axis and an Euler-Lagrange equation of the form $\bigl(Q(x)y''\bigr)''+\lambda y''= 0,\;y(0)= y'(0)=0,\;y(1) =y'(1)=0. \tag{1}$ If the volume of the column is assumed to be fixed, then $\int^1_0Q^{1 \over 2} (x)dx=1. \tag{2}$ Lagrange’s solution to the problem was incorrect, and it wasn’t until 1962 that a successful investigation was made by J. B. Keller and I. Tadjbakhsh [J. Appl. Mech. 29, 159-164 (1962; Zbl 0106.38301)]. However, some of their results were criticized by a number of authors, and the proof is deficient. A correct proof is given in this paper. However, this paper contains much more: in particular it contains the first proof of the existence of an optimal solution. A number of other extremal problems of considerable interest (related to Sturm-Liouville problems) are studied. Estimates of the eigenvalues of self-adjoint boundary-value problems for an ordinary differential equation of order $$m$$ are also given.

##### MSC:
 34B24 Sturm-Liouville theory 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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