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.

Reviewer: W.D.Evans (Cardiff)

##### MSC:

34B24 | Sturm-Liouville theory |

34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |