Extremal Problems for Eigenvalues with Applications to Buckling, Vibration and Sloshing - Barnes, David C. (SIAM journal on mathematical analysis, 1985)
Let $\lambda _n $ denote the $n$th eigenvalue of the equation $[R(x)y']' + [\lambda P(x) + Q(x)]y = 0$ subject to self-adjoins boundary conditions. Many applications of this equation involve calculating extremal values of $\lambda _n $ when the coefficients are subjected to some kind of additional constraints. For example the shape of the strongest column can be determined by maximizing an eigenvalue $\lambda _n $. In this work we will give a new method for solving extremal problems for $\lambda _n $ which unifies many previous works and provides (in some cases for the first time) a mathematically rigorous approach to these extremal properties. As an example of our method we determine the shape of the strongest "Profile" column. Our method can also be used to study fourth (and higher) order equations. We reduce the problem of maximizing or minimizing $\lambda _n $ to an elementary problem of minimizing or maximizing a real valued function of one real variable. One salient feature of this work is that it is completely independent of the theory of Rayleigh quotients.