Lagrange's problem of the strongest column

Problem

Consider elastic columns of given length $L$, given volume $V$ and given material. Among them, find the distribution of cross-section along the axis that makes the buckling load as large as possible. Let $A(x)$ be the cross-section area at $x \in [0, L]$. In the classical form the cross-sections are geometrically similar, so the second moment of area is $I(x) = \alpha A(x)^2$ for a constant $\alpha > 0$. For pinned ends the buckling load is the smallest eigenvalue $P$ of $$E \alpha A(x)^2 w''(x) + P w(x) = 0, \qquad w(0) = w(L) = 0,$$ and the problem is to maximize it over $A \ge 0$ with $\int_0^L A(x) \, dx = V$.