Dear MO contributors,

let $r > 0, L > 0$. I am interested in maximizing the integral: $$ \int_0^{2\pi} \frac{f(\alpha)^2 f'(\alpha)^2}{\sqrt{f(\alpha)^2 + f'(\alpha)^2}} \ \mathrm{d} \alpha $$ over continuous and piecewise smooth functions $f : [0,2\pi] \to \mathbf{R}$, subject to three the constraints: $$ f(0) = f(2\pi), \quad f \ge r, \quad \int_0^{2\pi} \sqrt{f(\alpha)^2 + f'(\alpha)^2} \ \mathrm{d} \alpha = L. $$ Although not directly relevant to the question, let me mention that the function $f$ represents a curve in the plane, where $f(\alpha)$ is the distance from the origin at angle $\alpha$ from one axis, and the aim is to maximize a certain torque for a given length of curve.

When I try to write down the Euler-Lagrange equation, I end up with a not-so-inspiring differential equation that is a polynomial of degree 7 in $f$, $f'$ and $f''$.

Heuristically, I expect an extremal function to develop a number of peaks, and I conjecture that the family of extremal functions can be indexed by the number of peaks (by peaks, I mean local maxima, probably with a discontinuous derivative). More precisely, is it true that there exists at most one extremal function with a given number of peaks ? How does the set $$ \{n \in \mathbf{N} : \text{there exists an extremal function with } n \text{ peaks} \} $$ look like ? How many peaks does the global maximizer have ? What is its rough shape ? How much do all this depend on $r$ and $L$ ? (of course, if $L < 2\pi r$, then there are no solutions, and there is a unique solution for $L = 2 \pi r$). Are there good ways to investigate these questions numerically ? Any suggestion about the relevant litterature would also be very welcome.