Many problems in optimization (such as finding exact constants or maximizers to inequalities for linear operators) are equivalent to finding all solutions to associated Euler-Lagrange equations. Of course, these functional equations will involve the operator we started with, so they aren't elementary enough to be a satisfactory answer. However, sometimes these problems are solved by showing that the solutions (or related functions) must satisfy more elementary looking functional equations. As an example, here are some elementary looking functional equations: Find all complex-valued measurable functions that satisfy (almost everywhere) the equations:
1) $f(x)f(y) = F(|x|^2 +|y|^2, x+y)$ for $x,y \in R^2$. ,
2) $g(x)g(y) = G(|x|+|y|, x+y)$ for $x,y \in R^3$,
3) $h(x)h(y) = H(x^2 +y^2+z^2, x+y+z)$ for $x,y,z \in R$.
These and similar problems arise in the work of D. Foschi on maximizers to Strichartz inequalities in low dimensions.