  We show that the concept of $H^2$-gradient flow for the Willmore energy and
other functionals that depend at most quadratically on the second fundamental
form is well-defined in the space of immersions of Sobolev class $W^{2,p}$ from
a compact, $n$-dimensional manifold into Euclidean space, provided that $p \geq
2$ and $p>n$. We also discuss why this is not true for Sobolev class
$H^2=W^{2,2}$. In the case of equality constraints, we provide sufficient
conditions for the existence of the projected $H^2$-gradient flow and
demonstrate its usability for optimization with several numerical examples.
