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At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...

Let $\mathcal{M}$ be a real, compact, orientable manifold and let $X$ be a vector field on $\mathcal{M}$. Consider the functional $$E(f) = \int_{\mathcal{M}} X_p(f)^2 dV$$ where $X_p(f)$ is the ...