A fact about integration of $H^{1}(\mathbb R^{d})$ functions

Question

For any $v \in H^{1}(\mathbb R^{d})$ how to show that $\int_{\mathbb R^{d}} f(v).\nabla v dx = 0$ ; where $f: \mathbb R \to \mathbb R^{d}$ is a Lipschitz continuous function such that $f(0) = 0$ ??

Answer 1

The question is misleading in that if you expand the scalar product, then the integral over each term is zero, so it is really a statement about scalar functions I would argue.

To prove the scalar valued version, try for $f$ (now a scalar valued function) the identity and then generalize. Ask if you have problems in executing this plan, but please add more info about your efforts than in the original question.