Return to Revisions

Set $V$ to be the subbundle of $TM$ whose fiber at $p$ is the span of $\nabla\phi_1(p)$ and $\nabla\phi_2(p)$. Your equation on $u$ is equivalent to requiring that $du$ vanishes on $V$. Frobenius integrability theorem tells you it has (EDIT: non constant) local solutions if and only if $V$ is integrable (i.e. it is stable under Lie bracket). It is not too hard to find functions $\phi_1$ and $\phi_2$ such that $[\nabla\phi_1(p),\nabla\phi_2(p)]\notin V$. For instance on $\mathbb{R}^3$, take $\phi_1(x,y,z)=x$ and $\phi_2(x,y,z)=xyz$.