Solution of Poisson equation vanishing at the boundary of any order

Question

Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and

$\Delta u=f$ in $\Omega$

such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \geq 0$. Is $f$ necessarily zero?

Answer 1

No. Take any $u$ which is not zero, but compactly supported in $\Omega$. Then define $f=\Delta u$; it will be also compactly supported, and non-zero.