I've been told that the answer is no, but I'm having a hard time finding a reference. More precisely, I'm interested in the following. Let $\Omega \subset \mathbb{R}^n$ be a bonded open connected set, say with $C^{\infty}$ smooth boundary and

$L=-\sum_{i,j}\frac{\partial}{\partial_{x_j}}\left(a_{i,j}(x)\frac{\partial}{\partial_{x_i}}\right)+\sum_i a_i(x)\frac{\partial}{\partial_{x_i}}+a_0(x)$

a differential operator satisfiying $a_{i,j},~a_i \in C^{\infty}(\overline{\Omega})$ and the uniform ellipticity condition,

$\exists \lambda >0,~\forall x \in \Omega,~\forall \xi \in \mathbb{R}^n ,~\sum_{i,j}a_{i,j}(x)\xi_i \xi_j \ge \lambda |\xi|^2.$

Elliptic regularity estimates tell us that if $u$ is a solution to the Dirichlet problem $Lu=0$ in $\Omega$ and $u=0$ on $\partial \Omega$ then $u \in C^{\infty}(\overline{\Omega})$. What I would like to know is: assume that $x\in \Omega$ and that all derivatives of $u$ at $x$ are zero. Is $u$ zero on $\Omega$? And what if $x \in \partial \Omega$? My motivation is to understand the description of the nodal set of an eigenfuction of the Laplace operator.

Like I said, I'm quite sure this has been done, but I don't know where and I would be grateful if someone could point me in the right direction.