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.



Can the solution of an elliptic operator with smooth coefficients have zeros of infinite order?
Take a look at [this](http://www.jstor.org/pss/1993331) paper by Protter; there are certainly newer references (which you can easily find starting from this one), but basically the problem was already solved at the time.


