Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with **smooth** boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real analytic coefficients in the interior $\Omega$. $u$ and $f$ are real analytic in the interior $\Omega$ and both are smooth up to the boundary, with zero Dirichlet boundary data $u|_{\partial \Omega} = f|_{\partial \Omega} = 0$. Suppose in addition we assume that $u$ satisfies zero Neumann data $ \frac{\partial u}{\partial \eta}|_{\partial \Omega} = 0$.

I am looking for references which give unique continuation results like $u = f = 0$. I am not sure if we need additional restrictions on $L$ to ensure such a conclusion (I am happy with references that say something about $L = \Delta$). Any pointers will be highly appreciated.