Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that $Ls=0$, and $s|U=0$ for some nonempty open $U\subset X$. Then do we have $s=0$ identically?

One can assume $X$ is an open subset of $\mathbb{R}^n$ and $L$ is a matrix of differential operators that is elliptic if feel uncomfortable with vector bundles.

For the fact that this is true for Laplacian on an open subset of $\mathbb{R}^n$, see [here][1] 1.27 and 1.28.

[1]: https://books.google.com.hk/books?id=CYbdBwAAQBAJ&pg=PA21&lpg=PA21&dq=if+a+harmonic+function+is+0+on+an+open+subset,+then+it+is+identically+0&source=bl&ots=m3iRspbsvH&sig=OycB56TBaRPxcW0d7nPn72G_xi4&hl=en&sa=X&ved=0ahUKEwjelqjpo5zKAhWRHI4KHZiCCJE4ChDoAQguMAU#v=onepage&q=if%20a%20harmonic%20function%20is%200%20on%20an%20open%20subset%2C%20then%20it%20is%20identically%200&f=false