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.

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 1.27 and 1.28.

Let $X$ be a 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 should 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.

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.