I was studying Sobolev spaces. I saw the following statement, concerning PDEs.
Statement:
Let $\Omega$ be an open, bounded set in ${\mathbb{R}}^n$ and let $u$ be a continuous eigenvalue on $\bar{\Omega}$ of the operator $-\Delta$, where $\Delta$ denotes, as usually, the Laplace operator. If $\ \exists x_0\in \Omega$ such that $\partial^au(x_0)=0$ for all multi-indices $a$, then $u\equiv 0$ on $\bar{\Omega}.$
First of all, I assume that the derivatives are taken in the distributional sense. I am not very familiar with this theory and I don't see how to use the hypothesis about the vanishing of all partial derivatives (and the function itself for $a=0$) at $x_0.$ That's why I don't have any serious attempt of mine to present to you. I only know that the eigenvalues tend to $+\infty$, being positive, that the eigenfunctions are smooth and a relation between their $L^2$ and Sobolev norms.
Is it allowed to follow an argument with a Taylor series expansion of the eigenfunction $u$? I don't know. Could you provide me an idea or an important hint to construct a proof?
Thank you in advance for your valuable help!