Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $\{p_i\}_{i=1}^m$ for some finite $m\in\mathbb{N}$ be the overlapping open sets of $\Omega$ such that $\Omega =\bigcup_{i=1}^mp_i$.
Now, Define a function $f_i\in C^{r}$ on $p_i$ where $r\in \mathbb{N}$.
Question: What extra conditions on $f_i$ and $f_j$ do we have to impose so that $f_i=f_j$ on $p_i\cap p_j$ for $i\ne j$?
This question seems to suggest that we will require some vector space inner product on $f_i$'s to show them equal on the overlapping domain.
I appreciate any help you can provide.
The motivation behind this question comes from the Domain decomposition for solving the eigenvalue problem stated in this paper, section 2.
