Application on Partition of unity

Question

How can we prove that a function $f$ is in $H^1(\Omega)$ if there is a finite open cover $(U_i)_{i= 1}^N$ of $\Omega$ such that $f_{|U_i} \in H^1(U_i)$ for all $i$, by using partition of unity?

Answer 1

Let $(\gamma_i)_{i= 1}^N$ be a smooth partition of unity subordinate to $(U_i)_{i= 1}^N$. Then
\begin{align*} \|\gamma_i f\|_{H^1(\Omega)} = \| (\gamma_i f)_{|U_i}\|_{H^1(U_i)} \leq \|\gamma_i\|_{C^1} \|f_{|U_i}\|_{H^1(U_i)}, \end{align*} where the first step is since $\text{supp }\gamma_i \subset U_i$ and the last step is by the Leibniz rule. Thus, $\gamma_i f \in H^1(\Omega)$. Since $f = \sum_{i = 1}^N \gamma_i f$ and $H^1(\Omega)$ is a vector space, we have $f \in H^1(\Omega)$.