Sobolev space on union of two open sets

Question

Let $\Omega_1,\Omega_2 \subset \mathbb{R}^n$ be open sets. Let $p \in [1,\infty]$. Let $u: \Omega_1 \cup \Omega_2 \to \mathbb{R}$ be a function such that $u|_{\Omega_1} \in W^{1,p}(\Omega_1)$ and $u|_{\Omega_2} \in W^{1,p}(\Omega_2)$. I need to prove that $u \in W^{1,p}(\Omega_1 \cup \Omega_2)$.

I have already proved this for case $\Omega_1, \Omega_2$ disjoint, but there are some problems with the second case $\Omega_1 \cap \Omega_2 \neq \emptyset$. In the first case I defined a function $g_\alpha = (D^\alpha u|_{\Omega1})\chi_{\Omega_1} + (D^\alpha u|_{\Omega2})\chi_{\Omega_2}$ and used equalities

$\int_{\Omega_1} D^\alpha u|_{\Omega_1}\ \varphi = - \int_{\Omega_1} u|_{\Omega_1}\ D^\alpha \varphi\ \forall \varphi \in D(\Omega_1)$,

$\int_{\Omega_2} D^\alpha u|_{\Omega_2}\ \varphi = - \int_{\Omega_2} u|_{\Omega_2}\ D^\alpha \varphi\ \forall \varphi \in D(\Omega_2)$

(where $D(\Omega)$ denotes the space of smooth functions with compact support) to prove that $D^\alpha u = g_\alpha$, i.e.

$\int_{\Omega_1 \cup \Omega_2} g_\alpha\ \varphi = - \int_{\Omega_1 \cup \Omega_2} u\ D^\alpha \varphi\ \forall \varphi \in D(\Omega_1 \cup \Omega_2)$.

But I probably can't proceed the same way in the second case, since $D(\Omega_1 \cup \Omega_2)$ can contain much more other functions than linear combinations of these in $D(\Omega_1)$ and $D(\Omega_2)$. How can I proceed in this case?

Thank you!

Answer 1

Take $\phi\in\mathcal{C}^\infty_c(\Omega_1\cup\Omega_2)$ and let $K=\mathrm{supp}\phi$ be its support; let $U_i=\Omega_i\cap K$, $i=1,2$ and take $\delta$ the Lebesgue number of the covering $\{U_1, U_2\}$.

Now let $B_1,\ldots, B_m$ balls of radius $r$ such that $2r<\delta$ and $$\bigcup B_j\supseteq K\;.$$ Let also $\{\chi_j\}$ be a partition of unity such that $\mathrm{supp}\chi_j\Subset 2B_j$ and $\chi_j\vert_{B_j}\equiv1$.

The functions $D^a(u\vert_{\Omega_1})$ and $D^a(u\vert_{\Omega_2})$ have to coincide a.e. on $\Omega_1\cap \Omega_2$, which is an open set, because there they are the differential of the same function $u\vert_{\Omega_1\cap\Omega_2}$. So you can define a.e. on $\Omega_1\cup \Omega_2$ a function $g^a$ such that $g^a\vert_{\Omega_i}\equiv D^a(u\vert_{\Omega_i})$ a.e. on $\Omega_i$ for $i=1,2$.

Now, $$\int_{\Omega_1\cup\Omega_2}g_a\phi=\sum_j \int_{2B_j}g_a(\chi_j \phi)$$ every such integral involves a function supported either in $\Omega_1$ or $\Omega_2$, so we can integrate by parts and get $$=\sum_j -\int_{2B_j}uD^a(\chi_j \phi)=-\int_{\Omega_1\cup \Omega_2}uD^a(\phi)\;.$$

Answer 2

Apparently your problem is verifying that if a distribution $v$ ( really your $D^{\alpha}u$) is of function type when restricted to $\Omega_1$ and $\Omega_2$ then so is on $\Omega_1\cup \Omega_2$ --the "sheaf property" of the distributions.

It is enough to show that any compactly supported function $\phi$ on $\Omega_1\cup \Omega_2$ is a sum of two functions $\phi = \phi_1 + \phi_2$ with $\chi_i$ supported on $\Omega_i$. Use a partition of unity associated to the cover $(\Omega_1, \Omega_2)$. It is a pair of smooth functions $\chi_1$, $\chi_2$ defined on $\Omega_1\cup \Omega_2$ such that $\text{supp} u_i \subset \Omega_i$ and $\chi_1 + \chi_2 = 1 $ on $\Omega_1 \cup \Omega_2$. If $\Omega_1$, $\Omega_2$ are disjoint these $\it{are}$ the characteristic functions of $\Omega_i$, but otherwise you want some smooth attenuations of them on the intersections. For every $\phi$ compactly supported on $\Omega_1 \cup \Omega_2$ consider $$\phi= \phi\cdot \chi_1 + \phi\cdot \chi_2$$