In chapter 11 of Jost partial differential equations he wants to prove the following theorem of interior regularity for the Poisson equation:
Theorem 11.2.1: Let $u\in W^{1,2}(\Omega)$ be a weak solution of $\Delta u=f$, with $f\in L^2(\Omega)$. For any $\Omega'\subset\subset\Omega$, then $u\in W^{2,2}(\Omega')$ and
$$\lVert u \rVert_{W_{2,2}(\Omega')}\leq const(\lVert u \rVert_{L_2(\Omega)}+\lVert f \rVert_{L_2(\Omega)})$$
where $\Omega \subseteq \mathbb{R}^n$ is a bounded open domain. To prove this theorem he uses this cutoff function:
$$\eta(x)=
\begin{cases}
1 & x \in \Omega'\\
0 & d(x,\Omega')\geq \delta\\
1-\frac{1}{\delta}d(x,\Omega') & 0\leq d(x,\Omega')\leq \delta
\end{cases}$$
where $\delta=d(\Omega',\partial \Omega)$. It is easy to prove that $\eta$ satisfies the following properties:
- $0\leq \eta \leq 1$
- $\eta=1$ on $\Omega'$
- $\eta\in W_0^{1,2}(\Omega)$
- $\lvert \nabla\eta\rvert\leq \frac{2}{\delta}$
Then he takes $v=\eta^2 u$ and he plugs it in the definition of weak solution:
$$\int_\Omega \nabla u \cdot \nabla v=-\int_\Omega fv \qquad \forall v\in H_0^1(\Omega)=W_0^{1,2}(\Omega)$$
I cannot see why $v=\eta^2 u\in H_0^1(\Omega)$. I know that $\eta, u\in H_0^1(\Omega)$, but $H_0^1(\Omega)$ is not an algebra.
I know that I can take a $C^\infty_c(\Omega)$ cutoff function using mollifications, but I would like to understand why we can use this simpler cutoff function.
Thank you
Edit: Since $\eta$ is lipschitz $\eta\in W^{1,\infty}_0(\Omega)$. Now from this lemma (Differentiation of a product of Sobolev functions) I deduce that also $\eta^2\in W^{1,\infty}_0(\Omega)$. I don't know if it can help
Best Answer
The point is that you have more information about $\eta$ than just the fact that it is in $H^1_0(\Omega)$. Applying the product rule, we have \begin{align} |\nabla(\eta^2 u)| =& |2 \eta u \nabla \eta + \eta^2 \nabla u |\\ \leq & 2| \eta| |u|| \nabla \eta| + |\eta|^2| \nabla u | \\ \leq & \frac{4}{\delta}|u| + |\nabla u| \, . \end{align} Thus \begin{equation} \int_\Omega|\nabla(\eta^2 u)|^2 \leq C \lVert u \rVert_{H^1_0(\Omega)}^2 \, , \end{equation} for some constant $C$ dependent on $\delta$.