We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and let $\chi\_\omega$ denotes the indicator function on a subdomain $\omega$ of $\Omega$.


Do we have
$$\chi\_\omega\Delta=\Delta\chi\_\omega?$$



Does the Laplacian commutes with the indicator function
This fails even in the case of the one dimensional case on an interval since the rhs picks up $\delta$ functions at the endpoints due to the jump singularities there and these are abseent in the lhs.


