Norm of laplacian in dual space $H^{-2}$

Question

how we can prove the estimate $\|\Delta u\|_{H^{-2}} \leq C \|u\|_{L^2(\Omega)}$, where $\Omega$ is a bounded open set of $\mathbb{R}^n$, and $H^{-2}$ is the dual space of $H^{2}\cap H_0^1$. Can we get this estimate without using extrapolation spaces ? Thank for any help.

Answer 1

Let suppose your laplacian is with Dirichlet boundaries conditions (i.e $H^2$ is in fact $H^2$ with $0$ as Dirichlet trace).

Let $v \in C^\infty(\bar{\Omega})$ such that $v_{|\partial \Omega}=0$, by integration by parts: $$|\langle \Delta u, v \rangle|=\left| \int_\Omega \Delta u v \right|=\left|\int_\Omega u \Delta v \right|\leq \|u|_{L^2} \|\Delta v\|_{L^2}$$ but $\|\Delta v\|_{L^2} \leq \|v\|_{H^2}$.

You obtain that for any sufficiently regular $v$: $$|\langle \Delta u,v\rangle| \leq\|u\|_{L^2} \|v\|_{H_2} $$ which gives the result by density.