Domain of the adjoint of the Laplacian

Given a domain $$\Omega \subset\Bbb R^n$$, consider the set $$D := \{u \in L^2(\Omega)| \Delta u \in L^2(\Omega)\}$$, where $$-\Delta$$ is the Laplacian. I think this is the domain of the adjoint of $$-\Delta$$.

My question: is the set $$D$$ always identical to the Sobolev space $$H^2(\Omega)$$ ?

Perhaps this is dependent on the boundedness or smoothness of $$\Omega$$. I didn't found anything about this problem in functional analysis textbooks. Are such sources known to anybody ?

• you can find such things in Taylor's book Partial Differential Equations I amazon.com/Partial-Differential-Equations-Mathematical-Sciences/…
– shu
Commented Jan 16, 2014 at 13:28
• The book of Renardy-Rogers gives you also information on this. If the boundary is smooth, then yes, in general no. Commented Jan 16, 2014 at 13:30
• When you speak of adjoint of $-\Delta$ you first need to specify the domain of $-\Delta$. How do you define the domain of the Laplacian? Commented Jan 16, 2014 at 14:34
• It u and $\Delta u$ are in $L^2$, it certainly does not follow that $u\in H^2$, unless some restriction is imposed at the boundary. Commented Jan 16, 2014 at 14:47
• Thanks to all for the helpful answers. With respect to the question of Liviu, the domain of the Laplacian should be the space of smooth functions with compact support. Commented Jan 17, 2014 at 11:17