Definition : The Distribution of laplacian of a function?

Question

Let $u\in H_0^1(\Omega)$(but i think we can be way more general than than).

How is $\langle\Delta u,\phi\rangle$ (distribution) defined for $\phi \in D(\Omega)$ a test function ?

Answer 1

As with most definitions in distribution theory, it boils down to "formal integration by parts":

$$\langle \Delta u,\phi \rangle = -\langle \nabla u,\nabla \phi \rangle = \langle u,\Delta \phi \rangle.$$

In general when doing this you need to be careful to be sure that the boundary terms in each of these integrations actually vanish in your setting. In the case described in the question this is trivial since $\phi \in D(\Omega)$.

Answer 2

$$ \langle \Delta u, \phi \rangle := \langle u, \Delta\phi \rangle. $$