Weak convergence and Laplace's equation

Question

Suppose $\{u_i\}$ is a sequence of distributions that satisfies Laplace's equation

$$\Delta u_i = 0, \forall i$$

in the weak sense. If $\{u_i\} \to u$ in the sense of distribution, show that $u$ also satisfies Laplace's equation.

I am stuck. I have the following for any test function $\phi$:

$$\int u_i \phi \to \int u \phi$$ and $$\int \Delta u_i \phi = \int u_i \Delta \phi =0$$

How do we conclude $\int \Delta u \phi =0$?

Answer 1

HINT: Just remember that if $\phi$ is a test function, $\Delta \phi$ is a test function as well.