I'm trying to solve the following excercise:

Let $v \in C(D)$ be a nonnegative subharmonic function in an open set $D$. Prove that $v \in W_{loc}^{1,2}(D)$

The problem has the following hint:

Mollifications $v_\epsilon$ of $v$ satisfy the inequality $$ \int_D \nabla v_\epsilon \cdot \nabla \phi \;dx \leq 0 $$ for any nonnegative $\phi \in C_0^\infty(D)$. Take $\phi = v_\epsilon \zeta^2$ with $\zeta \in C_0^\infty(D)$ and let $\epsilon \to 0+$.

I got confused trying to integrate by parts and use the Poincaré inequality to prove a uniform bound, but I guess and don't really understand how to use the hint. I know that under the hypotheses the convergence $v_\epsilon \to v$ is in $L_{loc}^2(D)$ and uniform on compact subsets of $D$, but not more than that.

I will really appreciate any comment.


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