Regularity of biharmonic problem with solution $u \in H_0^2(\Omega)$

Question

Supose $f \in L^2(\Omega)$, $\Omega$ bounded and $u \in H_0^2(\Omega)$ weak solution for biharmonic problem $$\left\{\begin{array}{l} \Delta^2u=f\\ u=\partial_{\nu} u=0 \ \mbox{in} \ \partial \Omega \end{array}\right. $$ Can I prove that $u \in H^4(\Omega)$? I am trying use the theorems of regularity elliptc of Evans but I can't solve this.