Supposed that $u, f\in L^2(\Omega)$ satisfies

$$\int_{\Omega}u\Delta^2 \phi^2 = \int_{\Omega} f\phi dx,$$

for all $\phi \in A = \{u\in H^4\cap H^1_0 (\Omega) : ~\Delta u \in H^1_0(\Omega)\}$. Prove that $u\in A$.

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