Solution to Problem 3

So this problem comes from the functional analysis course I took last semester. Although simple, I thought was pretty cool. In the actual exercise they had broken it down into three parts, which I did not show in the problem post. Here I will show the steps as claims....

Problem of the Week 3

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 $u\in A$.

Analysis Tricks Part 1

I recently learnt a really neat trick when it comes to proving two quantities are equivalent. It started of with my supervisor claiming that for some $\xi \in \mathbb{R}^n$ and $m\in \mathbb{R}$, \begin{equation} \Sigma_{\abs{\alpha}\leq m} \abs{\xi^{\alpha}}^2 \simeq...