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I am attempting to solve some problems from Evans, I need some help with the following question.

Suppose $u\in H^2_0(\Omega)$, where $\Omega$ is open, bounded subset of $\mathbb{R}^n$.

  • How can I solve the biharmonic equation $$\begin{cases} \Delta^2u=f \quad\text{in } \Omega, \\ u =\frac {\partial u } {\partial n }=0\quad \text{on }\partial\Omega. \end{cases} $$ where $n$ is the normal vector such that $\int _\Omega \Delta u \Delta v \, \,dx =\int _\Omega fv $ for all $v\in H^2_0(\Omega)$.

  • Given $f \in L^2(\Omega)$ , and prove that the weak solution is unique.

Any kind of help would be great.

5 Answers 5