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Let $\Omega\in\mathbb{R}^n$ be a bounded open set with smooth boundary. How to prove the invertibility of $$- \triangle:H^2_0(\Omega) \to L²(\Omega) $$ The injectivity is easy. But how to prove surjectivity without the use of weak notion of solution (when the domain becomes $H^1_0(\Omega)$ and this can be easily found in books)? |
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The operator $$- \triangle:H^1_0(\Omega) \to L²(\Omega) $$ is "surjective" in the weak sense (as a direct use of Riesz representation theorem). Then you can use regularity theorems to prove that, in fact, this weak solution is in $H^2(\Omega)$. A good reference to a result like this is the Brezis book "functional analysis, Sobolev spaces and PDE". See theorem 9.25 and note that a $C^2$ domain $\Omega$ is required. So your result is valid from $H^1_0(\Omega)\cap H^2(\Omega) $ to $L²(\Omega) $. |
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