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maximal monotone operator in $H_{0}^{1}(\Omega)$

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I am considering an operator\begin{align}A:H_{0}^{1}(\Omega)&\to H^{-1}(\Omega)\\u&\mapsto \int_{\Omega}\nabla u\cdot\nabla v {dx}\mbox{ for all }v\in H_{0}^{1}(\Omega)\end{align}I already showed that this operator is monotone, so my question is whether this operator is also maximal. Because if it is a maximal monotone operator, then I can have that the sum of this operator and another maximal monotone operator is also invertibel.


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