Let $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following system has a unique solution in $H^1(\Omega)$ : $$ \begin{cases} (-\Delta + q) v= f&\mbox{ in }\Omega\\ \frac{\partial v}{\partial n}=0&\mbox{ on }\partial \Omega \end{cases} $$ where $f\in H^{-1}(\Omega)$ is a given data.

If we choose $q>0$ on $\Omega$, we obtain the result using the Lax-Milgram Lemma. But what if $q$ is not positive?



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