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Thank you, this is already quite helpful. Though I'm noticing that "weakly coupled" generally refers to the fact that derivatives don't act of coupling terms. But in this case we have a gradient acting on v in the equation for u. Oh well, it's not everyday that you find exactly what you want I suppose. Thanks again.
Thanks, looks good. Just a sanity check here: if we were simply going for a statement like $\|u\|_{L^2(\Omega)}\le C(\|\nabla u\|_{L^2(\Omega)}+\|u\|_{L^2(\partial\Omega)})$, then I think we can take $X=H^1(\Omega)$?
Strictly speaking, I think the space $\dot{H^1}$ is defined in terms of equivalence classes where functions that differ by polynomials are identified. In particular, all constant functions are equivalent to the zero function.
