Continuity of Gradient operator

Question

I know that the gradient operator is continuous when it is defined from $H^1_0(\Omega)$ to $\mathbb L^2(\Omega)$

But is it still continuous when it is defined from $L^2(\Omega)$ to $\mathbb H^{^-1}(\Omega)$ ?

Also, is there exists a positive constant $c$ such that $$ ||u||_{L^2(\Omega)} \le c ||\nabla u||_{H^{-1}(\Omega)}$$

I just need a short answer and I will work on proving or disproving this, thank you.

Answer 1

1)the gradient operator isn’t continuous from $L^2(\Omega)$ to $H^{-1}(\Omega)$

2)the inequality is true only if $u \in L^2_0(\Omega)$ this means that when it’s of zero mean.
$$m(u)=\frac{1}{|\Omega|}\int_{\Omega} u(x) dx$$