Equivalence of dual spaces of Sobolev Spaces

Question

I have a quick question: Is the following equivalence true for Sobolev Spaces $(W^{1,p}(\Omega))^{*} = W^{-1,p}(\Omega) = (W^{1,p}_{0}(\Omega))^{*}$ where $W^{1,p}_{0}(\Omega)$ is the closure of $C_{c}^{\infty}(\Omega)$ in $W^{1,p}(\Omega)$. Thanks for any assistance.

Answer 1

No, this is not true.

You only have $$(W^{1,p}_0(\Omega))^*=W^{-1,p}(\Omega) $$ However, the dual space of $W^{1,p}(\Omega)$ is not identified, although it is smaller then $W^{-1,p}(\Omega)$.

For more information, please read 10.4 in Leoni's book, it has a complete treatment of dual of Sobolev space.