Do we have the same characterization of $(H^1(\Omega))^*$ as that of $(H^1_0(\Omega))^*$?

Question

The following is the characterization theorem for $H^{-1}(U):=(H_0^1)^*(U)$in Evans's Partial Differential Equations:

Here is my question:

If one defines $H^{-1}(U)$ as the dual of $H^1(U)$ instead, is the theorem above still true?

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Here

• $U$ is an open subset of $\mathbb{R}^{n}$
• $H^{1}(U)$ is the Sobolev space of $L^{2}(U)$ functions with weak derivatives in $L^{2}(U)$
• $H_{0}^{1}(U)$ is the closure of the subspace $\mathcal{C}_{c}^{\infty}(U)$ of compactly supported smooth functions on $U$
• $H^{-1}(U)$ is the (continuous) dual of $H_{0}^{1}(U)$