Let $H^1$ be the Sobolev space of $L^2$ functions with $L^2$ generalized derivatives and let $H^1_0$ be the space of $H^1$ functions with zero trace. Are these two spaces isomorphic? If so is there an easy way to explicitly write down an isomorphism?
Since $H^1$ and $H^1_0$ are Hilbert spaces, they are isomorphic to their duals. I've read that the dual of both these spaces is $H^{-1}$, so that would define an isomorphism between $H^1$ and $H^1_0$. But is there a more explicit isomorphism?