Let $\Omega$ be a closed domain with smooth boundary in $\mathbb{R}^n$. Let $H^1_0(\Omega)$ be the closure of compactly supported smooth functions under the norm $\|u\|_1 = \int_\Omega u^2 + |\nabla u|^2\ dx$ and let $H^1(\Omega)$ be the closure of smooth, continuous functions under the same norm.
Any $H^1$ function which has nonvanishing trace cannot be approximated by any sequence of functions in $H^1_0$. So $H^1_0$ is a closed subspace of the Hilbert space $(H^1, \|\cdot\|_1)$, hence has an orthogonal complement.
What is a generating set of the orthogonal complement of $H^1_0$ in $H^1$?
Motivation is to get my hands on some concrete examples, rather than to just appeal to theorems that establish the existence of a right inverse to a trace operator.
Of course if anyone has references, I'm happy to follow them up. I've skimmed through Gilbarg-Trudinger and Evans and found nothing, but maybe I'm looking in the wrong place.
As reuns has stated in the comments, the answer depends on the inner product you choose in $H^1(\Omega)$. Let us fix the most common choice $$ (u,v)_{H^1(\Omega)} = \int_\Omega \nabla u \cdot \nabla v + u \, v \, \mathrm{d}x.$$
In this case, the orthogonal complement of $H_0^1(\Omega)$ consists precisely of the (weak) solutions $u \in H^1(\Omega)$ of $$ -\Delta u + u = 0$$ (without B.C.). Indeed, the weak formulation of this PDE is $$(u,v)_{H^1(\Omega)} = 0 \quad\forall v \in H_0^1(\Omega).$$
For different inner products, you get different PDEs.