Sobolev spaces on a compact Riemannian domain with boundary

Assume Countable Choice.

Let $n=\dim M\ge 0$ and let $(M,g)$ be a compact smooth Riemannian manifold with smooth boundary, with no orientability assumption. Fix a finite smooth atlas of interior and boundary charts and a smooth partition of unity subordinate to it. If $n=0$, define every $H^k(M)$ to be the space of scalar-valued functions on the finite set $M$ with the counting-measure ${\mathrm{\ell }}^2$ norm. For $n\ge 1$ and an integer $k\ge 0$, define $H^k(M)$ as the completion of $C^{\mathrm{\infty }}(M)$ in the sum of the Euclidean $W^{k,2}$ norms of the localized coordinate representatives; derivatives in a boundary chart are taken on the corresponding half-space. Define $H_0^1(M)$ as the closure of $C_c^{\mathrm{\infty }}(M^{\circ })$ in $H^1(M)$. If $\mathrm{\partial }M=\mathrm{\varnothing }$, set $H^{1\mathrm{/}2}(\mathrm{\partial }M)=\{0\}$. If $n=1$, set $H^{1\mathrm{/}2}(\mathrm{\partial }M)={\mathbb{K}}^{\mathrm{\partial }M}$ with its counting-measure ${\mathrm{\ell }}^2$ norm, where $\mathbb{K}$ is the scalar field. For $n\ge 2$, define $H^{1\mathrm{/}2}(\mathrm{\partial }M)$ by the fixed finite boundary atlas and partition of unity, using the Euclidean fractional Slobodeckij space $W^{1\mathrm{/}2,2}({\mathbb{R}}^{n-1})$ for each localized representative. For real scalars, define $H^{-1\mathrm{/}2}(\mathrm{\partial }M)$ as the continuous linear dual of $H^{1\mathrm{/}2}(\mathrm{\partial }M)$. For complex scalars, define it as the continuous conjugate-linear dual: its pairing with $h\in H^{1\mathrm{/}2}(\mathrm{\partial }M)$ is linear in the dual element and conjugate-linear in $h$, using the Riemannian boundary density. These are the fixed-atlas norms used on this page; no atlas-independence assertion is part of this definition.

Riemannian manifold and volume density

Smooth charts on a manifold with boundary

Euclidean integer-order Sobolev norms

Euclidean fractional Sobolev spaces

Smooth partition of unity

Existence of finite subordinate partitions