States compatible with the uncontrolled obstacle boundary

PDF p. 4: Theorem 1.1, definition of Xs(Ω0); PDF p. 17: Lemma 3.11

Assume the Axiom of Choice. Let $n\ge 2$ and let $\mathrm{\Theta }$, ${\mathrm{\Omega }}_0$ and ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$ have the smooth obstacle and truncated-exterior geometry specified in the exterior-domain definition. Let $1\le s\le 2$.

Use the complex $L^2$ Bessel-potential Sobolev space: $H^s({\mathbb{R}}^n)$ consists of tempered distributions $U$ for which $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2{)}^{s\mathrm{/}2}\widehat{U}$ is an $L^2$ function, with its corresponding $L^2$ norm, using the negative-sign $2\pi$ Fourier transform. Define $H^s({\mathrm{\Omega }}_0)$ as the restrictions of these distributions to ${\mathrm{\Omega }}_0$, with norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s({\mathrm{\Omega }}_0)}=\inf \{\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s({\mathbb{R}}^n)}:U{\mathrm{\mid }}_{{\mathrm{\Omega }}_0}=u\}$.

Let ${\gamma }_{{\mathrm{\Gamma }}_2}u$ be the Sobolev value trace on the obstacle boundary. Its target $H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_2)$ is defined by a finite smooth boundary atlas and subordinate partition of unity, using the $H^{s-1\mathrm{/}2}({\mathbb{R}}^{n-1})$ norms of the localized coordinate representatives. Define

$X^s({\mathrm{\Omega }}_0)=\{u\in H^s({\mathrm{\Omega }}_0):{\gamma }_{{\mathrm{\Gamma }}_2}u=0\},\mathrm{\parallel }u{\mathrm{\parallel }}_{X^s({\mathrm{\Omega }}_0)}=\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s({\mathrm{\Omega }}_0)}\mathrm{.}$

The zero condition is equality in $H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_2)$. No boundary condition is imposed on the outer boundary ${\mathrm{\Gamma }}_1$.

Exterior and truncated obstacle domains

Simultaneous Sobolev extension and localization

Collar traces and extension across an artificial boundary

Real-order Hˢ as weighted Fourier distributions

The Axiom of Choice

The exterior-domain definition fixes the disjoint smooth boundary pieces and the bounded smooth truncated domain. The whole-space Fourier characterization supplies the displayed Sobolev space and norm, and the Sobolev extension and localization result supplies the restriction-quotient convention and its norm on this bounded smooth domain. The Axiom of Choice includes the choice assumptions of these suppliers.

The collar-trace result supplies a bounded linear value trace from $H^s({\mathrm{\Omega }}_0)$ to the stated finite-atlas boundary space for $1\le s\le 2$, independent of the atlas up to equivalent norms. Apply it to the obstacle boundary components; compactness and the local smooth graph structure give finitely many such components. Thus the zero-trace condition is a definite linear condition, and $X^s({\mathrm{\Omega }}_0)$ is its kernel, a closed linear subspace with the inherited norm. The definition does not assert a normal-derivative condition or a zero condition on the artificial outer sphere.

The original article is cited for attribution of $X^s$ in Theorem 1.1, where $s\in \{1,2\}$, and for its use for $1\le s\le 2$ in Lemma 3.11. The mathematical meaning and well-definedness above use the independent suppliers.