Scalar conductivity and Schrödinger Dirichlet-to-Neumann maps

In dimension one the trace and lift conclusions follow from the endpoint argument in [F3], and integration by parts is the fundamental theorem of calculus with counting boundary density. In higher dimensions use the Euclidean suppliers as follows. Their difference $W=V_h^{(1)}-V_h^{(2)}$ has zero trace. For dimension one, [F3] gives $W\in H_0^1(M)$ directly; for dimension at least two, localize $W$ by the fixed finite partition in [F1]. On each boundary chart, [F7] shows that the localized function has zero Euclidean trace after flattening; [F2] then approximates it in $H^1$ by compactly supported interior smooth functions. On interior charts, ordinary compactly supported mollification gives the same approximation. Pulling back and summing the finitely many approximations shows $W\in H_0^1(M)$.

For the conductivity pairing, [F3] gives ${\int }_M\gamma g({\mathrm{\nabla }}_g{v}_f,{\mathrm{\nabla }}_g\bar{W})d{V}_g=0$, since $W\in H_0^1(M)$. Therefore replacing $V_h^{(1)}$ by $V_h^{(2)}$ changes the displayed pairing by zero. For the Schrödinger pairing, the same argument with the weak equation in [F4] gives ${\int }_M(g({\mathrm{\nabla }}_g{u}_f,{\mathrm{\nabla }}_g\bar{W})+q{u}_f\bar{W})d{V}_g=0$. Thus both definitions are independent of the chosen extension. For each $h$, choose a bounded lift $V_h$ by [F6]; Cauchy--Schwarz gives $\mathrm{\mid }⟨{\mathrm{\Lambda }}_{\gamma }f,h⟩\mathrm{\mid }\le C\mathrm{\parallel }{v}_f{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }h{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$ and $\mathrm{\mid }⟨{\mathrm{\Lambda }}_{q}f,h⟩\mathrm{\mid }\le C(1+\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\parallel }{u}_f{\mathrm{\parallel }}_{H^1}\mathrm{\parallel }h{\mathrm{\parallel }}_{H^{1\mathrm{/}2}}$. Hence each pairing is a continuous conjugate-linear functional of $h$, as required for the anti-dual in [F1].

By [F3] and [F4], smooth boundary data yield solutions smooth up to $\mathrm{\partial }M$. For smooth test data $h$, choose a smooth collar extension $V_h$ by [F8]. In a boundary chart, write ${\mathrm{div}}_g(\gamma {\mathrm{\nabla }}_{g}v)=\mathrm{\mid }g{\mathrm{\mid }}^{-1\mathrm{/}2}{\mathrm{\partial }}_i(\mathrm{\mid }g{\mathrm{\mid }}^{1\mathrm{/}2}\gamma g^{ij}{\mathrm{\partial }}_{j}v)$ and ${\mathrm{\Delta }}_{g}u=\mathrm{\mid }g{\mathrm{\mid }}^{-1\mathrm{/}2}{\mathrm{\partial }}_i(\mathrm{\mid }g{\mathrm{\mid }}^{1\mathrm{/}2}{g}^{ij}{\mathrm{\partial }}_{j}u)$. Apply the Euclidean Gauss-Green identity [F5] to the coordinate fluxes; summing over the finite partition gives the Riemannian boundary flux, while artificial chart boundaries cancel and the density transforms to $d{S}_g$. For the conductivity equation it yields $⟨{\mathrm{\Lambda }}_{\gamma }f,h⟩={\int }_{\mathrm{\partial }M}\gamma {\mathrm{\partial }}_{{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}}{v}_f\bar{h}d{S}_g,$ since ${\mathrm{div}}_g(\gamma {\mathrm{\nabla }}_g{v}_f)=0$. For the Schrödinger equation it yields $⟨{\mathrm{\Lambda }}_{q}f,h⟩={\int }_{\mathrm{\partial }M}{\mathrm{\partial }}_{{\nu }_{\mathrm{o}\mathrm{u}\mathrm{t}}}{u}_f\bar{h}d{S}_g,$ since $(-{\mathrm{\Delta }}_g+q)u_f=0$. The extensions required for the weak pairings exist by [F6], and smooth test extensions exist by [F8].