Time-dependent Dirichlet-to-Neumann map for the wave equation

Assume Countable Choice. Let $(M,g)$ be a compact, connected, nonempty smooth Riemannian manifold with smooth boundary, let $T>0$, and write $Q=(0,T)\times M$ and $\mathrm{\Sigma }=(0,T)\times \mathrm{\partial }M$. Let $q\in C^{\mathrm{\infty }}(M;\mathbb{R})$ be independent of time, and let $L_q={\mathrm{\partial }}_t^2-{\mathrm{\Delta }}_g+q$. Write $\mathcal{H}=C_c^{\mathrm{\infty }}(\mathrm{\Sigma })$ for smooth Dirichlet data compactly supported away from the temporal endpoints. Assume that for every $h\in \mathcal{H}$ the zero-initial-data Dirichlet problem $L_{q}u=0$, $u{\mathrm{\mid }}_{\mathrm{\Sigma }}=h$, $(u(0),u_t(0))=(0,0)$ has a unique solution $u_{q,h}\in C^2([0,T]\times M)$. Define the full time-dependent Dirichlet-to-Neumann map by ${\mathrm{\Lambda }}_q^{+}:\mathcal{H}\to L^2(\mathrm{\Sigma },d{S}_{g}dt),{\mathrm{\Lambda }}_q^{+}h={\mathrm{\partial }}_{{\nu }_{out}}{u}_{q,h}{\mathrm{\mid }}_{\mathrm{\Sigma }},$ where ${\nu }_{out}$ is the outward unit normal. Also assume that for every $k\in \mathcal{H}$ the terminal-data Dirichlet problem $L_{q}v=0$, $v{\mathrm{\mid }}_{\mathrm{\Sigma }}=k$, $(v(T),v_t(T))=(0,0)$ has a unique solution $v_{q,k}\in C^2([0,T]\times M)$, and define ${\mathrm{\Lambda }}_q^{-}:\mathcal{H}\to L^2(\mathrm{\Sigma },d{S}_{g}dt),{\mathrm{\Lambda }}_q^{-}k={\mathrm{\partial }}_{{\nu }_{out}}{v}_{q,k}{\mathrm{\mid }}_{\mathrm{\Sigma }}\mathrm{.}$ The Green boundary pairing is the bilinear pairing ${\int }_{\mathrm{\Sigma }}h{\mathrm{\partial }}_{{\nu }_{out}}vd{S}_{g}dt$; the classical normal trace is in $L^2(\mathrm{\Sigma })$ for each stated $C^2$ solution class. Equality of two forward maps means equality on every $h\in \mathcal{H}$ over the whole time interval $(0,T)$, with the analogous convention for terminal maps.

Energy well-posedness for the variable-metric wave equation

Riemannian Green identity, boundary trace, and wave lifting

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The energy theorem proves existence and uniqueness for the homogeneous-boundary lifted problem and, in its proof's final step, smoothness on the closed cylinder for every admitted smooth time-compact input. Adding the smooth lift therefore gives the asserted $C^2$ solution. Its energy uniqueness implies uniqueness among classical solutions. Time reversal gives the terminal solution. The Green-and-lifting lemma bounds its classical normal trace in $L^2(\mathrm{\Sigma })$. Thus both displayed maps are defined on every element of $\mathcal{H}$.