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Alessandrini identities for scalar conductivity and Schrödinger DN maps

Statement

Assume the Axiom of Choice and Countable Choice.

Let $(M, g)$ be a nonempty compact connected smooth Riemannian domain with nonempty smooth boundary. If $γ_{1}, γ_{2} \in C^{\infty} (M, R)$ are strictly positive and $u_{j}$ solves ${div}_{g} (γ_{j} \nabla_{g} u_{j}) = 0$ with smooth boundary value $f_{j}$ , then $⟨ (Λ_{γ_{1}} - Λ_{γ_{2}}) f_{1}, f_{2} ⟩_{\partial M} = \int_{M} (γ_{1} - γ_{2}) g (\nabla_{g} u_{1}, \nabla_{g} \overline{u_{2}}) d V_{g} .$ If instead $q_{1}, q_{2} \in C^{\infty} (M, R)$ have trivial Dirichlet kernels for $- Δ_{g} + q_{j}$ and $u_{j}$ solves $(- Δ_{g} + q_{j}) u_{j} = 0$ with smooth boundary value $f_{j}$ , then $⟨ (Λ_{q_{1}} - Λ_{q_{2}}) f_{1}, f_{2} ⟩_{\partial M} = \int_{M} (q_{1} - q_{2}) u_{1} \overline{u_{2}} d V_{g} .$

Statement references

Full-data conductivity and Schrödinger DN maps

Positive conductivity weak solvability

Nonresonant Schrödinger solvability

Facts & Assumptions

[A1]

The Axiom of Choice is inherited from the weak DN-map well-posedness suppliers [F2]--[F3].

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is inherited from the weak DN-map well-posedness suppliers [F2]--[F3].

Given: the compact smooth Riemannian domain, two positive conductivities or two real nonresonant potentials, and arbitrary smooth complex boundary values $f_{1}, f_{2}$ .

[F1]

Scalar conductivity and Schrödinger Dirichlet-to-Neumann maps: the conductivity DN map is the weak boundary form of ${div}_{g} (γ \nabla_{g})$ , and the Schrödinger DN map is the weak boundary form of $- Δ_{g} + q$ ; their smooth-data fluxes use $γ \partial_{ν_{o u t}}$ and $\partial_{ν_{o u t}}$ , respectively.

[F2]

Weak solvability of the positive scalar conductivity equation: positive smooth conductivity gives a unique weak solution for each $H^{1 / 2}$ boundary value.

[F3]

Smooth-manifold elliptic nonresonant solvability and regularity: a trivial Schrödinger Dirichlet kernel gives a unique weak solution for each $H^{1 / 2}$ boundary value, with smooth solutions for smooth boundary data.

[F4]

Riemannian volume density: in each chart the Riemannian volume is a positive density, so the integration-by-parts identities can be patched without assuming orientability.

Proof

Given: the data in the statement.

1.1A1A2F1F2F4givenalgebra

Write $a_{γ_{j}} (v, w) = \int_{M} γ_{j} g (\nabla v, \nabla \overset{ˉ}{w}) d V_{g}$ . By [F1], $⟨ Λ_{γ_{1}} f_{1}, f_{2} ⟩ = a_{γ_{1}} (u_{1}, u_{2})$ , since $u_{2}$ is an extension of $f_{2}$ . Let ${\tilde{u}}_{1}$ solve the $γ_{2}$ equation with boundary value $f_{1}$ , which exists by [F2]. Then $⟨ Λ_{γ_{2}} f_{1}, f_{2} ⟩ = a_{γ_{2}} ({\tilde{u}}_{1}, u_{2})$ . Since $u_{1} - {\tilde{u}}_{1} \in H_{0}^{1} (M)$ and $u_{2}$ solves the $γ_{2}$ weak equation, Hermitian symmetry gives $a_{γ_{2}} (u_{1} - {\tilde{u}}_{1}, u_{2}) = \overline{a_{γ_{2}} (u_{2}, u_{1} - {\tilde{u}}_{1})} = 0$ . Hence $a_{γ_{2}} ({\tilde{u}}_{1}, u_{2}) = a_{γ_{2}} (u_{1}, u_{2})$ . Subtracting the two equalities gives the conductivity identity.

1.2A1A2F1F3F4givenalgebra

Write $a_{q_{j}} (v, w) = \int_{M} (g (\nabla v, \nabla \overset{ˉ}{w}) + q_{j} v \overset{ˉ}{w}) d V_{g}$ . By [F1], $⟨ Λ_{q_{1}} f_{1}, f_{2} ⟩ = a_{q_{1}} (u_{1}, u_{2})$ because $u_{2}$ extends $f_{2}$ . Let ${\tilde{u}}_{1}$ be the $q_{2}$ solution with boundary value $f_{1}$ , which exists by [F3]. Then $⟨ Λ_{q_{2}} f_{1}, f_{2} ⟩ = a_{q_{2}} ({\tilde{u}}_{1}, u_{2})$ . Since $u_{1} - {\tilde{u}}_{1} \in H_{0}^{1} (M)$ and $u_{2}$ solves the $q_{2}$ weak equation, real-valuedness of $q_{2}$ gives $a_{q_{2}} (u_{1} - {\tilde{u}}_{1}, u_{2}) = \overline{a_{q_{2}} (u_{2}, u_{1} - {\tilde{u}}_{1})} = 0$ . Thus $⟨ Λ_{q_{2}} f_{1}, f_{2} ⟩ = a_{q_{2}} (u_{1}, u_{2})$ .

2.1step 1.2algebra∎

Subtract the identities in step 1.2. The gradient terms cancel and the remaining volume term is $\int_{M} (q_{1} - q_{2}) u_{1} \overline{u_{2}} d V_{g}$ , which proves the Schrödinger identity.

Depends on

Used by

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Sources

Mikko Salo, Applications of microlocal analysis to inverse problems (2019) (standard reference, not scraped)
Dos Santos Ferreira, Kurylev, Lassas, and Salo, The Calderon problem in transversally anisotropic geometries (2016) (standard reference, not scraped)