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Logarithmic L2 stability for Calderón conductivity classes

Statement

Assume the Axiom of Choice and Countable Choice. For each fixed integer $d \geq 2$ , let $Ω \subset (0, 1)^{d}$ be a nonempty bounded connected smooth domain, let $Ω^{'} ⋐ Ω$ , fix $m > 0$ , $M \geq \max {1, m}$ , $R > 0$ and $0 < α < 1$ , and let $γ, γ_{0} \in X_{d} (R)$ . Set $δ : = ∥ {\dot{Λ}}_{γ} - {\dot{Λ}}_{γ_{0}} ∥_{L (D (Ω) / C 1, (D (Ω) / C 1)^{*})} .$ There exist constants $C > 0$ , $ρ > 0$ and $t_{0} \in (0, 1)$ , depending only on $d, α, Ω, Ω^{'}, m, M, R$ , such that $∥ γ - γ_{0} ∥_{L^{2} (Ω)} \leq C ω (δ), ω (0) : = 0, ω (t) : = {[\log ⁣ (\frac{1}{\min {t, t_{0}}})]}^{- ρ} (t > 0) .$ Here $X_{d} (R)$ is the BV class when $d = 2$ and the bounded $C^{1, α}$ class when $d \geq 3$ , and ${\dot{Λ}}_{γ}$ is the induced weak conductivity Dirichlet-to-Neumann map on the trace quotient by constants.

Statement references

The explicit conductivity class and dimension split.

The weak DN map, trace quotient and operator norm convention.

The planar BV stability branch.

The higher-dimensional C1alpha stability branch.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; a fixed $d \geq 2$ and the domain and parameters in the statement; $γ, γ_{0} \in X_{d} (R)$ ; and $δ$ as defined there.

[F1]

$X_{2} (R)$ is the real conductivity class with $m \leq γ \leq M$ , background value $1$ outside $Ω^{'}$ , and the stated BV norm bound; for $d \geq 3$ , $X_{d} (R)$ has the stated $C^{1, α}$ bound. (Calderón admissible conductivity classes)

[F2]

$Λ_{γ}$ is the weak energy-form DN map, is well-defined and bounded on the trace quotient, kills constants and has range functionals annihilating constants; ${\dot{Λ}}_{γ}$ is its induced bounded map on the constant quotient. (Weak Dirichlet-to-Neumann map for bounded conductivities, Well-posedness of the weak conductivity Dirichlet-to-Neumann map, Boundary trace quotient and chart norm equivalence)

[L1]

In dimension $2$ , every pair in $X_{2} (R)$ satisfies an $L^{2}$ logarithmic stability bound with positive exponent and a positive threshold, including the zero-data case. (Planar BV logarithmic Calderón stability)

[L2]

In dimensions $d \geq 3$ , every pair in $X_{d} (R)$ satisfies an $L^{2}$ logarithmic stability bound with positive exponent and a positive threshold, including the zero-data case. (Higher-dimensional C1alpha logarithmic Calderón stability)

Proof

Given: The data in the statement and Facts & Assumptions [F1]–[L2].

1.1F1F2L1L2givencases

If $d = 2$ , apply [L1] and write its constants as $C_{2}, ρ_{2}, t_{2}$ . If $d \geq 3$ , apply [L2] and write its constants as $C_{d}, ρ_{d}, t_{d}$ . In either case, for the fixed dimension there are $C_{*} > 0$ , $ρ_{*} > 0$ , and $t_{*} > 0$ such that $∥ γ - γ_{0} ∥_{2} \leq C_{*} [\log (1 / \min {δ, t_{*}})]^{- ρ_{*}}$ when $δ > 0$ , and equality of the DN maps implies $γ = γ_{0}$ .

2.1step 1.1givenalgebra

Set $ρ : = ρ_{*}$ and $t_{0} : = \min {t_{*}, 1 / 2}$ . If $0 < δ < t_{0}$ , then $\min {δ, t_{0}} = δ$ and $\min {δ, t_{*}} = δ$ , so step 1.1 is exactly the asserted estimate with constant $C_{*}$ . If $δ = 0$ , step 1.1 gives $γ = γ_{0}$ , so the estimate holds with $ω (0) = 0$ .

3.1F1step 2.1givenalgebra∎

If $δ \geq t_{0}$ , [F1] gives $∥ γ - γ_{0} ∥_{2} \leq 2 M ∣ Ω ∣^{1 / 2}$ , while $ω (δ) = [\log (1 / t_{0})]^{- ρ} > 0$ . Choose $C \geq \max {C_{*}, 2 M ∣ Ω ∣^{1 / 2} [\log (1 / t_{0})]^{ρ}}$ . Then the desired inequality follows for $δ \geq t_{0}$ by this uniform coefficient bound and for $δ < t_{0}$ by step 2.1. The constants have only the listed class and geometry dependence.

Depends on

Used by

Dependency tree · two levels

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Sources

Maarten V. de Hoop, Nikola B. Kovachki, Matti Lassas and Nicholas H. Nelsen, Extension and neural operator approximation of the electrical impedance tomography inverse map (standard reference, not scraped)
Daniel Faraco and Martí Prats, Characterization for stability in planar conductivities (standard reference, not scraped)
Pedro Caro, Andoni García and Juan Manuel Reyes, Stability of the Calderón problem for less regular conductivities (standard reference, not scraped)