CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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A shrinking inclusion rules out uniform Lipschitz stability on the BV class

Statement refuted

Assume the Axiom of Choice and Countable Choice. Let $D = B (x_{0}, a) \subset (0, 1)^{2}$ with $x_{0} = (1 / 2, 1 / 2)$ and $a = 1 / 4$ , let $D^{'} = B (x_{0}, a / 2)$ , and set $m = 1 / 2$ , $M = 3 / 2$ , and $R = 1$ . There exists a finite constant $L$ such that $∥ γ_{1} - γ_{2} ∥_{L^{2} (D)} \leq L ∥ {\dot{Λ}}_{γ_{1}} - {\dot{Λ}}_{γ_{2}} ∥_{L (D (D) / C 1, (D (D) / C 1)^{*})}$ for every $γ_{1}, γ_{2} \in X_{2} (1)$ .

Statement references

The fixed planar BV class used in the refuted estimate.

The exact DN spectrum and quotient operator bound for a concentric inclusion.

Facts & Assumptions

Given: The Axiom of Choice and Countable Choice; $D = B (x_{0}, a)$ , $a = 1 / 4$ , $D^{'} = B (x_{0}, a / 2)$ , $m = 1 / 2$ , $M = 3 / 2$ , $R = 1$ , $γ_{0} \equiv 1$ , and $γ_{j} = 1 + \frac{1}{2} χ_{B (x_{0}, r_{j})}$ with $r_{j} = a / 2^{j + 2}$ .

[F1]

The class $X_{2} (1)$ uses the bounds $1 / 2 \leq γ \leq 3 / 2$ , background value $1$ off $D^{'}$ , and $\max {∥ γ ∥_{L^{1} (D)}, T V (γ; D)} \leq 1$ . (Calderón admissible conductivity classes)

[F2]

For each $0 < r \leq a / 2$ and $β > 0$ , the disk inclusion $1 + β χ_{B (x_{0}, r)}$ has DN eigenvalue difference from the background disk equal to $\frac{2 n}{a} \frac{η (r / a)^{2 n}}{1 - η (r / a)^{2 n}}$ on mode $n \geq 1$ , and the induced quotient-space operator norm is at most $\frac{2 η}{1 - η} (r / a)^{2}$ , where $η = β / (2 + β)$ . (DN spectrum of a concentric radial conductivity inclusion)

[F3]

The weak DN maps are bounded operators on the trace quotient and are uniquely defined for these uniformly elliptic conductivities. (Weak Dirichlet-to-Neumann map for bounded conductivities, Well-posedness of the weak conductivity Dirichlet-to-Neumann map)

[F4]

For every disk $B (x_{0}, r)$ , its indicator has total variation $2 π r$ in the plane. (Scaling the perimeter of a disk inclusion)

[A1]

The Axiom of Choice is assumed at the weak DN interface, and Countable Choice at the disk-perimeter scaling and Fourier trace-quotient interfaces. (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Counterexample

technique · direct

Given: The data in the Statement refuted.

1.1A1F1F4givenalgebra

For every $j \geq 1$ , $r_{j} \leq a / 8 < a / 2$ , so the inclusion lies in $D^{'}$ . Its values are $1$ and $3 / 2$ , and its $L^{1} (D)$ norm is $π a^{2} + \frac{1}{2} π r_{j}^{2} \leq π / 16 + π / 2048 < 1$ . By [F4], $T V (γ_{j}; D) = \frac{1}{2} (2 π r_{j}) = π r_{j} \leq π a / 8 < 1$ . Also $γ_{0}$ equals the background and has $L^{1} (D) = π / 16 < 1$ and zero variation. Thus [F1] gives $γ_{0}, γ_{j} \in X_{2} (1)$ .

1.2A1F2F3givenalgebra

The squared $L^{2} (D)$ difference is $\int_{B (x_{0}, r_{j})} (1 / 2)^{2} d x = π r_{j}^{2} / 4$ , hence its norm is $\frac{1}{2} \sqrt{π} r_{j}$ . Apply [F2] with $β = 1 / 2$ , so $η = (1 / 2) / (5 / 2) = 1 / 5$ and $2 η / (1 - η) = 1 / 2$ ; this gives the stated DN bound. The well-posedness and definition facts [F3] ensure the operator comparison is in the stated quotient-space norm.

2.1F3step 1.1step 1.2algebra∎

If the asserted finite constant $K$ existed, step 1.1 puts both coefficients in its class, while step 1.2 computes the data and DN differences; hence the estimate applied to $(γ_{j}, γ_{0})$ would give $\frac{1}{2} \sqrt{π} r_{j} \leq \frac{K}{2} (r_{j} / a)^{2}$ . Dividing by $r_{j} > 0$ yields $\sqrt{π} \leq K r_{j} / a^{2}$ for every $j$ . Since $r_{j} \to 0$ , the right-hand side tends to zero, contradicting $\sqrt{π} > 0$ . Therefore the refuted statement is false.

Depends on

Used by

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Dependency tree · two levels

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Sources

Mikko Salo, Calderón problem (University of Helsinki lecture notes, Spring 2008) (standard reference, not scraped)