OAI.ElasticityRegularity.exists_localized_derivative: intermediate lemma from OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem

Proved

OAI.ElasticityRegularity.exists_localized_derivative

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Elasticity.global_uniqueness, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, result family 372, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.ElasticityRegularity.exists_localized_derivative of lean/OAI/MathematicalPhysics/Elasticity/CGO.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.ElasticityRegularity, with open MeasureTheory TemperedDistribution, open scoped SchwartzMap LineDeriv Laplacian BigOperators):

theorem exists_localized_derivative (k : ℤ) {a η : E → ℂ}
    (ha : ContDiff ℝ (⊤ : ℕ∞) a) (hca : HasCompactSupport a)
    (hη : ContDiff ℝ (⊤ : ℕ∞) η) (hcη : HasCompactSupport η)
    (h1 : Set.EqOn η 1 (tsupport a)) (v : E) :
    ∃ L : Lp ℂ 2 (volume : Measure E) →L[ℂ] Lp ℂ 2 (volume : Measure E),
      ∀ u f, sobolevInjection (k : ℝ) f=smulLeftCLM ℂ η u →
        sobolevInjection ((k : ℝ)-1) (L f)=smulLeftCLM ℂ a (∂_{v} u)

Definitions. The statement is written with OAI.ElasticityRegularity.sobolevInjection (definitions bundle OAIQ6El002 of this split).

Used by. The proof of OAI.ElasticityDistribution.exists_lower_realization of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ6El002

Formal statement

section

attribute [-instance] _root_.OAI.ElasticityGeneralSlicing.cylinder_temperate
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

namespace ElasticityRegularity
open MeasureTheory TemperedDistribution
open scoped SchwartzMap LineDeriv Laplacian BigOperators
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

theorem exists_localized_derivative (k : ℤ) {a η : E → ℂ}
    (ha : ContDiff ℝ (⊤ : ℕ∞) a) (hca : HasCompactSupport a)
    (hη : ContDiff ℝ (⊤ : ℕ∞) η) (hcη : HasCompactSupport η)
    (h1 : Set.EqOn η 1 (tsupport a)) (v : E) :
    ∃ L : Lp ℂ 2 (volume : Measure E) →L[ℂ] Lp ℂ 2 (volume : Measure E),
      ∀ u f, sobolevInjection (k : ℝ) f=smulLeftCLM ℂ η u →
        sobolevInjection ((k : ℝ)-1) (L f)=smulLeftCLM ℂ a (∂_{v} u) := by
  sorry

end ElasticityRegularity

end

end OAI
end

Source

OpenAI, Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 372; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/MathematicalPhysics/Elasticity/CGO.lean (declaration OAI.ElasticityRegularity.exists_localized_derivative)

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