OAI/Analysis/AffineBernstein/ActualFlatWeakEuler.lean

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1 declarations, 0 contain sorry, 0 declared as axioms

1. theorem OAI.AffineBernstein.affineMaximal_flat_tube_weak_euler line 23

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   The original affine-maximal PDE gives the literal flat tube weak Euler identity. Both Hessian blocks, the support variation and the local stationary chart are produced from the actual affine epigraph.

   theorem affineMaximal_flat_tube_weak_euler {n : ℕ} {Ω : Set (Space n)}
       (hΩ : IsOpen Ω) (hcv : Convex ℝ Ω) {u : Space n → ℝ}
       (hu : ContDiffOn ℝ ∞ u Ω) (hp : ∀ x ∈ Ω, (hessian u x).PosDef)
       (hm : AffineMaximalOn Ω u)
       (a : Space n × ℝ) (L : (S × E) ≃L[ℝ] (Space n × ℝ))
       {B : Set S} (hB : IsOpen B)
       (hK : ∀ s ∈ B, IsCompact {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       (hzero : ∀ s ∈ B, (0 : E) ∈ interior {y | (s,y) ∈ affineEpigraphPullback Ω u a L})
       (bS : Module.Basis ι ℝ S) (bE : OrthonormalBasis (κ ⊕ Unit) ℝ E)
       (q₀ : S × E) (hd : inner ℝ q₀.2 (bE (Sum.inr ())) = 1)
       (J : Space n →L[ℝ] (S × E)) (hi : Function.Injective J)
       (hJ : ∀ v, inner ℝ (J v).2 (bE (Sum.inr ())) = 0)
   …

   Mentions: OAI.AffineBernstein.Space, OAI.AffineBernstein.hessian, OAI.AffineBernstein.AffineMaximalOn, OAI.AffineBernstein.affineEpigraphPullback