| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
|
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory |
| open scoped ENNReal |
| open Matrix Filter Topology |
| open Set MeasureTheory Filter Topology |
| open scoped BigOperators |
| open Set MeasureTheory Filter Topology |
| open scoped Manifold |
| open Set Filter |
| open scoped Topology |
| namespace ScalarConductivity |
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|
| def completedPotential {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| (u : E → F) (D : E →L[ℝ] F) (Q : F →L[ℝ] E) (y : E) : E := |
| Q (u y) + y - Q (D y) |
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| lemma completedPotential_pairing {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| (u : E → F) (D : E →L[ℝ] F) (Q : F →L[ℝ] E) |
| (hQ : Function.RightInverse Q D) (y : E) : |
| D (completedPotential u D Q y) = u y := by |
| simp only [completedPotential, map_sub, map_add, hQ (u y), hQ (D y)] |
| abel |
|
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| lemma completedPotential_smooth {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {u : E → F} (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) |
| (D : E →L[ℝ] F) (Q : F →L[ℝ] E) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (completedPotential u D Q) := |
| ((Q.contDiff.comp hu).add contDiff_id).sub (Q.contDiff.comp D.contDiff) |
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| lemma completedPotential_derivative {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] |
| {u : E → F} {D : E →L[ℝ] F} {a : E} (hu : HasFDerivAt u D a) |
| (Q : F →L[ℝ] E) : |
| HasFDerivAt (completedPotential u D Q) (ContinuousLinearMap.id ℝ E) a := by |
| convert ((Q.hasFDerivAt.comp a hu).add (hasFDerivAt_id a)).sub |
| (Q.hasFDerivAt.comp a D.hasFDerivAt) using 1 |
| · rfl |
| · abel |
|
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| theorem exists_potential_chart |
| {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] |
| {u : E → F} (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u) (a : E) |
| (hD : Function.Surjective (fderiv ℝ u a)) {U : Set E} (hU : IsOpen U) (ha : a ∈ U) : |
| ∃ X : OpenPartialHomeomorph E E, |
| a ∈ X.source ∧ X.source ⊆ U ∧ |
| ContDiff ℝ (↑(⊤ : ℕ∞)) X ∧ |
| ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target ∧ |
| (∀ y, (fderiv ℝ u a) (X y) = u y) ∧ |
| fderiv ℝ X a = ContinuousLinearMap.id ℝ E := by |
| let D := fderiv ℝ u a |
| obtain ⟨Q, hQ⟩ := ContinuousLinearMap.HasRightInverse.of_surjective_of_finiteDimensional hD |
| let f := completedPotential u D Q |
| have hf := completedPotential_smooth hu D Q |
| have hfa : HasFDerivAt f (ContinuousLinearEquiv.refl ℝ E : E →L[ℝ] E) a := |
| completedPotential_derivative ((hu.differentiable (by simp) a).hasFDerivAt) Q |
| let X₀ := hf.contDiffAt.toOpenPartialHomeomorph f hfa (by simp) |
| let V := U ∩ {y | IsUnit (fderiv ℝ f y)} |
| have hV : IsOpen V := hU.inter (Units.isOpen.preimage (hf.continuous_fderiv (by simp))) |
| have haV : a ∈ V := ⟨ha, by change IsUnit (fderiv ℝ f a); rw [hfa.fderiv]; exact isUnit_one⟩ |
| let X := X₀.restr V |
| have hXeq : (X : E → E) = f := rfl |
| have hsource : X.source = X₀.source ∩ V := by |
| simp only [X, OpenPartialHomeomorph.restr_source, hV.interior_eq] |
| have hinv : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target := by |
| intro y hy |
| have hx : X.symm y ∈ V := (hsource ▸ X.map_target hy).2 |
| obtain ⟨v, hv⟩ := hx.2 |
| let e : E ≃L[ℝ] E := ContinuousLinearEquiv.ofUnit v |
| have he : (e : E →L[ℝ] E) = fderiv ℝ X (X.symm y) := by |
| change (v : E →L[ℝ] E) = _ |
| simpa only [hXeq] using hv |
| have hd : HasFDerivAt X (e : E →L[ℝ] E) (X.symm y) := by |
| rw [he, hXeq] |
| exact (hf.differentiable (by simp) _).hasFDerivAt |
| exact (X.contDiffAt_symm hy hd (hXeq ▸ hf.contDiffAt)).contDiffWithinAt |
| refine ⟨X, ?_, ?_, ?_, hinv, ?_, ?_⟩ |
| · rw [hsource] |
| exact ⟨hf.contDiffAt.mem_toOpenPartialHomeomorph_source hfa (by simp), haV⟩ |
| · rw [hsource] |
| exact fun _ hx => hx.2.1 |
| · rw [hXeq] |
| exact hf |
| · intro y |
| exact completedPotential_pairing u D Q hQ y |
| · exact hfa.fderiv |
|
|
| end ScalarConductivity |
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| end |
|
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| end OAI |
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