| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.LocalPiola |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
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| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma localPiolaSource_support (X : OpenPartialHomeomorph E E) (r : E → ℝ) : |
| Function.support (localPiolaSource X r)⊆X '' (Function.support r∩X.source) := by |
| intro y hy |
| by_cases hyt : y∈X.target |
| · refine ⟨X.symm y,⟨?_,X.map_target hyt⟩,X.right_inv hyt⟩ |
| intro hz |
| exact hy (by simp [localPiolaSource,hyt,hz]) |
| · exact False.elim (hy (by simp [localPiolaSource,hyt])) |
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| omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in |
| lemma localPiolaSource_compact (X : OpenPartialHomeomorph E E) |
| (r : E → ℝ) (hc : HasCompactSupport r) (hs : tsupport r⊆X.source) : |
| HasCompactSupport (localPiolaSource X r) ∧ tsupport (localPiolaSource X r)⊆X.target := by |
| have hci : IsCompact (X '' tsupport r) := hc.image_of_continuousOn (X.continuousOn.mono hs) |
| have ht : tsupport (localPiolaSource X r)⊆X '' tsupport r := |
| closure_minimal ((localPiolaSource_support X r).trans |
| (image_mono (fun _ hx => subset_closure hx.1))) hci.isClosed |
| exact ⟨hci.of_isClosed_subset isClosed_closure ht,ht.trans (by |
| rintro y ⟨x,hx,rfl⟩; exact X.map_source (hs hx))⟩ |
| |
| omit [MeasurableSpace E] [BorelSpace E] in |
| lemma localPiolaSource_smooth (X : OpenPartialHomeomorph E E) |
| (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| (r : E → ℝ) (hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r) |
| (hc : HasCompactSupport r) (hs : tsupport r⊆X.source) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (localPiolaSource X r) := by |
| apply contDiff_of_contDiffOn_tsupport X.open_target _ (localPiolaSource_compact X r hc hs).2 |
| apply X.open_target.contDiffOn_iff.mpr |
| intro y hy |
| have hxi := hXi.contDiffAt (X.open_target.mem_nhds hy) |
| have hx := hX.contDiffAt (X.open_source.mem_nhds (X.map_target hy)) |
| have hd : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun z => fderiv ℝ X (X.symm z)) y := |
| (hx.fderiv_right (by simp)).comp y hxi |
| have hd0 := local_fderiv_det_ne_zero X (hX.differentiableOn (by simp)) |
| (hXi.differentiableOn (by simp)) (X.map_target hy) |
| have hdet := ((contDiff_clm_det (E:=E) (↑(⊤ : ℕ∞))).contDiffAt.comp y hd).abs hd0 |
| have hformula := (hdet.inv (abs_ne_zero.mpr hd0)).mul (hr.contDiffAt.comp y hxi) |
| apply hformula.congr_of_eventuallyEq |
| filter_upwards [X.open_target.mem_nhds hy] with z hz |
| simp only [localPiolaSource,ite_eq_left hz] |
| rfl |
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| lemma localPiolaSource_coordinate_pairing (μ : Measure E) [μ.IsAddHaarMeasure] |
| (X : OpenPartialHomeomorph E E) (hX : DifferentiableOn ℝ X X.source) |
| (hXi : DifferentiableOn ℝ X.symm X.target) |
| (r : E → ℝ) (hs : Function.support r⊆X.source) (φ : E → ℝ) : |
| (∫ y,φ (X.symm y)*localPiolaSource X r y ∂μ)=(∫ x,φ x*r x ∂μ) := by |
| rw [localPiolaSource_pairing μ X hX |
| (fun _ hx => local_fderiv_det_ne_zero X hX hXi hx) r hs (fun y => φ (X.symm y))] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| by_cases hx : x∈Function.support r |
| · simp only [X.left_inv (hs hx)] |
| · rw [Function.notMem_support.mp hx,mul_zero,mul_zero] |
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| end ScalarConductivity |
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| end |
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| end OAI |
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