| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.LocalPiolaFlux |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| 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 |
| namespace ScalarConductivity |
|
|
| lemma contDiff_clm_det {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] (n : WithTop ℕ∞) : |
| ContDiff ℝ n (fun L : E →L[ℝ] E => L.det) := by |
| classical |
| let b := Module.finBasis ℝ E |
| simp_rw [ContinuousLinearMap.det, ← LinearMap.det_toMatrix b, Matrix.det_apply', |
| LinearMap.toMatrix_apply] |
| apply ContDiff.sum |
| intro σ _ |
| apply contDiff_const.mul |
| apply contDiff_prod |
| intro i _ |
| change ContDiff ℝ n (fun L : E →L[ℝ] E => (b.coord (σ i)).toContinuousLinearMap (L (b i))) |
| exact (b.coord (σ i)).toContinuousLinearMap.contDiff.comp ((ContinuousLinearMap.apply ℝ E (b i)).contDiff) |
|
|
| lemma contDiff_of_contDiffOn_tsupport {E V : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup V] [NormedSpace ℝ V] {n : WithTop ℕ∞} {f : E → V} {s : Set E} |
| (hs : IsOpen s) (hf : ContDiffOn ℝ n f s) (hfS : tsupport f ⊆ s) : ContDiff ℝ n f := by |
| rw [contDiff_iff_contDiffAt] |
| intro x |
| by_cases hx : x ∈ tsupport f |
| · exact hf.contDiffAt (hs.mem_nhds (hfS hx)) |
| · exact contDiffAt_const.congr_of_eventuallyEq (notMem_tsupport_iff_eventuallyEq.mp hx) |
|
|
| lemma localPiolaFlux_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| (X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| (F : E → E) (hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F) (hc : HasCompactSupport F) |
| (hs : tsupport F ⊆ X.source) : ContDiff ℝ (↑(⊤ : ℕ∞)) (localPiolaFlux X F) := by |
| apply contDiff_of_contDiffOn_tsupport X.open_target _ (localPiolaFlux_tsupport X F hc hs) |
| 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 : (fderiv ℝ X (X.symm y)).det ≠ 0 := |
| 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)).smul |
| (hd.clm_apply (hF.contDiffAt.comp y hxi)) |
| apply hformula.congr_of_eventuallyEq |
| filter_upwards [X.open_target.mem_nhds hy] with z hz |
| simp [localPiolaFlux, hz] |
|
|
| lemma exists_smooth_extension_near_compact {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] {U K : Set E} (hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) |
| (φ : E → ℝ) (hφ : ContDiffOn ℝ (↑(⊤ : ℕ∞)) φ U) : |
| ∃ g : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) g ∧ ∀ x ∈ K, g =ᶠ[𝓝 x] φ := by |
| obtain ⟨χ, hχ₀, hχ₁, _⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed |
| (𝓘(ℝ, E)) hU.isClosed_compl hK.isClosed |
| (disjoint_left.mpr (fun x hx hxK => hx (hKU hxK))) (n := ⊤) |
| have hχ : ContDiff ℝ (↑(⊤ : ℕ∞)) (χ : E → ℝ) := contMDiff_iff_contDiff.mp χ.contMDiff |
| refine ⟨fun x => χ x * φ x, ?_, ?_⟩ |
| · rw [contDiff_iff_contDiffAt] |
| intro x |
| by_cases hx : x ∈ U |
| · exact hχ.contDiffAt.mul (hφ.contDiffAt (hU.mem_nhds hx)) |
| · have hzero : ∀ᶠ z in 𝓝 x, χ z = 0 := hχ₀.filter_mono (nhds_le_nhdsSet hx) |
| apply (contDiffAt_const : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun _ : E => (0 : ℝ)) x).congr_of_eventuallyEq |
| filter_upwards [hzero] with z hz |
| simp [hz] |
| · intro x hx |
| have hone : ∀ᶠ z in 𝓝 x, χ z = 1 := hχ₁.filter_mono (nhds_le_nhdsSet hx) |
| filter_upwards [hone] with z hz |
| simp [hz] |
|
|
| lemma compactFlux_local_test |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] |
| (μ : Measure E) {U : Set E} (hU : IsOpen U) |
| (F : E → E) (hc : HasCompactSupport F) (hs : tsupport F ⊆ U) |
| (hF : ∀ ψ : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x, fderiv ℝ ψ x (F x) ∂μ) = 0) |
| (φ : E → ℝ) (hφ : ContDiffOn ℝ (↑(⊤ : ℕ∞)) φ U) : |
| (∫ x, fderiv ℝ φ x (F x) ∂μ) = 0 := by |
| obtain ⟨g, hg, hge⟩ := exists_smooth_extension_near_compact hU hc hs φ hφ |
| rw [← hF g hg] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| by_cases hx : x ∈ tsupport F |
| · rw [(hge x hx).fderiv_eq] |
| · rw [image_eq_zero_of_notMem_tsupport hx, map_zero, map_zero] |
|
|
| lemma localPiolaFlux_zero_pairing |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
| (X : OpenPartialHomeomorph E E) (hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source) |
| (hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target) |
| (F : E → E) (hc : HasCompactSupport F) (hs : tsupport F ⊆ X.source) |
| (hF : ∀ ψ : E → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → |
| (∫ x, fderiv ℝ ψ x (F x) ∂μ) = 0) |
| (ψ : E → ℝ) (hψ : ContDiff ℝ (↑(⊤ : ℕ∞)) ψ) : |
| (∫ y, fderiv ℝ ψ y (localPiolaFlux X F y) ∂μ) = 0 := by |
| rw [localPiolaFlux_pairing μ X (hX.differentiableOn (by simp)) |
| (fun _ hx => local_fderiv_det_ne_zero X (hX.differentiableOn (by simp)) |
| (hXi.differentiableOn (by simp)) hx) |
| F (subset_closure.trans hs) ψ (hψ.differentiable (by simp))] |
| exact compactFlux_local_test μ X.open_source F hc hs hF _ (hψ.comp_contDiffOn hX) |
|
|
| end ScalarConductivity |
|
|
| end |
|
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| end OAI |
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