| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.CentralJoinCover |
| import OAI.Analysis.Conductivity.Geometry.CutSlopeGradient |
| import OAI.Analysis.Conductivity.Variational.WholeBallJetFormula |
| import OAI.Analysis.Conductivity.Branching.ChildTransportFormula |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| def collarSlopeJet (χ : (Fin 3 → ℝ) → ℝ) (a b κ : ℝ) (y : Fin 3 → ℝ) : JetFiber := |
| WithLp.toLp 2 (Fin.cases |
| (κ*(χ y*max 0 (a*(sourceCollarTime y-b)))) |
| (fun i => κ*(fderiv ℝ χ y (Pi.single i 1)*max 0 (a*(sourceCollarTime y-b))+ |
| χ y*(if 0<a*(sourceCollarTime y-b) then |
| fderiv ℝ (fun z => a*(sourceCollarTime z-b)) y (Pi.single i 1) else 0)))) |
|
|
| lemma collarSlopeJet_eq_zero {χ : (Fin 3 → ℝ) → ℝ} {a b κ : ℝ} {y : Fin 3 → ℝ} |
| (ht : a*(sourceCollarTime y-b)≤0) : collarSlopeJet χ a b κ y=0 := by |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i <;> |
| simp [collarSlopeJet,max_eq_left ht,not_lt.mpr ht] |
|
|
| lemma collarSlopeJet_zero_not_tsupport {χ : (Fin 3 → ℝ) → ℝ} {a b κ : ℝ} {y : Fin 3 → ℝ} |
| (hy : y∉tsupport χ) : collarSlopeJet χ a b κ y=0 := by |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i <;> |
| simp [collarSlopeJet,image_eq_zero_of_notMem_tsupport hy,fderiv_of_notMem_tsupport ℝ hy] |
|
|
| lemma compact_collar_slope_jet {χ : (Fin 3 → ℝ) → ℝ} |
| (hχ : ContDiff ℝ (↑(⊤:ℕ∞)) χ) (hc : HasCompactSupport χ) (hχb : ∀ x,|χ x|≤1) |
| {a : ℝ} (ha : a≠0) {b : ℝ} (hb : b∈Icc (-(1:ℝ)/100) (1/100)) |
| (κ l r : ℝ) (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) |
| (hχs : tsupport χ⊆sourceClosedCollarBand l r) : |
| ∃ w : H1,w∈H10 ∧ w.val=ᵐ[ballMeasure] |
| (fun x => collarSlopeJet χ a b κ (WithLp.ofLp x)) := by |
| obtain ⟨w,hw,hwe⟩ := compact_collar_slope_value_gradient hχ hc hχb ha hb κ l r hl hr hχs |
| refine ⟨w,hw,?_⟩ |
| filter_upwards [hwe] with x hx |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact hx.1 |
| · exact hx.2 j |
|
|
| lemma ballWholePiJetCLM_slope_ae {z : JetSpace} {χ : (Fin 3 → ℝ) → ℝ} {a b κ : ℝ} |
| (hs : ∀ y∈tsupport χ,WithLp.toLp 2 y∈ball) |
| (he : z=ᵐ[ballMeasure] (fun x => collarSlopeJet χ a b κ (WithLp.ofLp x))) : |
| ballWholePiJetCLM z=ᵐ[volume] collarSlopeJet χ a b κ := by |
| apply (ballWholePiJetCLM_ae_of_ae z (collarSlopeJet χ a b κ) he).trans |
| exact Filter.Eventually.of_forall (fun y => by |
| dsimp only |
| split_ifs with hb |
| · rfl |
| · exact (collarSlopeJet_zero_not_tsupport (fun hy => hb (hs y hy))).symm) |
|
|
| def physicalChildSlopeJet (k : Fin 2) (a κ : ℝ) (y : Fin 3 → ℝ) : JetFiber := |
| let z := collarSlopeJet (centralJoinChildCutoff k) a (-centralThickness) κ |
| ((sourceChildHomeomorph (actualChildSign k)).symm y) |
| WithLp.toLp 2 (Fin.cases (z 0) (fun j => sourceScale⁻¹*z (childAxis j).succ)) |
|
|
| lemma exists_parentSlopeExtension {a : ℝ} (ha : a<0) (κ : ℝ) : |
| ∃ w : H1,w∈H10 ∧ w.val=ᵐ[ballMeasure] (fun x => |
| collarSlopeJet (centralJoinPartition 1) a centralThickness κ (WithLp.ofLp x)) := |
| compact_collar_slope_jet (centralJoinPartition_smooth 1) |
| centralJoinPartition_parent_compact (centralJoinPartition_bound 1) ha.ne |
| (by norm_num [centralThickness]) κ 0 (2*centralThickness) |
| (by norm_num) (by norm_num [centralThickness]) centralJoinPartition_parent_support |
|
|
| lemma exists_childSlopeExtension {a : ℝ} (ha : 0<a) (κ : ℝ) (k : Fin 2) : |
| ∃ w : H1,w∈H10 ∧ w.val=ᵐ[ballMeasure] |
| (fun x => physicalChildSlopeJet k a κ (WithLp.ofLp x)) := by |
| obtain ⟨w,hw,hwe⟩ := compact_collar_slope_jet (centralJoinChildCutoff_smooth k) |
| (centralJoinChildCutoff_compact k) (centralJoinChildCutoff_bound k) ha.ne' |
| (show -centralThickness∈Icc (-(1:ℝ)/100) (1/100) by norm_num [centralThickness]) |
| κ (-2*centralThickness) 0 (by norm_num [centralThickness]) (by norm_num) |
| (centralJoinChildCutoff_support k) |
| refine ⟨childH10Transport k ⟨w,hw⟩,childH10Transport_mem_H10 k _,?_⟩ |
| have hB : ∀ y∈tsupport (centralJoinChildCutoff k),WithLp.toLp 2 y∈ball := by |
| intro y hy |
| have hb := centralJoinChildCutoff_support k hy |
| exact sourceBand_mem_ball ⟨le_trans (by norm_num [centralThickness]) hb.1, |
| le_trans hb.2 (by norm_num)⟩ |
| exact childTransportJetCLM_ae_of_ae k w.val _ (ballWholePiJetCLM_slope_ae hB hwe) |
| |
| lemma parentSlopeJet_zero_central {a : ℝ} (ha : a<0) (κ : ℝ) {y : Fin 3 → ℝ} |
| (hy : y∈centralPhysical) : collarSlopeJet (centralJoinPartition 1) a centralThickness κ y=0 := |
| collarSlopeJet_eq_zero (mul_nonpos_of_nonpos_of_nonneg ha.le |
| (sub_nonneg.mpr ((centralPhysical_time_iff y).mp hy).1)) |
| |
| lemma physicalChildSlopeJet_zero_central {a : ℝ} (ha : 0<a) (κ : ℝ) (k : Fin 2) |
| {y : Fin 3 → ℝ} (hy : y∈centralPhysical) : physicalChildSlopeJet k a κ y=0 := by |
| have hz := collarSlopeJet_eq_zero (χ:=centralJoinChildCutoff k) (κ:=κ) |
| (mul_nonpos_of_nonneg_of_nonpos ha.le |
| (sub_nonpos.mpr (((centralPhysical_time_iff y).mp hy).2 k))) |
| dsimp only [physicalChildSlopeJet] |
| rw [hz] |
| ext i |
| refine Fin.cases ?_ (fun j => ?_) i <;> simp |
| |
| def centralSlopeJet (a κ : Fin 3 → ℝ) (y : Fin 3 → ℝ) : JetFiber := |
| collarSlopeJet (centralJoinPartition 1) (a 0) centralThickness (κ 0) y+ |
| physicalChildSlopeJet 0 (a 1) (κ 1) y+physicalChildSlopeJet 1 (a 2) (κ 2) y |
| |
| theorem centralSlopeExtension_exists (a κ : Fin 3 → ℝ) |
| (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ) : |
| ∃ w : H1,w∈H10 ∧ w.val=ᵐ[ballMeasure] |
| (fun x => centralSlopeJet a κ (WithLp.ofLp x)) ∧ |
| (∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical → w.val x=0) := by |
| obtain ⟨wp,hp,hep⟩ := exists_parentSlopeExtension ha₀ (κ 0) |
| obtain ⟨w₀,h₀,he₀⟩ := exists_childSlopeExtension (ha 0) (κ 1) 0 |
| obtain ⟨w₁,h₁,he₁⟩ := exists_childSlopeExtension (ha 1) (κ 2) 1 |
| have he : (wp+w₀+w₁).val=ᵐ[ballMeasure] (fun x => centralSlopeJet a κ (WithLp.ofLp x)) := by |
| filter_upwards [Lp.coeFn_add wp.val w₀.val,Lp.coeFn_add (wp.val+w₀.val) w₁.val,hep,he₀,he₁] |
| with x h₀ h₁ hp hc₀ hc₁ |
| change (wp.val+w₀.val+w₁.val) x=_ |
| simp only [Pi.add_apply] at h₀ h₁ |
| rw [h₁,h₀,hp,hc₀,hc₁] |
| rfl |
| refine ⟨wp+w₀+w₁,H10.add_mem (H10.add_mem hp h₀) h₁,he,?_⟩ |
| filter_upwards [he] with x hx |
| intro hy |
| rw [hx] |
| dsimp only [centralSlopeJet] |
| have hzero₀ : physicalChildSlopeJet 0 (a 1) (κ 1) (WithLp.ofLp x)=0 := by |
| exact physicalChildSlopeJet_zero_central (show 0<a 1 from ha 0) _ 0 hy |
| have hzero₁ : physicalChildSlopeJet 1 (a 2) (κ 2) (WithLp.ofLp x)=0 := by |
| exact physicalChildSlopeJet_zero_central (show 0<a 2 from ha 1) _ 1 hy |
| rw [parentSlopeJet_zero_central ha₀ _ hy,hzero₀,hzero₁,add_zero,add_zero] |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
