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159357f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.SobolevGaugeTransform
import OAI.Analysis.Conductivity.Walls.PhysicalTerminalCutoff
import OAI.Analysis.Conductivity.Sources.PhysicalPatching
import OAI.Analysis.Conductivity.Sources.PhysicalTrace
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology MeasureTheory Matrix UnitAddTorus
attribute [local instance] Classical.propDecidable
local instance physicalRootMeasureSpaceUnitAddCircle : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance physicalRootIsProbabilityMeasure : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
namespace PhysicalFiniteEndingData
variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s)
theorem source_patch_trace_exists
(hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2)
{η : ℝ} (hη : 0<η) (hηc : 2*η<centralThickness)
(hT : ∀ i : Fin 3,∀ t : ℝ,|t|≤η → (z.ending i).R<z.compression i*(t-terminalBase i))
(w q : Fin 2 → H1) (he : ∀ j i,H1JetOn (w j) (physicalEndRegion i) (z.correctedEndJet j i))
(Z : VoltageJetSpace volume physicalOpenBlock)
(hZ : Z∈zeroVoltageJets volume physicalOpenBlock)
(hq : ∀ j,(q j-w j).val=voltageOriginalJetCLM physicalOpenBlock_open.measurableSet j Z)
(j : Fin 2) :
∃ W : H1,W∈H10 ∧ (∀ᵐ x∂ballMeasure,weakValue W x=
if WithLp.ofLp x∈physicalOpenBlock then weakValue (q j) x-z.terminalValue 0 j
else z.terminalProfile η j (WithLp.ofLp x)) ∧
∀ i : Fin 3,physicalRealTraceCLM i W=ᵐ[volume] fun _ => z.terminalValue i j-z.terminalValue 0 j := by
obtain ⟨f,hf,hfv⟩ := z.terminalProfile_H10 hη hηc j
obtain ⟨χ,hχ,hχc,hχs,hχ1⟩ := terminal_cutoff_exists hη
let E : R3 ≃L[ℝ] Coord3 := PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)
have hχd : ContDiff ℝ (↑(⊤:ℕ∞)) (χ ∘ E) := hχ.comp E.contDiff
have hχcc : HasCompactSupport (χ ∘ E) := hχc.comp_homeomorph E.toHomeomorph
have hχblock : tsupport (χ ∘ E)⊆E ⁻¹' physicalOpenBlock := by
have hh : tsupport (χ ∘ E)⊆E ⁻¹' tsupport χ :=
closure_minimal (fun y hy => subset_tsupport χ hy) ((isClosed_tsupport χ).preimage E.continuous)
intro x hx
exact hχs (hh hx)
have hχss : tsupport (χ ∘ E)⊆ball := fun _ hx => physicalOpenBlock_subset_ball (hχblock hx)
let u := w j-constantH1 (z.terminalValue 0 j)-f
obtain ⟨v,hv,hvv⟩ := original_smooth_mul_exists hχd hχcc hχss u
have hd : q j-w j∈H10 := by
change (q j-w j).val∈zeroTraceAmbient
rw [hq]
exact voltageOriginalJetCLM_zeroTrace physicalOpenBlock_open.measurableSet
(fun _ => physicalOpenBlock_subset_ball) j hZ
have hband := ae_restrict_of_ae (s:=ball) ((PiLp.volume_preserving_ofLp (Fin 3)).quasiMeasurePreserving.ae
(z.terminal_band_value hs hη hηc hT w he j))
have hdv : ∀ᵐ x∂ballMeasure,WithLp.ofLp x∉physicalOpenBlock → weakValue (q j-w j) x=0 := by
have hz := voltageOriginalJetCLM_ae physicalOpenBlock_open.measurableSet j Z
rw [←hq j] at hz
filter_upwards [hz] with x hx hxn
have hh := congrArg (fun f : JetFiber => f 0) hx
change (q j-w j).val x 0=0
rw [hx,ite_eq_right hxn]
rfl
refine ⟨v+f+(q j-w j),H10.add_mem (H10.add_mem hv hf) hd,?_,?_⟩
· filter_upwards [hvv,hfv,hband,hdv,weakValue_add (v+f) (q j-w j),weakValue_add v f,
weakValue_sub (w j-constantH1 (z.terminalValue 0 j)) f,
(H1_sub_constant_spec (w j) (z.terminalValue 0 j)).1,weakValue_sub (q j) (w j)]
with x hmv hfv hband hdv hsum hadd hu1 hu2 hd1
rw [hsum]
simp only [Pi.add_apply]
rw [hadd]
simp only [Pi.add_apply]
have hu : weakValue u x=weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x) := by
change weakValue (w j-constantH1 (z.terminalValue 0 j)-f) x=_
rw [hu1,Pi.sub_apply,hu2,hfv]
rw [hmv,hu,hfv]
change χ (WithLp.ofLp x)*(weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x))+
z.terminalProfile η j (WithLp.ofLp x)+weakValue (q j-w j) x=_
by_cases hx : WithLp.ofLp x∈physicalOpenBlock
· rw [ite_eq_left hx,hd1]
simp only [Pi.sub_apply]
by_cases hc : WithLp.ofLp x∈terminalCore η
· rw [hχ1 _ hc]; ring
· have hh := hband hx hc
simp only [WithLp.toLp_ofLp] at hh
rw [show weakValue (w j) x-z.terminalValue 0 j-z.terminalProfile η j (WithLp.ofLp x)=0 by linarith]
linarith
· rw [ite_eq_right hx,hdv hx]
have hz : χ (WithLp.ofLp x)=0 := image_eq_zero_of_notMem_tsupport (fun hn => hx (hχs hn))
rw [hz]
ring
· intro i
have htv := physicalOuterTrace_compact_mul hχd hχcc hχblock u v hvv i
have htd := voltageOriginal_outer_trace_zero j hZ (q j-w j) (hq j) i
have hrzero (u : H1) (hu : physicalOuterTraceCLM i u=0) : physicalRealTraceCLM i u=0 := by
change Complex.reCLM.compLpL 2 volume ((mFourierBasis (d:=Fin 2)).repr.symm (physicalOuterTraceCLM i u))=0
rw [hu,map_zero,map_zero]
rw [map_add,map_add,hrzero v htv,hrzero (q j-w j) htd,zero_add,add_zero]
exact z.terminalProfile_trace hη hηc j f hfv i
end PhysicalFiniteEndingData
end ScalarConductivity
end
end
end OAI
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