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a458723 | 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 | import Mathlib
import OAI.Analysis.Conductivity.Scalarization.OriginalScalarizationCauchy
import OAI.Analysis.Conductivity.Variational.PhysicalOpenVoltage
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology Matrix
open scoped Matrix.Norms.Elementwise ENNReal
namespace PhysicalFiniteEndingData
variable {s : Fin 3 → ℝ} (z : PhysicalFiniteEndingData s)
theorem scalar_pair_green_exists
(hs : ∀ x y : ℝ,(1/2)*(x^2+y^2)≤ s 0*x^2+2*s 1*x*y+s 2*y^2) :
∃ (σ : Coord3 → ℝ) (c C : ℝ) (w q : Fin 2 → H1),
Measurable σ ∧ MemLp σ ∞ (volume.restrict physicalOpenBlock) ∧
0<c ∧ c<C ∧ (∀ᵐ y∂volume.restrict physicalOpenBlock,σ y∈Icc c C) ∧
(∀ j,q j∈H10) ∧
(∀ j,H1JetOn (w j) centralPhysical (centralFullJetCLM s (z.p j).val)) ∧
(∀ j i,H1JetOn (w j) (physicalEndRegion i) (z.correctedEndJet j i)) ∧
(∃ Z : VoltageJetSpace volume physicalOpenBlock,
Z-originalPairVoltageJet physicalOpenBlock w∈zeroVoltageJets volume physicalOpenBlock ∧
∀ j,(q j-w j).val=voltageOriginalJetCLM physicalOpenBlock_open.measurableSet j
(Z-originalPairVoltageJet physicalOpenBlock w)) ∧
∀ j (h : H1),(∫ y in physicalOpenBlock,σ y*(originalPiGradient (q j) y ⬝ᵥ
originalPiGradient h y))=constantTerminalFluxCLM (centralBasisSlopes j) h := by
let : IsFiniteMeasure (volume.restrict physicalOpenBlock) :=
isFiniteMeasure_restrict.mpr physicalOpenBlock_bounded.measure_lt_top.ne
let : Fact ((2:ℝ≥0∞)≠∞) := ⟨by norm_num⟩
obtain ⟨w,hw,hc,he,hgreen⟩ := z.corrected_pair_green_exists hs
obtain ⟨v,hv,hd,hr⟩ := z.corrected_pair_representative hs w hc he
have hv' : ∀ᵐ y : Coord3,y∈physicalOpenBlock → v y=fun j => weakValue (w j) (WithLp.toLp 2 y) := by
filter_upwards [hv] with y hy hyU
exact hy (physicalOpenBlock_subset hyU)
have hd' : ∀ᵐ y : Coord3,y∈physicalOpenBlock → ∀ i : Fin 3,∀ j : Fin 2,
fderiv ℝ (fun x => v x j) y (Pi.single i 1)=weakGradient (w j) (WithLp.toLp 2 y) i := by
filter_upwards [hd] with y hy hyU
exact hy (physicalOpenBlock_subset hyU)
have hr' := regularRegion_conull_open_restrict physicalOpenBlock_open physicalOpenBlock_subset hr
obtain ⟨hu,hE,hjet,hH1⟩ := original_pair_voltageH1 physicalOpenBlock_open.measurableSet
(fun _ => physicalOpenBlock_subset_ball) physicalOpenBlock_bounded w v (z.cartesianTensor hs) hv' hd' hr'
have hg := voltageGradient_original_ae physicalOpenBlock_open.measurableSet hd' hr'
have hF := voltageFlux_memLp v (z.cartesianTensor hs) hE
(z.cartesianTensor_entry_memLp hs physicalOpenBlock)
obtain ⟨l,L,hl,hlL,hb⟩ := z.cartesianTensor_dot_elliptic hs
obtain ⟨Z,F,σ,hσm,hσL,hzero,hσ,hcau,_⟩ := finite_field_scalarization volume hl hlL.le
physicalOpenBlock_bounded physicalOpenBlock_open.measurableSet v (z.cartesianTensor hs)
(z.cartesianTensor_measurable hs) hu hE hF hH1 (smoothFluxIntegrable_of_L2 v _ hF)
(fun j ψ hψ _ hψs => z.corrected_voltage_divergence hs w hc he v hg j hψ hψs)
hr' (Filter.Eventually.of_forall (fun y _ => hb y))
rw [hjet] at hzero
obtain ⟨q,hq,hqjet,hqv,hqg⟩ := original_pair_zeroVoltage_extension physicalOpenBlock_open.measurableSet
(fun _ => physicalOpenBlock_subset_ball) w hw Z.val hzero
refine ⟨σ,laminateLower l L,laminateUpper L,w,q,hσm,hσL,
(uniform_laminate_coverage hl hlL.le).1,(uniform_laminate_coverage hl hlL.le).2.1,
hσ.mono (fun _ h => h.1),hq,hc,he,⟨Z.val,hzero,hqjet⟩,?_⟩
intro j h
have hcc := scalarization_original_weak_gradient_cauchy physicalOpenBlock_open.measurableSet
(fun _ => physicalOpenBlock_subset_ball) physicalOpenBlock_bounded hcau h j
have hlhs : (∫ y,∑ i,weakGradient h (WithLp.toLp 2 y) i*F y (i,j) ∂volume.restrict physicalOpenBlock)=
∫ y in physicalOpenBlock,σ y*(originalPiGradient (q j) y ⬝ᵥ originalPiGradient h y) := by
apply integral_congr_ae
filter_upwards [hσ,hqg] with y hσ hqg
rw [hσ.2]
simp only [PiLp.smul_apply,smul_eq_mul,hqg,dotProduct,Finset.mul_sum,originalPiGradient]
apply Finset.sum_congr rfl
intro i _
ring
have hrhs : (∫ y,∑ i,weakGradient h (WithLp.toLp 2 y) i*(hF.toLp _) y (i,j) ∂volume.restrict physicalOpenBlock)=
∫ y in physicalOpenBlock,originalPiGradient (w j) y ⬝ᵥ
(variableBlockTensor z.flatTensor z.compression y*ᵥoriginalPiGradient h y) := by
apply integral_congr_ae
filter_upwards [hF.coeFn_toLp,hg] with y hF hg
rw [hF]
simp_rw [voltageFlux_component,hg]
change originalPiGradient h y ⬝ᵥ((z.cartesianTensor hs y).val*ᵥoriginalPiGradient (w j) y)=_
have hsym : (z.cartesianTensor hs y).valᵀ=(z.cartesianTensor hs y).val := (z.cartesianTensor hs y).property
have hh := Matrix.dotProduct_transpose_mulVec (z.cartesianTensor hs y).val
(originalPiGradient h y) (originalPiGradient (w j) y)
rw [hsym] at hh
exact hh
rw [hlhs,hrhs] at hcc
exact hcc.trans (by simpa only [physicalOpenBlock,physicalBlockRegion_openBlock_restrict] using hgreen j h)
end PhysicalFiniteEndingData
end ScalarConductivity
end
end
end OAI
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