| import Mathlib |
| import OAI.Analysis.Conductivity.Scalarization.OriginalScalarizationCauchy |
| import OAI.Analysis.Conductivity.Variational.PhysicalOpenVoltage |
|
|
| namespace OAI |
|
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| section |
|
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| 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 |
|
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| end OAI |
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