| import Mathlib |
| import OAI.Analysis.Conductivity.Sources.WeightedGaugeDN |
| import OAI.Analysis.Conductivity.Sobolev.CriticalWallParticularC0 |
| import OAI.Analysis.Conductivity.Fourier.PoissonGreen |
| import OAI.Analysis.Conductivity.Walls.SeamCubicRate |
| import OAI.Analysis.Conductivity.Walls.PhysicalCollarCutoff |
| import OAI.Analysis.Conductivity.Geometry.CylinderPhysicalEquiv |
| import OAI.Analysis.Conductivity.Branching.ChildFullEndEnergy |
| import OAI.Analysis.Conductivity.Variational.BoundedPiolaCorrection |
| import OAI.Analysis.Conductivity.Geometry.RegularPatchLinear |
| import OAI.Analysis.Conductivity.Variational.PhysicalCentralRegularPair |
| import OAI.Analysis.Conductivity.Fourier.GeneralModeRecombinedEnding |
| import OAI.Analysis.Conductivity.Geometry.RegularPatchCoordinates |
| import OAI.Analysis.Conductivity.Variational.CentralCorrectedRepresentative |
| import OAI.Analysis.Conductivity.Branching.PhysicalBlockRepresentative |
| import OAI.Analysis.Conductivity.Sources.PhysicalPatching |
| import OAI.Analysis.Conductivity.Sources.IteratedAverages |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma WeightedAnnihilation.smul {a : R3 → ℝ} {w u : H1} {c : ℝ} |
| (hw : WeightedAnnihilation a w u c) (δ : ℝ) : |
| WeightedAnnihilation a (δ • w) u c := by |
| intro f hf hc hs |
| calc |
| (∫ x,a x*inner ℝ (weakGradient (δ • w) x) |
| ((weakValue u x-c) • gradient f x+f x • weakGradient u x) ∂ballMeasure)= |
| ∫ x,δ*(a x*inner ℝ (weakGradient w x) |
| ((weakValue u x-c) • gradient f x+f x • weakGradient u x)) ∂ballMeasure := by |
| apply integral_congr_ae |
| filter_upwards [weakGradient_smul δ w] with x hx |
| rw [hx,Pi.smul_apply,real_inner_smul_left] |
| ring |
| _=0 := by rw [integral_const_mul,hw f hf hc hs,mul_zero] |
|
|
| theorem actual_nested_weighted_source_exists : |
| ∃ (a : R3 → ℝ) (c C : ℝ) (w : H1), |
| Measurable a ∧ 0<c ∧ c<C ∧ |
| (∀ᵐ x∂ballMeasure,c ≤ a x ∧ a x ≤ C) ∧ EqualOneNearBoundary a ∧ |
| w∈H10 ∧ (∀ᵐ x∂ballMeasure,abs (weakValue w x) ≤ 1/2) ∧ |
| 0<ballMeasure {x | weakValue w x≠0} ∧ |
| (∀ᵐ x∂ballMeasure,(5/2:ℝ)<‖x‖ → weakValue w x=0 ∧ weakGradient w x=0) ∧ |
| ∀ u : H1,Harmonic a u → (∃ M : ℝ,∀ᵐ x∂ballMeasure,abs (weakValue u x) ≤ M) → |
| ∃ d : ℝ,WeightedAnnihilation a w u d := by |
| let r : BoundedRootSources := Classical.choice boundedRootSources_nonempty |
| obtain ⟨w,hw,⟨M,hM,hMb⟩,ht⟩ := r.nested_source_limit_exists |
| let δ : ℝ := 1/(2*M) |
| have hδ : 0<δ := by dsimp [δ]; positivity |
| have hδM : δ*M=1/2 := by dsimp [δ]; field_simp |
| have hwne := r.nested_source_nonzero ht |
| have hvne : δ • w.val≠(0:H1) := smul_ne_zero hδ.ne' hwne.1 |
| refine ⟨r.coefficient,min r.c 1,max r.C 1+1,δ • w.val, |
| r.coefficient_measurable,lt_min r.c_pos (by norm_num),?_,?_, |
| copiedScalarCoefficient_collar r.b,H10.smul_mem δ w.property,?_, |
| nonzero_h1_positive_value_measure hvne,sourceSupported_collar (hw.smul δ),?_⟩ |
| · have hmin := min_le_left r.c 1 |
| have hmax := le_max_left r.C 1 |
| have hcc := r.c_lt_C |
| linarith |
| · exact Eventually.of_forall (fun x => ⟨(copiedScalarCoefficient_bounds r.bounds x).1, |
| (copiedScalarCoefficient_bounds r.bounds x).2.trans (by linarith)⟩) |
| · have hh := weakValue_bound_smul hMb δ |
| simpa only [abs_of_pos hδ,hδM] using hh |
| · intro u hu hb |
| exact ⟨sourceMeanCLM 0 u,(r.nested_source_weighted ht hu hb).smul δ⟩ |
| |
| theorem main_nonuniqueness : MainStatement := by |
| obtain ⟨a,c,C,w,ha,hc,hcC,hab,hnear,hw0,hw,hwne,hsupp,hweighted⟩ := |
| actual_nested_weighted_source_exists |
| exact MainStatement_of_weighted_source a ha c C hc hcC hab hnear w hw0 hw hwne |
| (5/2) (by norm_num) (by norm_num) hsupp hweighted |
| |
| end ScalarConductivity |
| |
| end |
| end |
| |
| end OAI |
| |
