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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 | 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
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