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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import Mathlib
import OAI.Analysis.Conductivity.Geometry.CentralSlopeJet
import OAI.Analysis.Conductivity.Variational.CentralLocalAgreement
namespace OAI
noncomputable section
namespace ScalarConductivity
section
open Set MeasureTheory Filter Topology
variable (s : Fin 3 → ℝ)
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
{a : Fin 3 → ℝ} (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ)
def centralBallSet : Set R3 := WithLp.ofLp ⁻¹' centralPhysical
lemma centralBallSet_measurable : MeasurableSet centralBallSet :=
centralPhysical_compact.measurableSet.preimage
(PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).continuous.measurable
lemma centralNeighborhoodRawCLM_smooth_restrict (f : centralSmoothFunctions) :
lpRestrictionCLM centralBallSet (centralNeighborhoodRawCLM s hs ha₀ ha (centralEmbedL s f))=
lpRestrictionCLM centralBallSet (centralFullJetCLM s (centralEmbedL s f)) := by
apply Lp.ext
filter_upwards [lpRestrictionCLM_ae centralBallSet (centralNeighborhoodRawCLM s hs ha₀ ha (centralEmbedL s f)),
lpRestrictionCLM_ae centralBallSet (centralFullJetCLM s (centralEmbedL s f)),
ae_restrict_of_ae (centralNeighborhoodRawCLM_smooth_ae s hs ha₀ ha f),
ae_restrict_of_ae (centralFullJetCLM_smooth_ae s f),
ae_restrict_mem centralBallSet_measurable] with x h₁ h₂ hn hc hx
exact h₁.trans ((hn hx).trans ((hc hx).symm.trans h₂.symm))
theorem centralNeighborhoodCompletion_restrict (u : centralEnergySpace s) :
lpRestrictionCLM centralBallSet (centralNeighborhoodCompletion s hs ha₀ ha u).val=
lpRestrictionCLM centralBallSet (centralFullJetCLM s u.val) := by
let L := (lpRestrictionCLM centralBallSet).comp (centralNeighborhoodRawCLM s hs ha₀ ha)-
(lpRestrictionCLM centralBallSet).comp (centralFullJetCLM s)
have hr : LinearMap.range (centralEmbedL s)≤LinearMap.ker L.toLinearMap := by
rintro _ ⟨f,rfl⟩
change lpRestrictionCLM centralBallSet (centralNeighborhoodRawCLM s hs ha₀ ha (centralEmbedL s f))-
lpRestrictionCLM centralBallSet (centralFullJetCLM s (centralEmbedL s f))=0
exact sub_eq_zero.mpr (centralNeighborhoodRawCLM_smooth_restrict s hs ha₀ ha f)
have hu := Submodule.topologicalClosure_minimal _ hr L.isClosed_ker u.property
rw [centralNeighborhoodCompletion_val]
exact sub_eq_zero.mp hu
lemma centralNeighborhoodCompletion_central_ae (u : centralEnergySpace s) :
∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical →
(centralNeighborhoodCompletion s hs ha₀ ha u).val x=centralFullJetCLM s u.val x := by
have he := centralNeighborhoodCompletion_restrict s hs ha₀ ha u
apply (ae_restrict_iff' centralBallSet_measurable).mp
filter_upwards [lpRestrictionCLM_ae centralBallSet (centralNeighborhoodCompletion s hs ha₀ ha u).val,
lpRestrictionCLM_ae centralBallSet (centralFullJetCLM s u.val)] with x h₁ h₂
rw [←h₁,he,h₂]
end
open Set MeasureTheory Filter Topology
lemma H1_add_jet_ae (u v : H1) :
(u+v).val=ᵐ[ballMeasure] (fun x => u.val x+v.val x) := by
exact Lp.coeFn_add u.val v.val
lemma central_neighborhood_with_slopes (s : Fin 3 → ℝ)
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
(a slopes : Fin 3 → ℝ) (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ)
(p : centralEnergySpace s) :
∃ w : H1,w∈H10 ∧
(w.val=ᵐ[ballMeasure] (fun x =>
(centralNeighborhoodCompletion s hs ha₀ ha p).val x+
centralSlopeJet a slopes (WithLp.ofLp x))) ∧
(∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical →
w.val x=centralFullJetCLM s p.val x) := by
obtain ⟨v,hv,hve,hvz⟩ := centralSlopeExtension_exists a slopes ha₀ ha
let z : H1 := centralNeighborhoodCompletion s hs ha₀ ha p
have hsum := H1_add_jet_ae z v
have hz : z∈H10 := centralNeighborhoodCompletion_mem_H10 s hs ha₀ ha p
refine ⟨z+v,H10.add_mem hz hv,?_,?_⟩
· filter_upwards [hsum,hve] with x hadd he
exact hadd.trans (congrArg (fun b : JetFiber => z.val x+b) he)
· have hcentral := centralNeighborhoodCompletion_central_ae s hs ha₀ ha p
filter_upwards [hsum,hvz,hcentral] with x hadd hvzero hc
intro hx
exact hadd.trans ((congrArg (fun b : JetFiber => z.val x+b) (hvzero hx)).trans
((add_zero _).trans (hc hx)))
theorem central_variational_neighborhood_exists (s : Fin 3 → ℝ)
(hs : ∀ u v : ℝ,(1/2)*(u^2+v^2) ≤ s 0*u^2+2*s 1*u*v+s 2*v^2)
(a slopes : Fin 3 → ℝ) (ha₀ : a 0<0) (ha : ∀ k : Fin 2,0<a k.succ)
(hslopes : ∑ i,slopes i=0) :
∃ p : centralEnergySpace s,∃ w : H1,w∈H10 ∧ centralM s p=0 ∧
(w.val=ᵐ[ballMeasure] (fun x =>
(centralNeighborhoodCompletion s hs ha₀ ha p).val x+
centralSlopeJet a slopes (WithLp.ofLp x))) ∧
(∀ᵐ x∂ballMeasure,WithLp.ofLp x∈centralPhysical →
w.val x=centralFullJetCLM s p.val x) ∧
(∀ v : centralEnergySpace s,
inner ℝ (centralD s p) (centralD s v)+
angularArea*(∑ i : Fin 3,inner ℝ
(spectralGraphWeight (torusRate s) (centralT s i p))
(spectralGraphWeight (torusRate s) (centralT s i v)))=
angularArea*(∑ i : Fin 3,slopes i*
spectralGraphMean (torusRate s) (centralT s i v))) := by
obtain ⟨p,hp,hvar,_⟩ := central_unnormalized_variational_exists s slopes hslopes
obtain ⟨w,hw,he,hc⟩ := central_neighborhood_with_slopes s hs a slopes ha₀ ha p
exact ⟨p,w,hw,hp,he,hc,hvar⟩
end ScalarConductivity
end
end OAI
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