| 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 |
|
|
