| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CentralEnergySpace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory |
|
|
| def centralAmbientEnergy (s : Fin 3 → ℝ) (u : CentralAmbient s) : ℝ := |
| ‖centralAmbientD s u‖^2+ |
| ∑ i : Fin 3,‖spectralGraphWeight (torusRate s) (centralAmbientT s i u)‖^2 |
|
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| lemma centralAmbientEnergy_nonneg (s : Fin 3 → ℝ) (u : CentralAmbient s) : |
| 0≤centralAmbientEnergy s u := add_nonneg (sq_nonneg _) (Finset.sum_nonneg fun _ _ => sq_nonneg _) |
|
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| lemma continuous_centralAmbientEnergy (s : Fin 3 → ℝ) : Continuous (centralAmbientEnergy s) := by |
| apply Continuous.add ((centralAmbientD s).continuous.norm.pow 2) |
| exact continuous_finsetSum _ (fun i _ => |
| (((spectralGraphWeight (torusRate s)).comp (centralAmbientT s i)).continuous.norm.pow 2)) |
|
|
| lemma centralAmbientD_shift (s : Fin 3 → ℝ) (u : CentralAmbient s) (a : ℝ) : |
| centralAmbientD s (u-a • centralAmbientOne s)=centralAmbientD s u := by simp only [map_sub,map_smul,centralAmbientD_one,smul_zero,sub_zero] |
|
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| lemma centralAmbientWeight_shift (s : Fin 3 → ℝ) (u : CentralAmbient s) (a : ℝ) (i : Fin 3) : |
| spectralGraphWeight (torusRate s) (centralAmbientT s i (u-a • centralAmbientOne s))= |
| spectralGraphWeight (torusRate s) (centralAmbientT s i u) := by |
| simp only [map_sub,map_smul,centralAmbientWeight_one,smul_zero,sub_zero] |
|
|
| lemma centralAmbientEnergy_shift (s : Fin 3 → ℝ) (u : CentralAmbient s) (a : ℝ) : |
| centralAmbientEnergy s (u-a • centralAmbientOne s)=centralAmbientEnergy s u := by |
| simp only [centralAmbientEnergy,centralAmbientD_shift,centralAmbientWeight_shift] |
|
|
| lemma centralAmbientQ_shift (s : Fin 3 → ℝ) (u : CentralAmbient s) (a : ℝ) : |
| centralAmbientQ s (u-a • centralAmbientOne s)=centralAmbientQ s u := by |
| simp only [map_sub,map_smul,centralAmbientQ_one,smul_zero,sub_zero] |
|
|
| def centralCoercivityBase : ℝ := |
| 1+centralPoincareConstant+3*centralTraceConstant*(192+2*centralPoincareConstant) |
|
|
| lemma centralCoercivityBase_ge : 1≤centralCoercivityBase := by |
| have hP := centralPoincareConstant_nonneg |
| have hT := centralTraceConstant_pos |
| dsimp [centralCoercivityBase] |
| nlinarith |
|
|
| lemma centralCoercivityBase_pos : 0<centralCoercivityBase := lt_of_lt_of_le zero_lt_one centralCoercivityBase_ge |
|
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| lemma central_smooth_anchor_norm (s : Fin 3 → ℝ) (f : centralSmoothFunctions) : |
| ‖centralEmbedL s (f-centralAnchorAverage 0 f • centralSmoothOne)‖^2≤ |
| centralCoercivityBase*centralAmbientEnergy s (centralEmbedL s f) := by |
| let a := centralAnchorAverage 0 f |
| let g : centralSmoothFunctions := f-a • centralSmoothOne |
| have hg : centralEmbedL s g=centralEmbedL s f-a • centralAmbientOne s := by |
| dsimp [g,centralAmbientOne] |
| simp only [map_sub,map_smul] |
| have hD : centralJetD (centralSmoothJet g)=centralAmbientD s (centralEmbedL s f) := by |
| rw [←centralAmbientD_embed s g,hg,centralAmbientD_shift] |
| have hP : (∫ z in centralClosed,(g z)^2)≤ |
| centralPoincareConstant*‖centralAmbientD s (centralEmbedL s f)‖^2 := by |
| rw [centralAmbientD_embed,centralSmoothJet_gradient_norm] |
| simpa [g,a,centralSmoothOne,localVariance] using central_physical_poincare (central_smooth f) |
| have ht (i : Fin 3) : ‖centralTraceFourier g i‖^2≤ |
| (centralTraceConstant*(192+2*centralPoincareConstant))*‖centralAmbientD s (centralEmbedL s f)‖^2 := by |
| have hh := centralTrace_anchor_bound g i |
| rw [hD] at hh |
| apply hh.trans |
| calc |
| _≤centralTraceConstant*(192*‖centralAmbientD s (centralEmbedL s f)‖^2+ |
| 2*(centralPoincareConstant*‖centralAmbientD s (centralEmbedL s f)‖^2)) := by |
| apply mul_le_mul_of_nonneg_left _ centralTraceConstant_pos.le |
| linarith only [hP] |
| _=_ := by ring |
| have hsum := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => ht i) |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at hsum |
| norm_num only [Nat.cast_ofNat] at hsum |
| have hw (i : Fin 3) : |
| spectralGraphWeight (torusRate s) (centralSmoothTrace s g i)= |
| spectralGraphWeight (torusRate s) (centralAmbientT s i (centralEmbedL s f)) := by |
| rw [←centralAmbientT_embed s g i,hg,centralAmbientWeight_shift] |
| change ‖centralEmbedL s g‖^2≤_ |
| rw [centralEmbed_norm,hD] |
| simp only [hw,Finset.sum_add_distrib] |
| have hW : 0≤∑ i : Fin 3,‖spectralGraphWeight (torusRate s) (centralAmbientT s i (centralEmbedL s f))‖^2 := |
| Finset.sum_nonneg fun _ _ => sq_nonneg _ |
| have hge := centralCoercivityBase_ge |
| dsimp only [centralAmbientEnergy] |
| have hcoef : centralCoercivityBase=1+centralPoincareConstant+ |
| 3*(centralTraceConstant*(192+2*centralPoincareConstant)) := by unfold centralCoercivityBase; ring |
| have hWW := mul_le_mul_of_nonneg_right hge hW |
| rw [one_mul] at hWW |
| rw [hcoef] at hWW ⊢ |
| nlinarith only [hP,hsum,hWW] |
|
|
| lemma central_smooth_projected_bound (s : Fin 3 → ℝ) (f : centralSmoothFunctions) : |
| ‖centralAmbientQ s (centralEmbedL s f)‖^2≤ |
| (‖centralAmbientQ s‖^2*centralCoercivityBase)*centralAmbientEnergy s (centralEmbedL s f) := by |
| let g : centralSmoothFunctions := f-centralAnchorAverage 0 f • centralSmoothOne |
| have hh : centralAmbientQ s (centralEmbedL s g)=centralAmbientQ s (centralEmbedL s f) := by |
| have hg : centralEmbedL s g=centralEmbedL s f-centralAnchorAverage 0 f • centralAmbientOne s := by |
| dsimp [g,centralAmbientOne] |
| simp only [map_sub,map_smul] |
| rw [hg,centralAmbientQ_shift] |
| have hn := (centralAmbientQ s).le_opNorm (centralEmbedL s g) |
| have hn2 := mul_self_le_mul_self (norm_nonneg _) hn |
| rw [hh] at hn2 |
| have hb := central_smooth_anchor_norm s f |
| have hm := mul_le_mul_of_nonneg_left hb (sq_nonneg ‖centralAmbientQ s‖) |
| nlinarith only [hn2,hm] |
|
|
| end ScalarConductivity |
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| end |
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| end OAI |
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