| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.BoxEnergyPoincare |
|
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| namespace OAI |
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| noncomputable section |
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| namespace ScalarConductivity |
| open Set MeasureTheory |
|
|
| lemma RectFootprint.prism_volume (Q : RectFootprint) (lo hi : ℝ) : |
| volume (Q.prism lo hi)=ENNReal.ofReal (Q.xu-Q.xl)*ENNReal.ofReal (Q.yu-Q.yl)* |
| ENNReal.ofReal (hi-lo) := by |
| simp only [RectFootprint.prism,RectFootprint.carrier,Measure.volume_eq_prod, |
| Measure.prod_prod,Real.volume_Icc] |
|
|
| lemma RectFootprint.prism_volume_pos (Q : RectFootprint) (lo hi : ℝ) : |
| 0 < volume (Q.prism lo hi) ↔ Q.xl<Q.xu ∧ Q.yl<Q.yu ∧ lo<hi := by |
| rw [Q.prism_volume] |
| simp only [ENNReal.mul_pos_iff,ENNReal.ofReal_pos,sub_pos] |
| tauto |
|
|
| lemma RectFootprint.prism_real_pos (Q : RectFootprint) (lo hi : ℝ) |
| (hx : Q.xl<Q.xu) (hy : Q.yl<Q.yu) (hz : lo<hi) : |
| 0 < volume.real (Q.prism lo hi) := by |
| exact ENNReal.toReal_pos ((Q.prism_volume_pos lo hi).mpr ⟨hx,hy,hz⟩).ne' |
| (Q.prism_compact lo hi).measure_ne_top |
|
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| def RectFootprint.poincareConstant (Q : RectFootprint) (lo hi : ℝ) : ℝ := |
| 4*((Q.xu-Q.xl)^2+(Q.yu-Q.yl)^2+(hi-lo)^2) |
|
|
| def RectFootprint.average (Q : RectFootprint) (lo hi : ℝ) (f : Box3 → ℝ) : ℝ := |
| boxAverage ((Q.xl,Q.yl),lo) ((Q.xu-Q.xl,Q.yu-Q.yl),hi-lo) f |
|
|
| lemma RectFootprint.poincareConstant_nonneg (Q : RectFootprint) (lo hi : ℝ) : |
| 0 ≤ Q.poincareConstant lo hi := by unfold RectFootprint.poincareConstant; positivity |
|
|
| theorem RectFootprint.poincare (Q : RectFootprint) (lo hi : ℝ) {f : Box3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) : |
| localVariance volume (Q.prism lo hi) f (Q.average lo hi f) ≤ |
| Q.poincareConstant lo hi*(∫ z in Q.prism lo hi,cubeEnergyDensity f z) := by |
| by_cases h : Q.xl<Q.xu ∧ Q.yl<Q.yu ∧ lo<hi |
| · rw [Q.prism_closedBox] |
| exact smooth_box_volume_poincare hf _ _ (sub_pos.mpr h.1) (sub_pos.mpr h.2.1) (sub_pos.mpr h.2.2) |
| · have hz : volume (Q.prism lo hi)=0 := by |
| exact nonpos_iff_eq_zero.mp (le_of_not_gt (fun hp => h ((Q.prism_volume_pos lo hi).mp hp))) |
| simp [localVariance,Measure.restrict_eq_zero.mpr hz] |
|
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| lemma rectangle_energy_mono {Q R : Set Box3} {f : Box3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (hR : IsCompact R) (hQR : Q⊆R) : |
| (∫ z in Q,cubeEnergyDensity f z) ≤ ∫ z in R,cubeEnergyDensity f z := by |
| exact setIntegral_mono_set ((continuous_cubeEnergyDensity hf).continuousOn.integrableOn_compact hR) |
| (Filter.Eventually.of_forall (fun _ => cubeEnergyDensity_nonneg f _)) |
| (Filter.Eventually.of_forall (fun _ hx => hQR hx)) |
|
|
| theorem RectFootprint.poincare_in (Q : RectFootprint) (lo hi : ℝ) {R : Set Box3} |
| (hR : IsCompact R) (hQR : Q.prism lo hi⊆R) {f : Box3 → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) : |
| localVariance volume (Q.prism lo hi) f (Q.average lo hi f) ≤ |
| Q.poincareConstant lo hi*(∫ z in R,cubeEnergyDensity f z) := by |
| exact (Q.poincare lo hi hf).trans (mul_le_mul_of_nonneg_left (rectangle_energy_mono hf hR hQR) |
| (Q.poincareConstant_nonneg lo hi)) |
|
|
| open Set MeasureTheory |
|
|
| variable {X : Type*} [MeasurableSpace X] |
|
|
| def patchCluster (Q : ℕ → Set X) : ℕ → Set X |
| | 0 => Q 0 |
| | n+1 => patchCluster Q n ∪ Q (n+1) |
|
|
| omit [MeasurableSpace X] in |
| lemma patchCluster_mono (Q : ℕ → Set X) {n m : ℕ} (h : n ≤ m) : |
| patchCluster Q n ⊆ patchCluster Q m := by |
| induction h with |
| | refl => exact Subset.rfl |
| | @step m h ih => exact ih.trans (subset_union_left (t := Q (m+1))) |
|
|
| omit [MeasurableSpace X] in |
| lemma subset_patchCluster (Q : ℕ → Set X) (n : ℕ) : Q n ⊆ patchCluster Q n := by |
| cases n with |
| | zero => exact Subset.rfl |
| | succ n => exact subset_union_right |
|
|
| def patchPoincareConstant (μ : Measure X) (Q : ℕ → Set X) (A : ℕ → ℝ) : ℕ → ℝ |
| | 0 => A 0 |
| | n+1 => patchPoincareConstant μ Q A n+2*A (n+1)+ |
| 4*μ.real (Q (n+1))/μ.real (patchCluster Q n ∩ Q (n+1))* |
| (patchPoincareConstant μ Q A n+A (n+1)) |
|
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| lemma patchPoincareConstant_nonneg (μ : Measure X) (Q : ℕ → Set X) (A : ℕ → ℝ) |
| (hA : ∀ n, 0≤A n) (n : ℕ) : 0≤patchPoincareConstant μ Q A n := by |
| induction n with |
| | zero => exact hA 0 |
| | succ n ih => |
| dsimp [patchPoincareConstant] |
| have hQ : 0≤μ.real (Q (n+1)) := ENNReal.toReal_nonneg |
| have hO : 0≤μ.real (patchCluster Q n ∩ Q (n+1)) := ENNReal.toReal_nonneg |
| have hn := hA (n+1) |
| positivity |
|
|
| theorem connected_patch_poincare (μ : Measure X) (Q : ℕ → Set X) (A : ℕ → ℝ) |
| (hA : ∀ n, 0≤A n) {f : X → ℝ} (m : ℕ → ℝ) (E : ℝ) (hE : 0≤E) (N : ℕ) |
| (hfin : ∀ n≤N, μ (Q n)≠⊤) |
| (hover : ∀ n<N, 0<μ.real (patchCluster Q n ∩ Q (n+1))) |
| (hint : ∀ n≤N, ∀ c : ℝ, IntegrableOn (fun x => (f x-c)^2) (Q n) μ) |
| (hloc : ∀ n≤N, localVariance μ (Q n) f (m n) ≤ A n*E) : |
| localVariance μ (patchCluster Q N) f (m 0) ≤ patchPoincareConstant μ Q A N*E := by |
| have hi : ∀ n≤N, ∀ c : ℝ, IntegrableOn (fun x => (f x-c)^2) (patchCluster Q n) μ := by |
| intro n hn |
| induction n with |
| | zero => exact hint 0 (by omega) |
| | succ n ih => |
| intro c |
| exact (ih (by omega) c).union (hint (n+1) hn c) |
| have hfinite : ∀ n≤N, μ (patchCluster Q n)≠⊤ := by |
| intro n hn |
| induction n with |
| | zero => exact hfin 0 (by omega) |
| | succ n ih => |
| exact ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨ih (by omega),hfin (n+1) hn⟩) |
| (measure_union_le _ _) |
| have hall : ∀ n≤N, localVariance μ (patchCluster Q n) f (m 0) ≤ patchPoincareConstant μ Q A n*E := by |
| intro n hn |
| induction n with |
| | zero => exact hloc 0 (by omega) |
| | succ n ih => |
| exact overlap_poincare_glue μ hE (patchPoincareConstant_nonneg μ Q A hA n) (hA (n+1)) |
| (hfinite n (by omega)) (hfin (n+1) hn) (hover n (by omega)) |
| (hi n (by omega) (m 0)) (hint (n+1) hn (m (n+1))) (hint (n+1) hn (m 0)) |
| (ih (by omega)) (hloc (n+1) hn) |
| exact hall N le_rfl |
|
|
| end ScalarConductivity |
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| end |
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| end OAI |
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