| import Mathlib |
| import OAI.Analysis.Conductivity.Model |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology |
|
|
| lemma affine_layer_interval (ha : a≠0) (hε : 0≤ε) : |
| let l := min b (b+ε/a) |
| let r := max b (b+ε/a) |
| r-l=ε/|a| ∧ |
| b-ε/|a|≤l ∧ r≤b+ε/|a| ∧ |
| (∀ t,t∈Icc l r ↔ a*(t-b)∈Icc 0 ε) ∧ |
| (∀ t∈Icc l r,|t-b|≤ε/|a|) := by |
| dsimp only |
| have hc : a*(ε/a)=ε := by field_simp |
| rcases lt_or_gt_of_ne ha with hn|hp |
| · have hd : ε/a≤0 := div_nonpos_of_nonneg_of_nonpos hε hn.le |
| rw [min_eq_right (by linarith : b+ε/a≤b),max_eq_left (by linarith : b+ε/a≤b),abs_of_neg hn,div_neg] |
| refine ⟨by ring,by linarith,by linarith,?_,?_⟩ |
| · intro t |
| constructor |
| · intro ht |
| have h1 := mul_le_mul_of_nonpos_left ht.2 hn.le |
| have h2 := mul_le_mul_of_nonpos_left ht.1 hn.le |
| constructor <;> nlinarith |
| · intro ht |
| constructor |
| · apply (mul_le_mul_left_of_neg hn).mp |
| nlinarith [ht.2] |
| · apply (mul_le_mul_left_of_neg hn).mp |
| nlinarith [ht.1] |
| · intro t ht |
| apply abs_le.mpr |
| constructor <;> linarith [ht.1,ht.2] |
| · have hd : 0≤ε/a := div_nonneg hε hp.le |
| rw [min_eq_left (by linarith : b≤b+ε/a),max_eq_right (by linarith : b≤b+ε/a),abs_of_pos hp] |
| refine ⟨by ring,by linarith,le_refl _,?_,?_⟩ |
| · intro t |
| constructor |
| · intro ht |
| have h1 := mul_le_mul_of_nonneg_left ht.1 hp.le |
| have h2 := mul_le_mul_of_nonneg_left ht.2 hp.le |
| constructor <;> nlinarith |
| · intro ht |
| constructor |
| · apply (mul_le_mul_iff_right₀ hp).mp |
| nlinarith [ht.1] |
| · apply (mul_le_mul_iff_right₀ hp).mp |
| nlinarith [ht.2] |
| · intro t ht |
| apply abs_le.mpr |
| constructor <;> linarith [ht.1,ht.2] |
|
|
| lemma seamLayer_scaled_tendsto_of_cubic_pi |
| (hδ : 0<δ) |
| (hbound : ∀ ε,0<ε → ε≤δ → (∫ x in ,u x^2)≤A*ε^3) : |
| Tendsto (fun n : ℕ => ((n:ℝ)+1)^2*∫ x in ,u x^2) atTop (𝓝 0) := by |
| have hn (n : ℕ) : 0<(n:ℝ)+1 := by positivity |
| have ht : Tendsto (fun n : ℕ => ((n:ℝ)+1)⁻¹) atTop (𝓝 0) := |
| tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_right atTop (1:ℝ) tendsto_natCast_atTop_atTop) |
| have he : ∀ᶠ n : ℕ in atTop,2/((n:ℝ)+1)≤δ := by |
| have hh := ht.const_mul 2 |
| simp only [mul_zero] at hh |
| simpa only [←div_eq_mul_inv] using (hh.eventually (gt_mem_nhds hδ)).mono (fun _ h => h.le) |
| have hlim := ht.const_mul (8*A) |
| simp only [mul_zero] at hlim |
| apply squeeze_zero' (Eventually.of_forall (fun n => mul_nonneg (sq_nonneg _) (integral_nonneg (fun _ => sq_nonneg _)))) _ hlim |
| filter_upwards [he] with n hnδ |
| have hs : {x | 0<τ x ∧ ((n:ℝ)+1)*τ x≤2}={x | 0<τ x ∧ τ x≤2/((n:ℝ)+1)} := by |
| ext x |
| exact and_congr_right (fun _ => by rw [le_div_iff₀ (hn n),mul_comm]) |
| rw [hs] |
| have hb := mul_le_mul_of_nonneg_left (hbound _ (div_pos (by norm_num) (hn n)) hnδ) (sq_nonneg ((n:ℝ)+1)) |
| apply hb.trans_eq |
| field_simp |
| ring |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
