| import OAI.Analysis.HenonEmden.CapOscillation |
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| namespace OAI |
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| universe uE |
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| section CapContinuity |
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| open MeasureTheory Set Metric Filter Topology |
| noncomputable section |
| namespace HenonLaneEmden.AdaptiveCap |
| open Cutoff CapWeights |
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] |
| (μ : Measure E) [μ.IsAddHaarMeasure] |
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| lemma weight_short_oscillation {κ δ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) |
| (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1/2) {x y : E} (hd : 0 < d x) |
| (hxy : dist x y ≤ δ*(d x)^(m (Module.finrank ℝ E))) (e : sphere (0:E) 1) : |
| |weight μ κ e x-weight μ κ e y| ≤ |
| oscillationConstant μ κ*δ*(d x)^(N (Module.finrank ℝ E)) := by |
| let n := Module.finrank ℝ E |
| have hn : 1 ≤ n := Module.finrank_pos |
| have hdist : dist x y ≤ d x/2 := by |
| have hp := pow_d_le_self x (show 1 ≤ m n by unfold m; omega) |
| calc |
| _ ≤ δ*(d x)^(m n) := hxy |
| _ ≤ δ*d x := mul_le_mul_of_nonneg_left hp hδ |
| _ ≤ _ := by nlinarith |
| have hh := weight_close_oscillation μ hκ hκ1 hd hdist e |
| have hc := oscillationConstant_nonneg μ hκ.le |
| calc |
| _ ≤ oscillationConstant μ κ*(d x)^(N n-n-1)*dist x y := hh |
| _ ≤ oscillationConstant μ κ*(d x)^(N n-n-1)*(δ*(d x)^(m n)) := |
| mul_le_mul_of_nonneg_left hxy (mul_nonneg hc (pow_nonneg hd.le _)) |
| _ = oscillationConstant μ κ*δ*(d x)^((N n-n-1)+m n) := by rw [pow_add]; ring |
| _ ≤ _ := by |
| apply mul_le_mul_of_nonneg_left _ (mul_nonneg hc hδ) |
| exact pow_le_pow_of_le_one hd.le (d_le_one x) (by change (k n) + n + 2 ≤ (k n + n + 2 - n - 1) + (n+3); omega) |
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| lemma continuous_weight {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) (e : sphere (0:E) 1) : |
| Continuous (weight μ κ e) := by |
| rw [continuous_iff_continuousAt] |
| intro x |
| by_cases hd : 0 < d x |
| · rw [Metric.continuousAt_iff] |
| intro ε hε |
| let C := oscillationConstant μ κ*(d x)^(N (Module.finrank ℝ E)-Module.finrank ℝ E-1) |
| have hC : 0 ≤ C := mul_nonneg (oscillationConstant_nonneg μ hκ.le) (pow_nonneg hd.le _) |
| have hC1 : 0 < C+1 := by linarith |
| refine ⟨min (d x/2) (ε/(C+1)),lt_min (half_pos hd) (div_pos hε hC1),?_⟩ |
| intro y hy |
| have hdist : dist x y < ε/(C+1) := by simpa only [dist_comm x y] using (lt_min_iff.mp hy).2 |
| have hh := weight_close_oscillation μ hκ hκ1 hd (y := y) |
| (by simpa only [dist_comm] using (lt_min_iff.mp hy).1.le) e |
| change |weight μ κ e x-weight μ κ e y| ≤ C*dist x y at hh |
| rw [Real.dist_eq,abs_sub_comm] |
| have hb : (C+1)*dist x y < ε := by simpa only [mul_comm] using (lt_div_iff₀ hC1).mp hdist |
| have hr := dist_nonneg (x := x) (y := y) |
| nlinarith |
| · have hd0 : d x = 0 := le_antisymm (not_lt.mp hd) (d_nonneg x) |
| have hx : 2 ≤ ‖x‖ := not_lt.mp (fun hh => hd ((d_pos_iff x).mpr hh)) |
| have hN : N (Module.finrank ℝ E)-Module.finrank ℝ E ≠ 0 := by unfold N; omega |
| have hcont : Continuous (fun y : E => boundConstant μ κ*(d y)^(N (Module.finrank ℝ E)-Module.finrank ℝ E)) := |
| continuous_const.mul (continuous_d.pow _) |
| have ht : Tendsto (fun y : E => boundConstant μ κ*(d y)^(N (Module.finrank ℝ E)-Module.finrank ℝ E)) |
| (𝓝 x) (𝓝 0) := by simpa only [hd0,zero_pow hN,mul_zero] using hcont.tendsto x |
| change Tendsto _ (𝓝 x) (𝓝 (weight μ κ e x)) |
| rw [weight_outer μ κ e hx] |
| exact squeeze_zero (weight_nonneg μ κ e) (weight_bound μ hκ hκ1 e) ht |
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| omit [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [μ.IsAddHaarMeasure] in |
| lemma continuous_weight_direction (κ : ℝ) (x : E) : |
| Continuous (fun e : sphere (0:E) 1 => weight μ κ e x) := by |
| by_cases hx : x ≠ 0 ∧ 0 < d x |
| · simp only [weight,dite_eq_left hx,normalized] |
| exact continuous_const.mul ((continuous_profile _ _).div_const _) |
| · simp only [weight,dite_eq_right hx]; exact continuous_const |
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| lemma measurable_weight {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) : |
| Measurable (fun z : sphere (0:E) 1 × E => weight μ κ z.1 z.2) := by |
| have hh : Measurable (fun z : E × sphere (0:E) 1 => weight μ κ z.2 z.1) := |
| measurable_uncurry_of_continuous_of_measurable (continuous_weight μ hκ hκ1) |
| (fun x => (continuous_weight_direction μ κ x).measurable) |
| exact hh.comp measurable_swap |
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| lemma weight_le_constant {κ : ℝ} (hκ : 0 < κ) (hκ1 : κ ≤ 1) (e : sphere (0:E) 1) (x : E) : |
| weight μ κ e x ≤ boundConstant μ κ := by |
| apply (weight_bound μ hκ hκ1 e x).trans |
| have hp := pow_le_one₀ (d_nonneg x) (d_le_one x) (n := N (Module.finrank ℝ E)-Module.finrank ℝ E) |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hp (boundConstant_nonneg μ hκ.le) |
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| end HenonLaneEmden.AdaptiveCap |
| end |
| end CapContinuity |
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| end OAI |
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