| import OAI.Geometry.Perimeter.Regularity.EllipticGradientLimit |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
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| theorem gradientEnergy_integrable {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) |
| (p : Euc n) (r : ℝ) (q : Euc n) : |
| IntegrableOn (fun x ↦ ‖gradient f x-q‖^2) (Metric.ball p r) volume := |
| ((((gradient_continuous_C1 hf).sub continuous_const).norm.pow 2).continuousOn.integrableOn_compact |
| (isCompact_closedBall p r)).mono_set Metric.ball_subset_closedBall |
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| theorem gradientEnergy_mono {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) |
| {p p' : Euc n} {r r' : ℝ} (q : Euc n) (h : Metric.ball p r ⊆ Metric.ball p' r') : |
| gradientEnergy f p r q ≤ gradientEnergy f p' r' q := |
| setIntegral_mono_set (gradientEnergy_integrable hf p' r' q) |
| (ae_of_all _ (fun _ ↦ sq_nonneg _)) (ae_of_all _ (fun x ↦ show x∈Metric.ball p r → x∈Metric.ball p' r' from fun hx ↦ h hx)) |
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| theorem constant_slopes_energy_bound {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) |
| (p : Euc n) {r : ℝ} (hr : 0<r) (q q' : Euc n) : |
| ‖q-q'‖^2*volume.real (Metric.ball p r) ≤ |
| 2*gradientEnergy f p r q+2*gradientEnergy f p r q' := by |
| have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by |
| rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩ |
| calc |
| _ = ∫ x in Metric.ball p r, ‖q-q'‖^2 := by |
| simp [real_volume_ball_center, real_volume_ball_scale hr, mul_comm] |
| _ ≤ ∫ x in Metric.ball p r, 2*‖gradient f x-q‖^2+2*‖gradient f x-q'‖^2 := by |
| apply integral_mono (integrable_const _) (((gradientEnergy_integrable hf p r q).const_mul 2).add |
| ((gradientEnergy_integrable hf p r q').const_mul 2)) |
| intro x |
| have hh := norm_sub_le (q-gradient f x) (q'-gradient f x) |
| have he : (q-gradient f x)-(q'-gradient f x)=q-q' := by abel |
| rw [he,norm_sub_rev q (gradient f x),norm_sub_rev q' (gradient f x)] at hh |
| have hh' := pow_le_pow_left₀ (norm_nonneg (q-q')) hh 2 |
| change ‖q-q'‖^2 ≤ 2*‖gradient f x-q‖^2+2*‖gradient f x-q'‖^2 |
| nlinarith [sq_nonneg (‖gradient f x-q‖-‖gradient f x-q'‖)] |
| _ = _ := by rw [integral_add ((gradientEnergy_integrable hf p r q).const_mul 2) |
| ((gradientEnergy_integrable hf p r q').const_mul 2),integral_const_mul,integral_const_mul]; rfl |
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| theorem scaled_slope_comparison {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) |
| (p p' : Euc n) {r : ℝ} (hr : 0<r) (hd : dist p p'≤r/2) (q q' : Euc n) : |
| volume.real (Metric.ball (0:Euc n) 1)*‖q-q'‖^2 ≤ |
| 2^(n+1)*(scaledGradientEnergy f p r q+scaledGradientEnergy f p' r q') := by |
| have hb : Metric.ball p' (r/2) ⊆ Metric.ball p r := by |
| intro x hx |
| have hh := dist_triangle x p' p |
| rw [dist_comm p' p] at hh |
| exact Metric.mem_ball.mpr (by linarith [Metric.mem_ball.mp hx]) |
| have hb' : Metric.ball p' (r/2) ⊆ Metric.ball p' r := Metric.ball_subset_ball (by linarith) |
| have hh := constant_slopes_energy_bound hf p' (by positivity : 0<r/2) q q' |
| have h1 := gradientEnergy_mono hf q hb |
| have h2 := gradientEnergy_mono hf q' hb' |
| have hh' : ‖q-q'‖^2*volume.real (Metric.ball p' (r/2)) ≤ |
| 2*gradientEnergy f p r q+2*gradientEnergy f p' r q' := by linarith |
| rw [real_volume_ball_center,real_volume_ball_scale (by positivity : 0<r/2)] at hh' |
| have hm := mul_le_mul_of_nonneg_left hh' (by positivity : 0≤((r/2)^n)⁻¹) |
| convert hm using 1 <;> first | rfl | (simp only [scaledGradientEnergy,div_pow,pow_succ,inv_div]; field_simp [hr.ne']) |
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| end RPISingularHarnack |
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| end |
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| end OAI |
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