File size: 3,667 Bytes
2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.Geometry.Perimeter.Regularity.EllipticGradientLimit
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
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
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))
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
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'])
end RPISingularHarnack
end
end OAI
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