File size: 3,577 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 70 71 | import OAI.Geometry.Perimeter.Regularity.EllipticScaleStep
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem real_volume_ball_center (p : Euc n) (r : ℝ) :
volume.real (Metric.ball p r)=volume.real (Metric.ball (0:Euc n) r) := by
exact Measure.addHaar_real_ball_center volume p r
theorem scaledGradientEnergy_lower {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
(p : Euc n) {r b : ℝ} (hr : 0<r) (q : Euc n)
(hb : ∀ x∈Metric.ball p r, b≤‖gradient f x-q‖^2) :
b*volume.real (Metric.ball (0:Euc n) 1) ≤ scaledGradientEnergy f p r q := by
have hI : 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
have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by
rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
have hh : (∫ x in Metric.ball p r, b) ≤ gradientEnergy f p r q := by
apply integral_mono_ae (integrable_const _) hI
filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
exact hb x hx
simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul] at hh
rw [real_volume_ball_center,real_volume_ball_scale hr] at hh
have hh' := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (pow_nonneg hr.le n))
change _ ≤ (r^n)⁻¹*gradientEnergy f p r q
convert hh' using 1
field_simp [hr.ne']
theorem gradient_eq_of_scaled_energy {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
(p : Euc n) {r ε : ℕ → ℝ} {q : ℕ → Euc n} {w : Euc n}
(hr : ∀ j,0<r j) (hrlim : Tendsto r atTop (𝓝 0))
(hε : Tendsto ε atTop (𝓝 0)) (hq : Tendsto q atTop (𝓝 w))
(hE : ∀ j, scaledGradientEnergy f p (r j) (q j) ≤ ε j) : gradient f p=w := by
by_contra hne
let d := ‖gradient f p-w‖
have hd : 0<d := norm_pos_iff.mpr (sub_ne_zero.mpr hne)
obtain ⟨s,hs,hgs⟩ := Metric.continuousAt_iff.mp (gradient_continuous_C1 hf).continuousAt (d/4) (by positivity)
have hv : 0<volume.real (Metric.ball (0:Euc n) 1) := by
exact ENNReal.toReal_pos (by exact ne_of_gt (Metric.measure_ball_pos volume _ (by norm_num))) measure_ball_lt_top.ne
have hsmall := hε.eventually (gt_mem_nhds (show 0<(d/2)^2*volume.real (Metric.ball (0:Euc n) 1) by positivity))
have hrsmall := hrlim.eventually (gt_mem_nhds hs)
have hqsmall := (Metric.tendsto_atTop.mp hq) (d/4) (by positivity)
obtain ⟨N,hN⟩ := hqsmall
obtain ⟨j,hjE,hjr,hjq⟩ := (hsmall.and (hrsmall.and (eventually_ge_atTop N))).exists
have hlo := scaledGradientEnergy_lower hf p (hr j) (q j) (b := (d/2)^2) (fun x hx ↦ by
have hg : ‖gradient f x-gradient f p‖<d/4 := by
rw [←dist_eq_norm]
exact hgs (by exact (Metric.mem_ball.mp hx).trans hjr)
have hq' : ‖q j-w‖<d/4 := by simpa only [dist_eq_norm] using hN j hjq
have htri : d≤‖gradient f x-gradient f p‖+‖gradient f x-q j‖+‖q j-w‖ := by
have hh := norm_add_le (gradient f p-gradient f x) (gradient f x-q j)
have hh' := norm_add_le (gradient f p-q j) (q j-w)
simp only [sub_add_sub_cancel,norm_sub_rev (gradient f p) (gradient f x)] at hh hh'
dsimp [d]
linarith
have hn : d/2≤‖gradient f x-q j‖ := by linarith
exact pow_le_pow_left₀ (by positivity) hn 2)
linarith [hE j]
end RPISingularHarnack
end
end OAI
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