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9655094 | 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 | import OAI.Analysis.CircleDomains.Sobolev.SmoothCutoffLinePayment
namespace OAI
/-! The actual smooth balloon cutoffs pay for all unit-speed affine
line segments with their indicator densities. -/
open Set Function Filter MeasureTheory Metric
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
theorem smooth_affine_line_increment {f g : ℂ → ℝ}
(hf : ContDiff ℝ 1 f) (hg : ∀ z, ‖fderiv ℝ f z‖ ≤ g z)
(z e : ℂ) (he : ‖e‖ ≤ 1) {x y : ℝ}
(hgi : IntervalIntegrable (fun t => g (z+t • e)) volume x y) :
|f (z+x • e)-f (z+y • e)| ≤ ∫ t in uIoc x y, g (z+t • e) := by
apply smooth_line_increment_payment (hf.comp (by fun_prop)) hgi
intro t _
have hline : HasDerivAt (fun s : ℝ => z+s • e) e t := by
simpa using ((hasDerivAt_id t).smul_const e).const_add z
have hder := ((hf.differentiable_one (z+t • e)).hasFDerivAt.comp_hasDerivAt t hline).deriv
rw [hder,← Real.norm_eq_abs]
exact ((fderiv ℝ f (z+t • e)).unit_le_opNorm e he).trans (hg _)
theorem balloon_line_density_intervalIntegrable (c z e : ℂ) (d : ℝ) {C : ℝ}
(hC : 0 ≤ C) (x y : ℝ) :
IntervalIntegrable (fun t : ℝ => (ball c d).indicator (fun _ => C) (z+t • e)) volume x y := by
have hline : Continuous (fun t : ℝ => z+t • e) := by fun_prop
have hm : Measurable (fun t : ℝ => (ball c d).indicator (fun _ => C) (z+t • e)) :=
(measurable_const.indicator (s := ball c d) measurableSet_ball).comp hline.measurable
apply intervalIntegrable_iff.mpr
apply (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ => C) volume x y).def'.mono'
hm.aestronglyMeasurable.restrict
apply Eventually.of_forall
intro t
simpa only [Real.norm_of_nonneg hC] using
norm_indicator_le_norm_self (s := ball c d) (fun _ : ℂ => C) (z+t • e)
theorem balloon_cutoff_line_payment {f : ℂ → ℝ} (hf : ContDiff ℝ 1 f)
{c : ℂ} {d C : ℝ} (hC : 0 ≤ C) (hd : 0 < d)
(hder : ∀ z, ‖fderiv ℝ f z‖ ≤ (ball c d).indicator (fun _ => C/d) z)
(z e : ℂ) (he : ‖e‖ ≤ 1) (x y : ℝ) :
|f (z+x • e)-f (z+y • e)| ≤
∫ t in uIoc x y, (ball c d).indicator (fun _ => C/d) (z+t • e) :=
smooth_affine_line_increment hf hder z e he
(balloon_line_density_intervalIntegrable c z e d (div_nonneg hC hd.le) x y)
end CircleDomainRigidity
end
end OAI
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