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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 72 73 74 75 76 77 78 79 80 81 | import OAI.Geometry.Perimeter.Regularity.EllipticDifference
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem contDiff_cutoff {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U) {k : ℕ∞}
(hf : ContDiffOn ℝ k f U) {χ : Euc n → ℝ} (hχ : ContDiff ℝ k χ)
(hsχ : tsupport χ⊆U) : ContDiff ℝ k (fun x ↦ χ x•f x) := by
rw [contDiff_iff_contDiffAt]
intro x
by_cases hx : x∈tsupport χ
· exact hχ.contDiffAt.smul (hf.contDiffAt (hU.mem_nhds (hsχ hx)))
· apply (contDiffAt_const (c := (0:V))).congr_of_eventuallyEq
filter_upwards [(isClosed_tsupport χ).isOpen_compl.mem_nhds hx] with y hy
simp only [image_eq_zero_of_notMem_tsupport hy,zero_smul]
theorem local_compact_extension {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
{f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
{k : ℕ∞} (hf : ContDiffOn ℝ k f U) :
∃ r : ℝ, 0<r ∧ Metric.closedBall p (4*r)⊆U ∧ ∃ g : Euc n → V,
ContDiff ℝ k g ∧ HasCompactSupport g ∧ ∀ x∈Metric.ball p (2*r), g x=f x := by
obtain ⟨s,hs,hsU⟩ := Metric.isOpen_iff.mp hU p hp
let r := s/8
let χ : ContDiffBump p := ⟨s/4,s/2,by positivity,by linarith⟩
have hsc : tsupport χ⊆U := by
rw [χ.tsupport_eq]
exact (Metric.closedBall_subset_ball (by dsimp [χ]; linarith)).trans hsU
refine ⟨r,by dsimp [r]; positivity,?_,fun x ↦ χ x•f x,
contDiff_cutoff hU hf χ.contDiff hsc,χ.hasCompactSupport.smul_right,?_⟩
· exact (Metric.closedBall_subset_ball (by dsimp [r]; linarith)).trans hsU
· intro x hx
have hh : x∈Metric.closedBall p χ.rIn := by
rw [Metric.mem_closedBall]
have hd := Metric.mem_ball.mp hx
dsimp [χ,r] at *
linarith
change χ x•f x=f x
rw [χ.one_of_mem_closedBall hh,one_smul]
theorem compact_C1_gradient_bound {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
(hc : HasCompactSupport f) : ∃ B : ℝ, 0≤B ∧ ∀ x, ‖gradient f x‖≤B := by
have hg := gradient_continuous_C1 hf
have hgc := gradient_hasCompactSupport hc
obtain ⟨B,hB⟩ := hgc.exists_bound_of_continuousOn hg.continuousOn
refine ⟨max 0 B,le_max_left _ _,fun x ↦ ?_⟩
by_cases hx : x∈tsupport (gradient f)
· exact (hB x hx).trans (le_max_right _ _)
· rw [image_eq_zero_of_notMem_tsupport hx,norm_zero]
exact le_max_left _ _
theorem differenceQuotient_bound {f : Euc n → ℝ} (hf : Differentiable ℝ f)
{B : ℝ} (hB : 0≤B) (hb : ∀ x, ‖gradient f x‖≤B) (e : Euc n) (s : ℝ) (x : Euc n) :
|differenceQuotient f e s x|≤B*‖e‖ := by
have hnorm (y : Euc n) : ‖fderiv ℝ f y‖≤B := by
have he : ‖fderiv ℝ f y‖=‖gradient f y‖ := by
rw [gradient,LinearIsometryEquiv.norm_map]
rw [he]
exact hb y
have hh := (convex_univ : Convex ℝ (univ : Set (Euc n))).norm_image_sub_le_of_norm_fderiv_le
(fun y _ ↦ hf y) (fun y _ ↦ hnorm y) (mem_univ x) (mem_univ (x+s•e))
by_cases hs : s=0
· simp only [differenceQuotient,hs,inv_zero,zero_mul,abs_zero]
positivity
· rw [differenceQuotient,abs_mul,abs_inv]
rw [Real.norm_eq_abs,add_sub_cancel_left,norm_smul,Real.norm_eq_abs] at hh
have h := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (abs_nonneg s))
have he : |s|⁻¹*(B*(|s| *‖e‖))=B*‖e‖ := by field_simp [abs_ne_zero.mpr hs]
exact h.trans_eq he
end RPISingularHarnack
end
end OAI
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