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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 | import OAI.Geometry.Perimeter.Regularity.EllipticCutField
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
def gradientEnergy (f : Euc n → ℝ) (p : Euc n) (r : ℝ) (q : Euc n) : ℝ :=
∫ x in Metric.ball p r, ‖gradient f x-q‖^2
theorem constant_gradient_pairing_zero {U : Set (Euc n)} (φ : SmoothLocalTest U) (q : Euc n) :
(∫ x, inner ℝ q (gradient φ.val x))=0 := by
have h := (scalar_compact_integration_parts_C1 (u := fun _ : Euc n ↦ (1:ℝ))
contDiff_const (φ.property.1.of_le (by simp)) φ.property.2.1 q).2.2
simpa [inner_gradient_right] using h
theorem inner_gradient_pairing_integrable {W : Euc n → Euc n} (hW : Continuous W)
{U : Set (Euc n)} (φ : SmoothLocalTest U) :
Integrable (fun x ↦ inner ℝ (W x) (gradient φ.val x)) volume :=
(hW.inner (gradient_contDiff φ.property.1).continuous).integrable_of_hasCompactSupport
((gradient_hasCompactSupport φ.property.2.1).of_isClosed_subset (isClosed_tsupport _) (by
apply closure_minimal _ (isClosed_tsupport _)
intro x hx
by_contra hh
exact hx (by simp [image_eq_zero_of_notMem_tsupport hh])))
theorem gradient_sub_linear {f : Euc n → ℝ} (hf : Differentiable ℝ f) (q : Euc n) (x : Euc n) :
gradient (fun y ↦ f y-inner ℝ q y) x=gradient f x-q := by
apply ext_inner_right ℝ
intro y
simp only [inner_gradient_left,inner_sub_left]
have h := ((hf x).hasFDerivAt.sub (innerSL ℝ q).hasFDerivAt).fderiv
exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L y) h
theorem continuous_gradient_one_scale (hn : 0<n) :
∃ C : ℝ, 0≤C ∧ ∀ (f : Euc n → ℝ), ContDiff ℝ 1 f →
∀ (e : Euc n → Euc n), Continuous e →
(∀ φ : SmoothLocalTest (Metric.ball 0 1),
(∫ x, inner ℝ (gradient f x) (gradient φ.val x))=∫ x, inner ℝ (e x) (gradient φ.val x)) →
∃ q : Euc n, ‖q‖^2≤C^2*gradientEnergy f 0 1 0 ∧ ∀ θ : ℝ, θ≤1/2 →
gradientEnergy f 0 θ q ≤ 2*(∫ x in Metric.ball 0 1, ‖e x‖^2)+
2*C^2*gradientEnergy f 0 1 0*θ^2*volume.real (Metric.ball (0:Euc n) θ) := by
obtain ⟨C,hC,hB⟩ := weak_field_one_scale hn
refine ⟨C,hC,fun f hf e he h ↦ ?_⟩
let u := cutFieldLp (gradient f) (gradient_continuous_C1 hf) 0 1
let w := cutFieldLp e he 0 1
have heq : ∀ φ : SmoothLocalTest (Metric.ball (0:Euc n) 1),
(∫ x, inner ℝ (u x) (gradient φ.val x))=∫ x, inner ℝ (w x) (gradient φ.val x) := by
intro φ
rw [cutFieldLp_pairing _ _ _ _ ((tsupport_gradient_subset _).trans φ.property.2.2),
cutFieldLp_pairing _ _ _ _ ((tsupport_gradient_subset _).trans φ.property.2.2)]
exact h φ
obtain ⟨q,hq,hqE⟩ := hB u w (cutGradient_curlFree hf 0 1) heq
have huE : ‖u‖^2=gradientEnergy f 0 1 0 := by
simpa only [gradientEnergy,sub_zero] using cutFieldLp_norm_sq (gradient f) (gradient_continuous_C1 hf) 0 1
refine ⟨q,?_,fun θ hθ ↦ ?_⟩
· calc
‖q‖^2 ≤ (C*‖u‖)^2 := pow_le_pow_left₀ (norm_nonneg _) hq 2
_ = C^2*gradientEnergy f 0 1 0 := by rw [mul_pow,huE]
· have hE := hqE θ hθ
have heU : (∫ x in Metric.ball 0 θ, ‖u x-q‖^2)=gradientEnergy f 0 θ q := by
apply integral_congr_ae
filter_upwards [ae_restrict_of_ae (cutFieldLp_ae (gradient f) (gradient_continuous_C1 hf) 0 1),
ae_restrict_mem measurableSet_ball] with x hx hxB
have hx1 : x∈Metric.ball (0:Euc n) 1 := Metric.ball_subset_ball (by linarith) hxB
rw [hx,Set.indicator_of_mem hx1]
rw [heU,cutFieldLp_norm_sq] at hE
convert hE using 1
rw [mul_pow,mul_pow,huE]
ring
end RPISingularHarnack
end
end OAI
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