File size: 2,835 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 | import OAI.Geometry.Perimeter.Regularity.EllipticOneScale
import OAI.Geometry.Perimeter.Regularity.EllipticC1Parts
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem continuous_field_ball_memLp {W : Euc n → Euc n} (hW : Continuous W)
(p : Euc n) (r : ℝ) : MemLp ((Metric.ball p r).indicator W) 2 volume := by
rw [memLp_indicator_iff_restrict measurableSet_ball]
apply (memLp_two_iff_integrable_sq_norm hW.aestronglyMeasurable).mpr
exact ((hW.norm.pow 2).continuousOn.integrableOn_compact (isCompact_closedBall p r)).mono_set
Metric.ball_subset_closedBall
def cutFieldLp (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) : VectorL2 n :=
(continuous_field_ball_memLp hW p r).toLp ((Metric.ball p r).indicator W)
theorem cutFieldLp_ae (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) :
(cutFieldLp W hW p r : Euc n → Euc n)=ᵐ[volume] (Metric.ball p r).indicator W :=
MemLp.coeFn_toLp _
theorem cutFieldLp_norm_sq (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) :
‖cutFieldLp W hW p r‖^2 = ∫ x in Metric.ball p r, ‖W x‖^2 := by
rw [←vectorLp_integral_sq]
calc
(∫ x, ‖cutFieldLp W hW p r x‖^2) = ∫ x, (Metric.ball p r).indicator (fun x ↦ ‖W x‖^2) x := by
apply integral_congr_ae
filter_upwards [cutFieldLp_ae W hW p r] with x hx
rw [hx]
by_cases hb : x∈Metric.ball p r <;> simp [hb]
_ = _ := integral_indicator measurableSet_ball
theorem cutFieldLp_pairing (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ)
{g : Euc n → Euc n} (hg : tsupport g ⊆ Metric.ball p r) :
(∫ x, inner ℝ (cutFieldLp W hW p r x) (g x)) = ∫ x, inner ℝ (W x) (g x) := by
apply integral_congr_ae
filter_upwards [cutFieldLp_ae W hW p r] with x hx
rw [hx]
by_cases hb : x∈Metric.ball p r
· simp [hb]
· have hz : g x=0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hb (hg h))
simp [hz]
theorem cutGradient_curlFree {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) (p : Euc n) (r : ℝ) :
CurlFreeLpOn (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r) (Metric.ball p r) := by
intro φ a b
rw [MeasureTheory.L2.inner_def]
calc
(∫ x, inner ℝ (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r x) (curlTestLp φ a b x)) =
∫ x, inner ℝ (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r x) (curlTest φ.val a b x) := by
apply integral_congr_ae
filter_upwards [curlTestLp_ae φ a b] with x hx
rw [hx]
_ = _ := cutFieldLp_pairing _ _ _ _ ((curlTest_tsupport _ _ _).trans φ.property.2.2)
_ = 0 := C1_gradient_curl_pairing_zero hf φ a b
end RPISingularHarnack
end
end OAI
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