| import OAI.Geometry.Perimeter.Regularity.EllipticOneScale |
| import OAI.Geometry.Perimeter.Regularity.EllipticC1Parts |
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| namespace OAI |
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| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
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|
| 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 |
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| 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) |
|
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| 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 _ |
|
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| 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 |
|
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| 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 |
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|
| end RPISingularHarnack |
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| end |
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| end OAI |
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