| import OAI.Geometry.Perimeter.Regularity.EllipticHodge |
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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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| def scalarNormLp (u : VectorL2 n) : Lp ℝ 2 (volume : Measure (Euc n)) := |
| (Lp.memLp u).norm.toLp (fun x ↦ ‖u x‖) |
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| theorem scalarNormLp_ae (u : VectorL2 n) : |
| (scalarNormLp u : Euc n → ℝ)=ᵐ[volume] fun x ↦ ‖u x‖ := MemLp.coeFn_toLp _ |
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| theorem scalarNormLp_norm (u : VectorL2 n) : ‖scalarNormLp u‖=‖u‖ := by |
| rw [scalarNormLp,Lp.norm_toLp,eLpNorm_norm,Lp.norm_def] |
| exact (Lp.memLp u).aestronglyMeasurable |
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| def ballL2Mass (n : ℕ) (r : ℝ) : ℝ := |
| ‖indicatorConstLp (μ := (volume : Measure (Euc n))) 2 (measurableSet_ball : MeasurableSet (Metric.ball (0:Euc n) r)) |
| measure_ball_lt_top.ne (1:ℝ)‖ |
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| theorem localL1_le_L2 (u : VectorL2 n) (r : ℝ) : |
| (∫ x in Metric.ball (0:Euc n) r, ‖u x‖) ≤ ballL2Mass n r * ‖u‖ := by |
| rw [←scalarNormLp_norm u] |
| have he : (∫ x in Metric.ball (0:Euc n) r, ‖u x‖)= |
| ∫ x in Metric.ball (0:Euc n) r, scalarNormLp u x := |
| integral_congr_ae (ae_restrict_of_ae (scalarNormLp_ae u).symm) |
| rw [he,←MeasureTheory.L2.inner_indicatorConstLp_one measurableSet_ball measure_ball_lt_top.ne] |
| exact real_inner_le_norm _ _ |
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| theorem WeakHarmonicOn.congr_on {u w : Euc n → ℝ} {U : Set (Euc n)} |
| (hu : WeakHarmonicOn u U) (hw : EqOn u w U) : WeakHarmonicOn w U := by |
| intro φ hφ hcφ hsφ |
| have he : (fun x ↦ w x*divergence (gradient φ) x)= |
| (fun x ↦ u x*divergence (gradient φ) x) := by |
| funext x |
| by_cases hx : x∈U |
| · rw [hw hx] |
| · have hn : x∉tsupport φ := fun hh ↦ hx (hsφ hh) |
| have hz : fderiv ℝ (gradient φ) x=0 := |
| fderiv_of_notMem_tsupport ℝ (fun hh ↦ hn (tsupport_gradient_subset φ hh)) |
| simp [divergence,hz] |
| rw [he] |
| exact hu φ hφ hcφ hsφ |
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| def localScalarComponent (u : VectorL2 n) (a : Euc n) : Euc n → ℝ := |
| (Metric.ball 0 1).indicator (fun x ↦ inner ℝ a (u x)) |
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| theorem localScalarComponent_integrable (u : VectorL2 n) (a : Euc n) : |
| Integrable (localScalarComponent u a) volume := by |
| apply IntegrableOn.integrable_indicator _ measurableSet_ball |
| exact ((((innerSL ℝ a).comp_memLp u).locallyIntegrable (by norm_num)).integrableOn_isCompact |
| (isCompact_closedBall (0:Euc n) 1)).mono_set Metric.ball_subset_closedBall |
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| theorem localScalarComponent_L1 (u : VectorL2 n) (a : Euc n) : |
| (∫ x, ‖localScalarComponent u a x‖) ≤ ‖a‖*ballL2Mass n 1*‖u‖ := by |
| have he : (fun x ↦ ‖localScalarComponent u a x‖)= |
| (Metric.ball 0 1).indicator (fun x ↦ ‖inner ℝ a (u x)‖) := by |
| funext x |
| simp only [localScalarComponent] |
| by_cases hx : x∈Metric.ball (0:Euc n) 1 <;> simp [hx] |
| rw [he,integral_indicator measurableSet_ball] |
| have hI : IntegrableOn (fun x ↦ u x) (Metric.ball (0:Euc n) 1) volume := |
| integrableOn_Lp_of_measure_ne_top u (by norm_num) measure_ball_lt_top.ne |
| calc |
| _ ≤ ∫ x in Metric.ball (0:Euc n) 1, ‖a‖*‖u x‖ := integral_mono_ae |
| (((((innerSL ℝ a).comp_memLp u).locallyIntegrable (by norm_num)).integrableOn_isCompact |
| (isCompact_closedBall (0:Euc n) 1)).mono_set Metric.ball_subset_closedBall).norm |
| (hI.norm.const_mul _) (ae_of_all _ fun x ↦ norm_inner_le_norm a (u x)) |
| _ = ‖a‖*(∫ x in Metric.ball (0:Euc n) 1, ‖u x‖) := integral_const_mul _ _ |
| _ ≤ _ := by rw [mul_assoc]; exact mul_le_mul_of_nonneg_left (localL1_le_L2 u 1) (norm_nonneg a) |
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| theorem localScalarComponent_weakHarmonic {u : VectorL2 n} |
| (hd : DivFreeLpOn u (Metric.ball 0 1)) (hc : CurlFreeLpOn u (Metric.ball 0 1)) |
| (a : Euc n) : WeakHarmonicOn (localScalarComponent u a) (Metric.ball 0 1) := by |
| apply (div_curl_weak_harmonic hd hc a).congr_on |
| intro x hx |
| change inner ℝ a (u x)=(Metric.ball (0:Euc n) 1).indicator (fun z ↦ inner ℝ a (u z)) x |
| rw [Set.indicator_of_mem hx] |
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| end RPISingularHarnack |
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| end |
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| end OAI |
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