File size: 4,043 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import OAI.Geometry.Perimeter.Regularity.EllipticHodge
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
def scalarNormLp (u : VectorL2 n) : Lp ℝ 2 (volume : Measure (Euc n)) :=
(Lp.memLp u).norm.toLp (fun x ↦ ‖u x‖)
theorem scalarNormLp_ae (u : VectorL2 n) :
(scalarNormLp u : Euc n → ℝ)=ᵐ[volume] fun x ↦ ‖u x‖ := MemLp.coeFn_toLp _
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
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)‖
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 _ _
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
def localScalarComponent (u : VectorL2 n) (a : Euc n) : Euc n → ℝ :=
(Metric.ball 0 1).indicator (fun x ↦ inner ℝ a (u x))
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
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)
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]
end RPISingularHarnack
end
end OAI
|