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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Geometry.Perimeter.Regularity.EllipticHarmonicBounds
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal ContDiff InnerProductSpace
namespace RPISingularHarnack
variable {n : ℕ}
theorem weak_curl_harmonic_comparison_norm {U : Set (Euc n)} {u e : VectorL2 n}
(hu : CurlFreeLpOn u U)
(h : ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=
∫ x, inner ℝ (e x) (gradient φ.val x)) :
∃ g : VectorL2 n, DivFreeLpOn g U ∧ CurlFreeLpOn g U ∧ ‖u-g‖≤‖e‖ ∧ ‖g‖≤‖u‖ := by
let K := testGradientSpace U
let g := u-K.starProjection u
have hmem : u-g∈K := by
dsimp [g]
rw [sub_sub_cancel]
exact (K.orthogonalProjectionOnto u).property
have hcf := hu.sub (testGradientSpace_curlFree hmem)
refine ⟨g,orthogonal_testGradient_divFree (K.sub_starProjection_mem_orthogonal u),?_,?_,?_⟩
· simpa only [sub_sub_cancel] using hcf
· exact harmonic_projection_error_le K
(testGradient_pairing_ext (fun φ ↦ by simpa only [testGradient_inner] using h φ))
· rw [show g=Kᗮ.starProjection u from (K.starProjection_orthogonal_val u).symm]
exact Kᗮ.norm_starProjection_apply_le u
theorem vectorLp_integral_sq (u : VectorL2 n) : (∫ x, ‖u x‖^2)=‖u‖^2 := by
rw [←real_inner_self_eq_norm_sq,MeasureTheory.L2.inner_def]
simp only [real_inner_self_eq_norm_sq]
theorem vectorLp_integrable_sq (u : VectorL2 n) : Integrable (fun x ↦ ‖u x‖^2) volume := by
simpa only [real_inner_self_eq_norm_sq] using (MeasureTheory.L2.integrable_inner (𝕜 := ℝ) u u)
theorem vectorLp_setIntegral_sq_le (u : VectorL2 n) (s : Set (Euc n)) :
(∫ x in s, ‖u x‖^2)≤‖u‖^2 := by
rw [←vectorLp_integral_sq]
exact setIntegral_le_integral (vectorLp_integrable_sq u) (ae_of_all _ fun _ ↦ sq_nonneg _)
theorem norm_sub_sq_le_two {V : Type*} [SeminormedAddCommGroup V] (x y z : V) :
‖x-z‖^2≤2*‖x-y‖^2+2*‖y-z‖^2 := by
have h := norm_add_le (x-y) (y-z)
rw [sub_add_sub_cancel] at h
have h' : ‖x-z‖^2≤(‖x-y‖+‖y-z‖)^2 := pow_le_pow_left₀ (norm_nonneg _) h 2
nlinarith [sq_nonneg (‖x-y‖-‖y-z‖)]
theorem weak_field_one_scale (hn : 0<n) :
∃ C : ℝ, 0≤C ∧ ∀ u e : VectorL2 n,
CurlFreeLpOn u (Metric.ball 0 1) →
(∀ φ : SmoothLocalTest (Metric.ball 0 1),
(∫ x, inner ℝ (u x) (gradient φ.val x))=∫ x, inner ℝ (e x) (gradient φ.val x)) →
∃ q : Euc n, ‖q‖≤C*‖u‖ ∧ ∀ θ : ℝ, θ≤1/2 →
(∫ x in Metric.ball 0 θ, ‖u x-q‖^2) ≤
2*‖e‖^2+2*(C*‖u‖*θ)^2*volume.real (Metric.ball (0:Euc n) θ) := by
obtain ⟨C,hC,hB⟩ := harmonic_field_representative_bounds hn
refine ⟨C,hC,fun u e hu h ↦ ?_⟩
obtain ⟨g,hd,hc,he,hgn⟩ := weak_curl_harmonic_comparison_norm hu h
obtain ⟨G,hG,hGe,hG0,hGL⟩ := hB g hd hc
refine ⟨G 0,(hG0 0).trans (mul_le_mul_of_nonneg_left hgn hC),fun θ hθ ↦ ?_⟩
by_cases hθ0 : 0<θ
swap
· have hz : Metric.ball (0:Euc n) θ=∅ := Metric.ball_eq_empty.mpr (le_of_not_gt hθ0)
simp only [hz,Measure.restrict_empty,integral_zero_measure,measureReal_empty,mul_zero,add_zero]
positivity
have : IsFiniteMeasure (volume.restrict (Metric.ball (0:Euc n) θ)) := ⟨by
rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
have hiug : Integrable (fun x ↦ ‖u x-g x‖^2) volume :=
(vectorLp_integrable_sq (u-g)).congr (by filter_upwards [Lp.coeFn_sub u g] with x hx; rw [hx]; rfl)
have hiu : IntegrableOn (fun x ↦ ‖u x-G 0‖^2) (Metric.ball 0 θ) volume := by
have hcst : MemLp (fun _ : Euc n ↦ G 0) 2 (volume.restrict (Metric.ball 0 θ)) := memLp_const _
exact (((Lp.memLp u).mono_measure Measure.restrict_le_self).sub hcst).integrable_norm_pow (by norm_num)
have hib : IntegrableOn (fun x ↦ 2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2) (Metric.ball 0 θ) volume :=
(hiug.integrableOn.const_mul 2).add (integrable_const _)
have hb : ∀ᵐ x ∂volume.restrict (Metric.ball (0:Euc n) θ),
‖u x-G 0‖^2≤2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2 := by
filter_upwards [ae_restrict_of_ae (ae_restrict_iff' measurableSet_ball |>.mp hGe),
ae_restrict_mem measurableSet_ball] with x hx hxB
have hxH : x∈Metric.ball (0:Euc n) (1/2) := Metric.ball_subset_ball hθ hxB
have hnorm : ‖g x-G 0‖≤C*‖u‖*θ := by
rw [hx hxH]
have hb' := hGL x 0
rw [sub_zero] at hb'
exact hb'.trans (mul_le_mul (mul_le_mul_of_nonneg_left hgn hC)
(by exact (show ‖x‖<θ by simpa only [Metric.mem_ball,dist_zero_right] using hxB).le)
(norm_nonneg _) (mul_nonneg hC (norm_nonneg _)))
exact (norm_sub_sq_le_two (u x) (g x) (G 0)).trans (by
have hh := pow_le_pow_left₀ (norm_nonneg _) hnorm 2
linarith)
calc
_ ≤ ∫ x in Metric.ball 0 θ, 2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2 := integral_mono_ae hiu hib hb
_ = 2*(∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)+
2*(C*‖u‖*θ)^2*volume.real (Metric.ball (0:Euc n) θ) := by
rw [integral_add (hiug.integrableOn.const_mul 2) (integrable_const _),integral_const_mul,integral_const]
simp only [smul_eq_mul,measureReal_def,Measure.restrict_apply_univ]; ring
_ ≤ _ := by
have hsq : (∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)≤‖u-g‖^2 := by
have heq : (∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)=∫ x in Metric.ball 0 θ, ‖(u-g) x‖^2 :=
integral_congr_ae (ae_restrict_of_ae (by filter_upwards [Lp.coeFn_sub u g] with x hx; rw [hx]; rfl))
rw [heq]
exact vectorLp_setIntegral_sq_le _ _
have hsq' := pow_le_pow_left₀ (norm_nonneg _) he 2
linarith
end RPISingularHarnack
end
end OAI
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