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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Sobolev.SobolevTraceChain
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
lemma h1_limit_ae_closed {u : ℕ → H1} {w : H1}
(hu : Tendsto u atTop (𝓝 w)) (K : R3 → Set JetFiber)
(hK : ∀ x, IsClosed (K x)) (hm : ∀ n, ∀ᵐ x ∂ballMeasure, (u n).val x∈K x) :
∀ᵐ x ∂ballMeasure, w.val x∈K x := by
have ht : Tendsto (fun n => (u n).val) atTop (𝓝 w.val) :=
continuous_subtype_val.continuousAt.tendsto.comp hu
obtain ⟨φ,hφ,hφae⟩ := (tendstoInMeasure_of_tendsto_Lp ht).exists_seq_tendsto_ae
filter_upwards [hφae,ae_all_iff.mpr hm] with x hx hy
exact (hK x).mem_of_tendsto hx (Eventually.of_forall (fun n => hy (φ n)))
lemma h1_limit_bounded {u : ℕ → H1} {w : H1} {M : ℝ}
(hu : Tendsto u atTop (𝓝 w))
(hm : ∀ n, ∀ᵐ x ∂ballMeasure, |weakValue (u n) x|≤M) :
∀ᵐ x ∂ballMeasure, |weakValue w x|≤M := by
exact h1_limit_ae_closed hu (fun _ => {z | |jetValue z|≤M})
(fun _ => isClosed_le (jetValue.continuous.abs) continuous_const) hm
lemma h1_limit_supported {u : ℕ → H1} {w : H1} {r : ℝ}
(hu : Tendsto u atTop (𝓝 w))
(hm : ∀ n, ∀ᵐ x ∂ballMeasure, r<‖x‖ →
weakValue (u n) x=0 ∧ weakGradient (u n) x=0) :
∀ᵐ x ∂ballMeasure, r<‖x‖ → weakValue w x=0 ∧ weakGradient w x=0 := by
exact h1_limit_ae_closed hu
(fun x => {z | r<‖x‖ → jetValue z=0 ∧ jetGradient z=0})
(fun x => by
by_cases hx : r<‖x‖
· simpa only [hx,true_implies,Set.inter_def,Set.mem_ofPred_eq] using
(isClosed_eq jetValue.continuous continuous_const).inter
(isClosed_eq jetGradient.continuous continuous_const)
· simp only [hx,IsEmpty.forall_iff,Set.ofPred_true]
exact isClosed_univ) hm
lemma weakValue_sum (s : Finset ℕ) (u : ℕ → H1) :
weakValue (∑ n∈s, u n) =ᵐ[ballMeasure] fun x => ∑ n∈s, weakValue (u n) x := by
classical
induction s using Finset.induction_on with
| empty =>
simp only [Finset.sum_empty]
filter_upwards [Lp.coeFn_zero JetFiber 2 ballMeasure] with x hx
change jetValue ((0:JetSpace) x)=0
rw [hx,Pi.zero_apply,map_zero]
| @insert a s ha ih =>
simp only [Finset.sum_insert ha]
filter_upwards [weakValue_add (u a) (∑ i∈s,u i),ih] with x hx hy
simpa only [Pi.add_apply,hy] using hx
lemma weakGradient_sum (s : Finset ℕ) (u : ℕ → H1) :
weakGradient (∑ n∈s, u n) =ᵐ[ballMeasure] fun x => ∑ n∈s, weakGradient (u n) x := by
classical
induction s using Finset.induction_on with
| empty =>
simp only [Finset.sum_empty]
filter_upwards [Lp.coeFn_zero JetFiber 2 ballMeasure] with x hx
change jetGradient ((0:JetSpace) x)=0
rw [hx,Pi.zero_apply,map_zero]
| @insert a s ha ih =>
simp only [Finset.sum_insert ha]
filter_upwards [weakGradient_add (u a) (∑ i∈s,u i),ih] with x hx hy
simpa only [Pi.add_apply,hy] using hx
theorem source_series_limit (Z : ℕ → H1) (b : ℕ → ℝ)
(hnorm : Summable (fun n => ‖Z n‖)) (hb : Summable b)
(hb0 : ∀ n, 0≤b n) (hZ : ∀ n, Z n∈H10)
(hbound : ∀ n, ∀ᵐ x ∂ballMeasure, |weakValue (Z n) x|≤b n)
(r : ℝ) (hsupp : ∀ n, ∀ᵐ x ∂ballMeasure, r<‖x‖ →
weakValue (Z n) x=0 ∧ weakGradient (Z n) x=0) :
∃ w : H1, w∈H10 ∧
Tendsto (fun N => ∑ n∈Finset.range N, Z n) atTop (𝓝 w) ∧
(∀ᵐ x ∂ballMeasure, |weakValue w x|≤∑' n,b n) ∧
(∀ᵐ x ∂ballMeasure, r<‖x‖ → weakValue w x=0 ∧ weakGradient w x=0) := by
classical
have hsum : Summable Z := hnorm.of_norm
refine ⟨∑' n,Z n,?_,hsum.hasSum.tendsto_sum_nat,?_,?_⟩
· exact H10_isClosed.mem_of_tendsto hsum.hasSum.tendsto_sum_nat
(Eventually.of_forall (fun N => H10.sum_mem (fun n _ => hZ n)))
· apply h1_limit_bounded hsum.hasSum.tendsto_sum_nat
intro N
filter_upwards [weakValue_sum (Finset.range N) Z,ae_all_iff.mpr hbound] with x hx hy
rw [hx]
exact (Finset.abs_sum_le_sum_abs _ _).trans
((Finset.sum_le_sum (fun n _ => hy n)).trans (Summable.sum_le_tsum _ (fun n _ => hb0 n) hb))
· apply h1_limit_supported hsum.hasSum.tendsto_sum_nat
intro N
filter_upwards [weakValue_sum (Finset.range N) Z,weakGradient_sum (Finset.range N) Z,
ae_all_iff.mpr hsupp] with x hv hg hx
intro hr
simp only [hv,hg]
exact ⟨Finset.sum_eq_zero (fun n _ => (hx n hr).1),Finset.sum_eq_zero (fun n _ => (hx n hr).2)⟩
end ScalarConductivity
end
end
end OAI
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