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67aee1d | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import Mathlib
import OAI.Analysis.HardSphere.Operators.FreeResolventDenseRange
import OAI.Analysis.HardSphere.Geometry.GeneratorScalar
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
open ProbabilityTheory
namespace HardSphere
open MeasureTheory Set Filter ContinuousLinearMap
open scoped Topology
lemma smoothPartial_smooth {N : ℕ} {L : Length} (f : smoothAlgebra N L) (p : Direction N) :
smoothPartial f p∈smoothAlgebra N L := by
change ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞)
(fun X : LiftedConfiguration N ↦ coordDerivative N L f.val p (project N L X))
simp_rw [coordDerivative_project]
have hf : ContDiff ℝ ((⊤:ℕ∞):WithTop ℕ∞)
(fun X : LiftedConfiguration N ↦ f.val (project N L X)) := f.property
exact (hf.fderiv_right (by simp)).clm_apply contDiff_const
def partialAlgebra {N : ℕ} {L : Length} (f : smoothAlgebra N L) (p : Direction N) :
smoothAlgebra N L := ⟨smoothPartial f p,smoothPartial_smooth f p⟩
def smoothLaplacian {N : ℕ} {L : Length} (f : smoothAlgebra N L) : ScalarL2 N L :=
-∑ p : Direction N, ContinuousMap.toLp 2 volume ℂ (smoothPartial (partialAlgebra f p) p)
lemma smooth_form_pairing {N : ℕ} {L : Length} {a : ℝ} {v : Jet N L}
(hv : v∈dirichletDomain N L a) (f : smoothAlgebra N L) :
(∑ p : Direction N, inner ℂ (ContinuousMap.toLp 2 volume ℂ (smoothPartial f p)) (v.2 p))=
inner ℂ (smoothLaplacian f) v.1 := by
rw [smoothLaplacian,inner_neg_left,sum_inner]
rw [← Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro p _
exact eq_neg_of_add_eq_zero_left (weak_derivative hv (partialAlgebra f p) p)
def SmoothTest.asSmooth {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a) :
smoothAlgebra N L := ⟨⟨f.toFun,f.continuous⟩,f.smooth⟩
lemma SmoothTest.scalar_toLp {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a) :
f.jet.1=ContinuousMap.toLp 2 volume ℂ f.asSmooth.val := by
apply Lp.ext
exact f.memLp.coeFn_toLp.trans (ContinuousMap.coeFn_toLp (𝕜 := ℂ) (p := 2) volume f.asSmooth.val).symm
lemma SmoothTest.derivative_toLp {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a)
(p : Direction N) : f.jet.2 p=ContinuousMap.toLp 2 volume ℂ (smoothPartial f.asSmooth p) := by
apply Lp.ext
exact (f.partial_memLp p).coeFn_toLp.trans
(ContinuousMap.coeFn_toLp (𝕜 := ℂ) (p := 2) volume (smoothPartial f.asSmooth p)).symm
lemma SmoothTest.form_pairing {N : ℕ} {L : Length} {a b : ℝ} (f : SmoothTest N L a)
{v : Jet N L} (hv : v∈dirichletDomain N L b) :
(∑ p : Direction N, inner ℂ (f.jet.2 p) (v.2 p))=inner ℂ (smoothLaplacian f.asSmooth) v.1 := by
simp_rw [f.derivative_toLp]
exact smooth_form_pairing hv f.asSmooth
def SmoothTest.freeTest {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a) :
SmoothTest N L (-1) where
toFun := f.toFun
smooth := f.smooth
compact := f.compact
support_allowed := by rw [allowed_of_neg N L (by norm_num : (-1:ℝ)<0)]; exact subset_univ _
memLp := f.memLp
partial_memLp := f.partial_memLp
lemma SmoothTest.free_resolvent {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a) :
formResolvent N L (-1) (f.jet.1+smoothLaplacian f.asSmooth)=f.jet.1 := by
let ug : hilbertDomain N L (-1) := ⟨(hilbertJetEquiv N L).symm f.jet,by
change hilbertJetEquiv N L ((hilbertJetEquiv N L).symm f.jet)∈dirichletDomain N L (-1)
rw [ContinuousLinearEquiv.apply_symm_apply]
exact subset_closure ⟨f.freeTest,rfl⟩⟩
have heq : (scalarEmbedding N L (-1)).adjoint (f.jet.1+smoothLaplacian f.asSmooth)=ug := by
apply ext_inner_right ℂ
intro v
rw [adjoint_inner_left,inner_add_left]
change _=inner ℂ (ug : HilbertJet N L) (v : HilbertJet N L)
rw [hilbertJet_inner]
have hgu : hilbertJetEquiv N L (ug : HilbertJet N L)=f.jet := by simp [ug]
rw [hgu]
congr 1
exact (f.form_pairing v.property).symm
change scalarEmbedding N L (-1) ((scalarEmbedding N L (-1)).adjoint _)=_
rw [heq]
change (hilbertJetEquiv N L ((hilbertJetEquiv N L).symm f.jet)).1=f.jet.1
rw [ContinuousLinearEquiv.apply_symm_apply]
lemma SmoothTest.free_generator {N : ℕ} {L : Length} {a : ℝ} (f : SmoothTest N L a)
{ι : Type*} {l : Filter ι} {d : ι→ℝ} (hd : ∀ i,0<d i) (ht : Tendsto d l (𝓝 0)) :
Tendsto (fun i ↦ ((d i)⁻¹ : ℂ) • (f.jet.1-freeTransition N L (d i) f.jet.1))
l (𝓝 (smoothLaplacian f.asSmooth)) := by
rcases subsingleton_or_nontrivial (ScalarL2 N L) with h | h
· convert tendsto_const_nhds (x := smoothLaplacian f.asSmooth) using 1
have hlim := heat_difference_resolvent_tendsto hd ht (formResolvent_isPositive N L (-1))
(formResolvent_norm_le_one N L (-1)) (freeResolvent_denseRange N L (-1) (by norm_num))
(f.jet.1+smoothLaplacian f.asSmooth)
simp only [f.free_resolvent] at hlim
change Tendsto (fun i ↦ ((d i)⁻¹:ℂ) • (f.jet.1-formHeat N L (-1) (d i) f.jet.1))
l (𝓝 (f.jet.1+smoothLaplacian f.asSmooth-f.jet.1)) at hlim
simp_rw [← freeTransition_eq_formHeat N L (-1) (by norm_num) (hd _)] at hlim
simpa only [add_sub_cancel_left] using hlim
end HardSphere
end
end OAI
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