| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.NormSqIntegral | |
| import OAI.Analysis.HardSphere.Comparison.KilledLaplace | |
| import OAI.Analysis.HardSphere.FormDomain.DirichletScalarInjective | |
| 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 | |
| namespace HardSphere | |
| open MeasureTheory Set Filter | |
| open scoped Topology | |
| lemma killedLaplace_innerRadius_tendsto (N : ℕ) (L : Length) (a : ℝ) (f : ScalarL2 N L) : | |
| Tendsto (fun n ↦ killedLaplace N L (innerRadius a n) f) atTop (𝓝 (killedLaplace N L a f)) := by | |
| apply tendsto_integral_of_dominated_convergence (fun t : ℝ ↦ Real.exp (-t)*‖f‖) | |
| · intro n | |
| exact (laplaceIntegrand_integrable (killedTransition N L (innerRadius a n)) | |
| (killedTransition_clamped_continuous N L (innerRadius a n)) | |
| (fun t _ ↦ killedTransition_apply_contraction N L (innerRadius a n) t) f 0).aestronglyMeasurable | |
| · exact (integrableOn_exp_neg_Ioi 0).mul_const ‖f‖ | |
| · intro n | |
| apply Eventually.of_forall | |
| intro t | |
| rw [laplaceIntegrand_eq_complex] | |
| rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] | |
| exact mul_le_mul_of_nonneg_left (killedTransition_apply_contraction N L (innerRadius a n) _ f) | |
| (Real.exp_nonneg _) | |
| · apply Eventually.of_forall | |
| intro t | |
| simp_rw [laplaceIntegrand_eq_complex] | |
| exact tendsto_const_nhds.smul (killedTransition_innerRadius_tendsto N L a (max 0 t) f) | |
| def bufferedAllowed (N : ℕ) (L : Length) (b : ℝ) : Set (Configuration N L) := | |
| {X | ∀ i j : Fin N, i<j → b≤torusDistance L (X i) (X j)} | |
| lemma bufferedAllowed_closed (N : ℕ) (L : Length) (b : ℝ) : IsClosed (bufferedAllowed N L b) := by | |
| let P := {p : Fin N × Fin N // p.1 < p.2} | |
| have he : bufferedAllowed N L b = ⋂ p : P, | |
| {X : Configuration N L | b≤torusDistance L (X p.1.1) (X p.1.2)} := by | |
| ext X | |
| simp only [bufferedAllowed,mem_ofPred_eq,mem_iInter] | |
| exact ⟨fun h p ↦ h _ _ p.2,fun h i j hij ↦ h ⟨(i,j),hij⟩⟩ | |
| rw [he] | |
| apply isClosed_iInter | |
| intro p | |
| have hc : Continuous (fun X : Configuration N L ↦ torusDistance L (X p.1.1) (X p.1.2)) := by | |
| unfold torusDistance | |
| fun_prop | |
| exact isClosed_le continuous_const hc | |
| lemma allowed_subset_buffered (N : ℕ) (L : Length) (b : ℝ) : | |
| allowed N L b⊆bufferedAllowed N L b := fun _ h i j hij ↦ (h i j hij).le | |
| lemma bufferedAllowed_subset (N : ℕ) (L : Length) {a b : ℝ} (hab : a<b) : | |
| bufferedAllowed N L b⊆allowed N L a := fun _ h i j hij ↦ hab.trans_le (h i j hij) | |
| lemma killedLaplace_buffered_graph (N : ℕ) (L : Length) {a b : ℝ} (hab : a<b) (f : ScalarL2 N L) : | |
| ∃ v : hilbertDomain N L a, scalarEmbedding N L a v=killedLaplace N L b f ∧ | |
| ‖v‖^2≤(1+4*Real.exp 1)*‖f‖^2 := by | |
| obtain ⟨u,hu,hub⟩ := killedLaplace_free_graph N L b f | |
| have hj : hilbertJetEquiv N L (u : HilbertJet N L)∈dirichletDomain N L a := by | |
| apply dirichletDomain_of_closed_support (isOpen_allowed N L a) | |
| (bufferedAllowed_closed N L b) (bufferedAllowed_subset N L hab) u.property | |
| have he : (hilbertJetEquiv N L (u : HilbertJet N L)).1=killedLaplace N L b f := hu | |
| change ∀ᵐ X ∂(volume : Measure (Configuration N L)), X∉bufferedAllowed N L b → | |
| (hilbertJetEquiv N L (u : HilbertJet N L)).1 X=0 | |
| rw [he] | |
| filter_upwards [killedLaplace_support N L b f] with X hX | |
| intro hn | |
| exact hX (fun ha ↦ hn (allowed_subset_buffered N L b ha)) | |
| let v : hilbertDomain N L a := ⟨u.val,hj⟩ | |
| refine ⟨v,hu,?_⟩ | |
| have hn := killedLaplace_norm_le N L b f | |
| change ‖u‖^2≤_ | |
| nlinarith [norm_nonneg (killedLaplace N L b f),norm_nonneg f] | |
| lemma killedLaplace_graph (N : ℕ) (L : Length) (a : ℝ) (f : ScalarL2 N L) : | |
| ∃ v : hilbertDomain N L a, scalarEmbedding N L a v=killedLaplace N L a f ∧ | |
| ‖v‖^2≤(1+4*Real.exp 1)*‖f‖^2 := by | |
| choose v hv hb using fun n ↦ killedLaplace_buffered_graph N L (innerRadius_gt a n) f | |
| let C := (1+4*Real.exp 1)*‖f‖^2 | |
| have hC : 0≤C := by dsimp [C]; positivity | |
| have hvb : ∀ n, ‖v n‖≤Real.sqrt C := by | |
| intro n | |
| exact (Real.le_sqrt (norm_nonneg _) hC).mpr (hb n) | |
| have ht : Tendsto (fun n ↦ scalarEmbedding N L a (v n)) atTop (𝓝 (killedLaplace N L a f)) := by | |
| simp_rw [hv] | |
| exact killedLaplace_innerRadius_tendsto N L a f | |
| obtain ⟨u,hub,hu⟩ := exists_bounded_lift_of_tendsto (scalarEmbedding N L a) v hvb ht | |
| refine ⟨u,hu,?_⟩ | |
| calc | |
| ‖u‖^2 ≤ (Real.sqrt C)^2 := sq_le_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _) |>.mpr hub | |
| _ = C := Real.sq_sqrt hC | |
| end HardSphere | |
| open ProbabilityTheory | |
| end | |
| end OAI | |