| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.CoordinateLine |
| import OAI.Analysis.HardSphere.Geometry.NormSqIntegral |
|
|
| 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 Filter Set |
|
|
| lemma smooth_insertNth {n : ℕ} (j : Fin (n+1)) (x : Fin 3 → ℝ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun Y : LiftedConfiguration n ↦ Fin.insertNth (α := fun _ ↦ Fin 3 → ℝ) j x Y) := by |
| apply contDiff_pi.mpr |
| intro i |
| induction i using j.succAboveCases with |
| | x => simpa only [Fin.insertNth_apply_same] using |
| (contDiff_const : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun _ : LiftedConfiguration n ↦ x)) |
| | p i => simpa only [Fin.insertNth_apply_succAbove] using |
| (contDiff_apply ℝ (Fin 3 → ℝ) i) |
|
|
| lemma smooth_slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a) |
| (j : Fin (n+1)) (x : Torus L) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun Y : LiftedConfiguration n ↦ f.toFun (j.insertNth x (project n L Y))) := by |
| have h := f.smooth.comp (smooth_insertNth j (representative 1 L (fun _ ↦ x) 0)) |
| convert h using 1 |
| funext Y |
| congr 1 |
| ext i q |
| induction i using j.succAboveCases with |
| | x => |
| simp only [project,Fin.insertNth_apply_same] |
| exact (congr_fun (congr_fun (project_representative 1 L (fun _ ↦ x)) 0) q).symm |
| | p i => simp only [project,Fin.insertNth_apply_succAbove] |
|
|
| lemma insertNth_coordinateLine {n : ℕ} {L : Length} (j : Fin (n+1)) (x : Torus L) |
| (Y : Configuration n L) (p : Direction n) (t : AddCircle L.val) : |
| j.insertNth x (coordinateLine Y p t)=coordinateLine (j.insertNth x Y) (j.succAbove p.1,p.2) t := by |
| ext i q |
| induction i using j.succAboveCases with |
| | x => simp [coordinateLine] |
| | p i => simp [coordinateLine,Pi.single_apply,Fin.succAbove_inj] |
|
|
| lemma coordDerivative_slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a) |
| (j : Fin (n+1)) (x : Torus L) (p : Direction n) : |
| coordDerivative n L (fun Y ↦ f.toFun (j.insertNth x Y)) p= |
| fun Y ↦ coordDerivative (n+1) L f.toFun (j.succAbove p.1,p.2) (j.insertNth x Y) := by |
| funext Y |
| have h1 := hasDerivAt_coordinateLine (f := fun Y ↦ f.toFun (j.insertNth x Y)) ((smooth_slice f j x).differentiable (by simp)) Y p 0 |
| have h2 := hasDerivAt_coordinateLine (f.smooth.differentiable (by simp)) |
| (j.insertNth x Y) (j.succAbove p.1,p.2) 0 |
| simp only [insertNth_coordinateLine] at h1 |
| have h := h1.unique h2 |
| simpa [coordinateLine] using h |
|
|
| def SmoothTest.slice {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest (n+1) L a) |
| (j : Fin (n+1)) (x : Torus L) : SmoothTest n L a where |
| toFun Y := f.toFun (j.insertNth x Y) |
| smooth := smooth_slice f j x |
| compact := HasCompactSupport.of_compactSpace _ |
| support_allowed := by |
| intro Y hY i k hik |
| have h := f.support_allowed (tsupport_comp_subset_preimage f.toFun |
| (Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id) hY) |
| simpa only [Fin.insertNth_apply_succAbove,id_eq] using h (j.succAbove i) (j.succAbove k) |
| (Fin.strictMono_succAbove j hik) |
| memLp := (f.continuous.comp (Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id)).memLp_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| partial_memLp := fun p ↦ by |
| rw [coordDerivative_slice] |
| exact ((f.continuous_partial (j.succAbove p.1,p.2)).comp |
| (Continuous.finInsertNth j (continuous_const : Continuous (fun _ : Configuration n L ↦ x)) continuous_id)).memLp_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
|
|
| lemma smoothTest_norm_sq {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest n L a) : |
| ‖f.jet.1‖^2=∫ X : Configuration n L, ‖f.toFun X‖^2 := by |
| rw [IndicatorStrong.norm_sq_integral] |
| exact integral_congr_ae ((MemLp.coeFn_toLp f.memLp).fun_comp (fun z : ℂ ↦ ‖z‖^2)) |
|
|
| lemma smoothTest_partial_norm_sq {n : ℕ} {L : Length} {a : ℝ} (f : SmoothTest n L a) |
| (p : Direction n) : |
| ‖f.jet.2 p‖^2=∫ X : Configuration n L, ‖coordDerivative n L f.toFun p X‖^2 := by |
| rw [IndicatorStrong.norm_sq_integral] |
| exact integral_congr_ae ((MemLp.coeFn_toLp (f.partial_memLp p)).fun_comp (fun z : ℂ ↦ ‖z‖^2)) |
|
|
| end HardSphere |
|
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| end |
|
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| end OAI |
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|
