| import Mathlib |
| import OAI.Analysis.HardSphere.FormDomain.Slice |
|
|
| 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 integrable_comp_insertNth {n : ℕ} {L : Length} {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Configuration (n+1) L → E} |
| (hF : Integrable F volume) (j : Fin (n+1)) : |
| Integrable (fun z : Torus L × Configuration n L ↦ F (j.insertNth z.1 z.2)) (volume.prod volume) := by |
| let e := MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) j |
| have he := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) j).symm |
| have hi := (he.integrable_comp_emb e.symm.measurableEmbedding).mpr hF |
| simpa only [MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv, |
| Equiv.coe_fn_mk,Function.comp_def,MeasureTheory.Measure.volume_eq_prod] using hi |
|
|
| lemma integral_configuration_insertNth {n : ℕ} {L : Length} {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] (F : Configuration (n+1) L → E) |
| (hF : Integrable F volume) (j : Fin (n+1)) : |
| ∫ X, F X = ∫ x : Torus L, ∫ Y : Configuration n L, F (j.insertNth x Y) := by |
| have he := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) j).symm |
| rw [← he.integral_comp' F] |
| simp only [MeasurableEquiv.piFinSuccAbove_symm_apply,Fin.insertNthEquiv, |
| Equiv.coe_fn_mk] |
| exact integral_prod _ (integrable_comp_insertNth hF j) |
|
|
| lemma SmoothTest.slice_norm_sq_integrable {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) : |
| Integrable (fun x : Torus L ↦ ‖(f.slice j x).jet.1‖^2) volume := by |
| simp only [smoothTest_norm_sq,SmoothTest.slice] |
| exact (integrable_comp_insertNth (f.memLp.integrable_norm_pow (by decide : (2:ℕ)≠0)) j).integral_prod_left |
|
|
| lemma SmoothTest.integral_slice_norm_sq {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) : |
| (∫ x : Torus L, ‖(f.slice j x).jet.1‖^2)=‖f.jet.1‖^2 := by |
| simp only [smoothTest_norm_sq,SmoothTest.slice] |
| exact (integral_configuration_insertNth _ (f.memLp.integrable_norm_pow (by decide : (2:ℕ)≠0)) j).symm |
|
|
| lemma SmoothTest.slice_partial_integrable {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) (p : Direction n) : |
| Integrable (fun x : Torus L ↦ ‖(f.slice j x).jet.2 p‖^2) volume := by |
| simp only [smoothTest_partial_norm_sq,SmoothTest.slice,coordDerivative_slice] |
| exact (integrable_comp_insertNth ((f.partial_memLp (j.succAbove p.1,p.2)).integrable_norm_pow |
| (by decide : (2:ℕ)≠0)) j).integral_prod_left |
|
|
| lemma SmoothTest.integral_slice_partial {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) (p : Direction n) : |
| (∫ x : Torus L, ‖(f.slice j x).jet.2 p‖^2)=‖f.jet.2 (j.succAbove p.1,p.2)‖^2 := by |
| simp only [smoothTest_partial_norm_sq,SmoothTest.slice,coordDerivative_slice] |
| exact (integral_configuration_insertNth _ ((f.partial_memLp (j.succAbove p.1,p.2)).integrable_norm_pow |
| (by decide : (2:ℕ)≠0)) j).symm |
|
|
| lemma SmoothTest.slice_energy_integrable {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) : |
| Integrable (fun x : Torus L ↦ energy (f.slice j x).jet) volume := |
| integrable_finsetSum _ (fun p _ ↦ f.slice_partial_integrable j p) |
|
|
| lemma SmoothTest.integral_slice_energy {n : ℕ} {L : Length} {a : ℝ} |
| (f : SmoothTest (n+1) L a) (j : Fin (n+1)) : |
| (∫ x : Torus L, energy (f.slice j x).jet)= |
| energy f.jet - ∑ q : Fin 3, ‖f.jet.2 (j,q)‖^2 := by |
| simp only [energy] |
| rw [integral_finsetSum _ (fun p _ ↦ f.slice_partial_integrable j p)] |
| simp only [f.integral_slice_partial,Fintype.sum_prod_type] |
| rw [Fin.sum_univ_succAbove (fun i : Fin (n+1) ↦ ∑ q : Fin 3, ‖f.jet.2 (i,q)‖^2) j] |
| ring |
|
|
| lemma rayleigh_successor {n : ℕ} {L : Length} {a c : ℝ} (hn : 0 < n) |
| (h : ∀ u : Jet n L, u∈dirichletDomain n L a → c*(n:ℝ)*‖u.1‖^2≤energy u) : |
| ∀ v : Jet (n+1) L, v∈dirichletDomain (n+1) L a → c*((n+1:ℕ):ℝ)*‖v.1‖^2≤energy v := by |
| have htest (f : SmoothTest (n+1) L a) : c*((n+1:ℕ):ℝ)*‖f.jet.1‖^2≤energy f.jet := by |
| have hi (j : Fin (n+1)) : c*(n:ℝ)*‖f.jet.1‖^2≤energy f.jet-∑ q : Fin 3, ‖f.jet.2 (j,q)‖^2 := by |
| have hh := integral_mono ((f.slice_norm_sq_integrable j).const_mul (c*(n:ℝ))) |
| (f.slice_energy_integrable j) (fun x ↦ h (f.slice j x).jet (subset_closure ⟨f.slice j x,rfl⟩)) |
| rw [integral_const_mul,f.integral_slice_norm_sq,f.integral_slice_energy] at hh |
| exact hh |
| have hs := Finset.sum_le_sum (s := (Finset.univ : Finset (Fin (n+1)))) (fun j _ ↦ hi j) |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul, |
| Finset.sum_sub_distrib] at hs |
| have he : (∑ j : Fin (n+1), ∑ q : Fin 3, ‖f.jet.2 (j,q)‖^2)=energy f.jet := by |
| rw [energy,Fintype.sum_prod_type] |
| rw [he] at hs |
| rw [Nat.cast_add,Nat.cast_one] at hs ⊢ |
| have hn' : (0:ℝ)<n := by exact_mod_cast hn |
| nlinarith |
| intro v hv |
| have hc : IsClosed {v : Jet (n+1) L | c*((n+1:ℕ):ℝ)*‖v.1‖^2≤energy v} := by |
| apply isClosed_le |
| · exact continuous_const.mul (continuous_fst.norm.pow 2) |
| · exact continuous_finsetSum _ (fun p _ ↦ ((continuous_apply p).comp continuous_snd).norm.pow 2) |
| exact hc.closure_subset_iff.mpr (by rintro _ ⟨f,rfl⟩; exact htest f) hv |
| |
| end HardSphere |
| |
| end |
| |
| end OAI |
| |
