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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 | 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
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