File size: 3,098 Bytes
159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Model
namespace OAI
noncomputable section
namespace ScalarConductivity
open MeasureTheory Set Filter Topology
open scoped ENNReal
abbrev SpectralL2 (ι : Type*) := lp (fun _ : ι => ℂ) 2
abbrev SpectralJet (ι : Type*) := PiLp 2 (fun _ : Fin 2 => SpectralL2 ι)
def spectralTraceGraph {ι : Type*} (w : ι → ℝ) : Submodule ℂ (SpectralJet ι) where
carrier := {z | ∀ i, (z 1) i=(Real.sqrt (w i):ℂ)*(z 0) i}
zero_mem' := by intro i; simp
add_mem' := by
intro x y hx hy i
change (x 1) i+(y 1) i=(Real.sqrt (w i):ℂ)*((x 0) i+(y 0) i)
rw [hx i,hy i,mul_add]
smul_mem' := by
intro c x hx i
change c*((x 1) i)=(Real.sqrt (w i):ℂ)*(c*((x 0) i))
rw [hx i]; ring
lemma spectralTraceGraph_closed {ι : Type*} (w : ι → ℝ) :
IsClosed (spectralTraceGraph w : Set (SpectralJet ι)) := by
change IsClosed {z : SpectralJet ι | ∀ i, (z 1) i=(Real.sqrt (w i):ℂ)*(z 0) i}
simp only [Set.ofPred_forall]
apply isClosed_iInter
intro i
exact isClosed_eq ((lp.evalCLM ℂ (fun _ : ι => ℂ) 2 i).continuous.comp (PiLp.continuous_apply 2 (fun _ : Fin 2 => SpectralL2 ι) 1))
(continuous_const.mul ((lp.evalCLM ℂ (fun _ : ι => ℂ) 2 i).continuous.comp (PiLp.continuous_apply 2 (fun _ : Fin 2 => SpectralL2 ι) 0)))
instance spectralTraceGraph_complete {ι : Type*} (w : ι → ℝ) :
CompleteSpace (spectralTraceGraph w) := (spectralTraceGraph_closed w).completeSpace_coe
lemma spectralTraceGraph_fst_injective {ι : Type*} (w : ι → ℝ) :
Function.Injective (fun z : spectralTraceGraph w => z.val 0) := by
intro x y h
apply Subtype.ext
apply PiLp.ext
intro j
fin_cases j
· exact h
· ext i
exact (x.property i).trans ((congrArg (fun f : SpectralL2 ι =>
(Real.sqrt (w i):ℂ)*f i) h).trans (y.property i).symm)
lemma sqrt_weight_norm_sq {ι : Type*} {w : ι → ℝ} (hw : ∀ i,0≤w i)
(f : SpectralL2 ι) (i : ι) :
‖(Real.sqrt (w i):ℂ)*f i‖^2=w i*‖f i‖^2 := by
rw [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (Real.sqrt_nonneg _),
Real.sq_sqrt (hw i)]
lemma spectralTraceGraph_exists_iff {ι : Type*} {w : ι → ℝ} (hw : ∀ i,0≤w i)
(f : SpectralL2 ι) :
(∃ z : spectralTraceGraph w, z.val 0=f) ↔ Summable (fun i => w i*‖f i‖^2) := by
constructor
· rintro ⟨z,hz⟩
have hh := (lp.memℓp (z.val 1)).summable (by norm_num : 0<(2:ℝ≥0∞).toReal)
simp only [ENNReal.toReal_ofNat,Real.rpow_two] at hh
have he (i : ι) : ‖(z.val 1) i‖^2=w i*‖f i‖^2 := by
rw [z.property i,hz]
exact sqrt_weight_norm_sq hw f i
simpa only [he] using hh
· intro hs
have hm : Memℓp (fun i => (Real.sqrt (w i):ℂ)*f i) 2 := by
apply (memℓp_gen_iff (by norm_num : 0<(2:ℝ≥0∞).toReal)).mpr
simpa only [ENNReal.toReal_ofNat,Real.rpow_two,sqrt_weight_norm_sq hw f] using hs
let g : SpectralL2 ι := ⟨_,hm⟩
refine ⟨⟨WithLp.toLp 2 ![f,g],?_⟩,?_⟩
· intro i; rfl
· rfl
end ScalarConductivity
end
end OAI
|