| 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 |
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| end ScalarConductivity |
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| end |
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| end OAI |
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