| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CentralVariational |
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| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory |
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| variable {V G : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [NormedAddCommGroup G] [InnerProductSpace ℝ G] |
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| def centralWeightedForm {ι : Type*} (κ : ℝ) (w : ι → ℝ) (D : V →L[ℝ] G) |
| (T : Fin 3 → V →L[ℝ] spectralTraceGraph w) : V →L[ℝ] V →L[ℝ] ℝ := |
| (((innerSL ℝ).comp D).flip.comp D).flip + κ • ∑ i : Fin 3, |
| (((innerSL ℝ).comp ((spectralGraphWeight w).comp (T i))).flip.comp |
| ((spectralGraphWeight w).comp (T i))).flip |
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| lemma centralWeightedForm_apply {ι : Type*} (κ : ℝ) (w : ι → ℝ) (D : V →L[ℝ] G) |
| (T : Fin 3 → V →L[ℝ] spectralTraceGraph w) (u v : V) : |
| centralWeightedForm κ w D T u v=inner ℝ (D u) (D v)+ |
| κ*(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i u)) (spectralGraphWeight w (T i v))) := by |
| simp [centralWeightedForm,smul_eq_mul] |
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| theorem central_variational_exists_graph_weighted [CompleteSpace V] {ι : Type*} [Zero ι] |
| (κ : ℝ) (hκ : 1≤κ) (w : ι → ℝ) (D : V →L[ℝ] G) |
| (T : Fin 3 → V →L[ℝ] spectralTraceGraph w) |
| (M : V →L[ℝ] ℝ) (oneV : V) (hM : M oneV=1) (hD : D oneV=0) |
| (hT : ∀ i,spectralGraphWeight w (T i oneV)=0) |
| (havg : ∀ i,spectralGraphMean w (T i oneV)=1) |
| {C : ℝ} (hC : 0<C) |
| (hP : ∀ u : M.ker,‖u‖^2≤C*(‖D u.val‖^2+ |
| ∑ i : Fin 3,‖spectralGraphWeight w (T i u.val)‖^2)) |
| (s : Fin 3 → ℝ) (hs : ∑ i,s i=0) : |
| ∃ p : V,M p=0 ∧ |
| (∀ v : V,inner ℝ (D p) (D v)+ |
| κ*(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i p)) |
| (spectralGraphWeight w (T i v)))= |
| κ*(∑ i : Fin 3,s i*spectralGraphMean w (T i v))) ∧ |
| (∀ q : V,(∀ v : V,inner ℝ (D q) (D v)+ |
| κ*(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i q)) |
| (spectralGraphWeight w (T i v)))= |
| κ*(∑ i : Fin 3,s i*spectralGraphMean w (T i v))) → q-p=M q • oneV) := by |
| let B := centralWeightedForm κ w D T |
| let F : V →L[ℝ] ℝ := κ • ∑ i : Fin 3,s i • (spectralGraphMean w).comp (T i) |
| have hFa (v : V) : F v=κ*(∑ i : Fin 3,s i*spectralGraphMean w (T i v)) := by |
| simp [F,smul_eq_mul] |
| have hF : F oneV=0 := by rw [hFa]; simp only [havg,mul_one,hs,mul_zero] |
| have hB (u : V) : B u oneV=0 := by |
| simp only [B,centralWeightedForm_apply,hD,inner_zero_right,hT,Finset.sum_const_zero,mul_zero,add_zero] |
| have hB' (v : V) : B oneV v=0 := by |
| simp only [B,centralWeightedForm_apply,hD,inner_zero_left,hT,Finset.sum_const_zero,mul_zero,zero_add] |
| have hcoer : IsCoercive (((B.comp M.ker.subtypeL).flip.comp M.ker.subtypeL).flip) := by |
| refine ⟨C⁻¹,inv_pos.mpr hC,?_⟩ |
| intro u |
| change C⁻¹*‖u‖*‖u‖≤B u.val u.val |
| have hn : 0≤∑ i : Fin 3,‖spectralGraphWeight w (T i u.val)‖^2 := |
| Finset.sum_nonneg (fun i _ => sq_nonneg _) |
| have hb : ‖D u.val‖^2+∑ i : Fin 3,‖spectralGraphWeight w (T i u.val)‖^2≤B u.val u.val := by |
| simp only [B,centralWeightedForm_apply,real_inner_self_eq_norm_sq] |
| nlinarith [mul_le_mul_of_nonneg_right hκ hn] |
| have hh := (hP u).trans (mul_le_mul_of_nonneg_left hb hC.le) |
| calc |
| _=‖u‖^2/C := by ring |
| _≤B u.val u.val := (div_le_iff₀ hC).mpr (by simpa only [mul_comm] using hh) |
| obtain ⟨p,hp,hfull,huniq⟩ := solve_form_mod_constants B M F oneV hM hF hB hB' hcoer |
| refine ⟨p,hp,?_,?_⟩ |
| · intro v |
| exact (centralWeightedForm_apply κ w D T p v).symm.trans ((hfull v).trans (hFa v)) |
| · intro q hq |
| apply huniq q |
| intro v |
| exact (centralWeightedForm_apply κ w D T q v).trans ((hq v).trans (hFa v).symm) |
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| end ScalarConductivity |
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| end |
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| end OAI |
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