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a458723 | 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 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CentralVariational
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory
variable {V G : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[NormedAddCommGroup G] [InnerProductSpace ℝ G]
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
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]
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)
end ScalarConductivity
end
end OAI
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