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import OAI.Analysis.Conductivity.Sources.WeightedSpectralSpace
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
open scoped ENNReal
def spectralGraphWeight {ι : Type*} (w : ι → ℝ) :
spectralTraceGraph w →L[ℝ] SpectralL2 ι :=
((PiLp.proj 2 (fun _ : Fin 2 => SpectralL2 ι) 1).comp
(spectralTraceGraph w).subtypeL).restrictScalars ℝ
def spectralGraphMean {ι : Type*} [Zero ι] (w : ι → ℝ) :
spectralTraceGraph w →L[ℝ] ℝ :=
Complex.reCLM.comp (((lp.evalCLM ℂ (fun _ : ι => ℂ) 2 0).comp
((PiLp.proj 2 (fun _ : Fin 2 => SpectralL2 ι) 0).comp
(spectralTraceGraph w).subtypeL)).restrictScalars ℝ)
variable {V G : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[NormedAddCommGroup G] [InnerProductSpace ℝ G]
def centralForm {ι : Type*} (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 + ∑ i : Fin 3,
(((innerSL ℝ).comp ((spectralGraphWeight w).comp (T i))).flip.comp
((spectralGraphWeight w).comp (T i))).flip
lemma centralForm_apply {ι : Type*} (w : ι → ℝ) (D : V →L[ℝ] G)
(T : Fin 3 → V →L[ℝ] spectralTraceGraph w) (u v : V) :
centralForm w D T u v=inner ℝ (D u) (D v)+
∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i u)) (spectralGraphWeight w (T i v)) := by
simp [centralForm]
lemma centralForm_lower {ι : Type*} (w : ι → ℝ) (D : V →L[ℝ] G)
(T : Fin 3 → V →L[ℝ] spectralTraceGraph w) (u : V) :
‖D u‖^2≤centralForm w D T u u := by
rw [centralForm_apply,real_inner_self_eq_norm_sq]
exact le_add_of_nonneg_right (Finset.sum_nonneg fun i _ => real_inner_self_nonneg)
theorem solve_form_mod_constants [CompleteSpace V]
(B : V →L[ℝ] V →L[ℝ] ℝ) (M F : V →L[ℝ] ℝ) (oneV : V)
(hM : M oneV=1) (hF : F oneV=0)
(hB : ∀ u : V,B u oneV=0) (hB' : ∀ v : V,B oneV v=0)
(hcoer : IsCoercive (((B.comp M.ker.subtypeL).flip.comp M.ker.subtypeL).flip)) :
∃ p : V, M p=0 ∧ (∀ v : V,B p v=F v) ∧
(∀ q : V,(∀ v : V,B q v=F v) → q-p=M q • oneV) := by
let K := M.ker
have hclosed : IsClosed (K : Set V) := M.isClosed_ker
let : CompleteSpace K := hclosed.completeSpace_coe
let A : K →L[ℝ] K →L[ℝ] ℝ := ((B.comp K.subtypeL).flip.comp K.subtypeL).flip
let L : K →L[ℝ] ℝ := F.comp K.subtypeL
let p : K := hcoer.continuousLinearEquivOfBilin.symm ((InnerProductSpace.toDual ℝ K).symm L)
have hp (v : K) : B p.val v.val=F v.val := by
change A p v=L v
rw [←hcoer.continuousLinearEquivOfBilin_apply]
dsimp only [p]
rw [hcoer.continuousLinearEquivOfBilin.apply_symm_apply]
exact congrArg (fun f : K →L[ℝ] ℝ => f v) ((InnerProductSpace.toDual ℝ K).apply_symm_apply L)
have hpM : M p.val=0 := p.property
have hfull (v : V) : B p.val v=F v := by
have hv : v-M v • oneV∈K := by
change M (v-M v • oneV)=0
simp [hM]
have hh := hp ⟨_,hv⟩
change B p.val (v-M v • oneV)=F (v-M v • oneV) at hh
simpa only [map_sub,map_smul,hB,hF,smul_zero,sub_zero] using hh
refine ⟨p.val,hpM,hfull,?_⟩
intro q hq
let d : V := q-p.val-M q • oneV
have hdK : d∈K := by
change M (q-p.val-M q • oneV)=0
simp [hpM,hM]
have hd : B d d=0 := by
dsimp only [d]
simp only [map_sub,map_smul,sub_apply,smul_apply,
hq,hfull,hB',smul_zero,sub_self]
have hz : (⟨d,hdK⟩ : K)=0 := by
apply hcoer.continuousLinearEquivOfBilin.injective
apply ext_inner_right ℝ
intro v
rw [map_zero,inner_zero_left,hcoer.continuousLinearEquivOfBilin_apply]
change B d v.val=0
dsimp only [d]
simp only [map_sub,map_smul,sub_apply,smul_apply,
hq,hfull,hB',smul_zero,sub_self]
have he : d=0 := congrArg (fun z : K => z.val) hz
exact sub_eq_zero.mp he
theorem central_variational_exists [CompleteSpace V] {ι : Type*} [Zero ι]
(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‖≤C*‖D u.val‖)
(s : Fin 3 → ℝ) (hs : ∑ i,s i=0) :
∃ p : V,M p=0 ∧
(∀ v : V,inner ℝ (D p) (D v)+
(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i p))
(spectralGraphWeight w (T i v)))=
∑ i : Fin 3,s i*spectralGraphMean w (T i v)) ∧
(∀ q : V,(∀ v : V,centralForm w D T q v=
∑ i : Fin 3,s i*spectralGraphMean w (T i v)) → q-p=M q • oneV) := by
let B := centralForm w D T
let F : V →L[ℝ] ℝ := ∑ i : Fin 3,s i • (spectralGraphMean w).comp (T i)
have hFa (v : V) : F v=∑ i : Fin 3,s i*spectralGraphMean w (T i v) := by
simp [F,smul_eq_mul]
have hF : F oneV=0 := by rw [hFa]; simpa only [havg,mul_one] using hs
have hB (u : V) : B u oneV=0 := by
simp only [B,centralForm_apply,hD,inner_zero_right,hT,Finset.sum_const_zero,add_zero]
have hB' (v : V) : B oneV v=0 := by
simp only [B,centralForm_apply,hD,inner_zero_left,hT,Finset.sum_const_zero,zero_add]
have hcoer : IsCoercive (((B.comp M.ker.subtypeL).flip.comp M.ker.subtypeL).flip) := by
refine ⟨C⁻¹^2,sq_pos_of_pos (inv_pos.mpr hC),?_⟩
intro u
change C⁻¹^2*‖u‖*‖u‖≤B u.val u.val
have hh := mul_self_le_mul_self (norm_nonneg u) (hP u)
have hlo := centralForm_lower w D T u.val
have hCC : 0<C^2 := sq_pos_of_pos hC
have hb : ‖u‖^2/C^2≤‖D u.val‖^2 := by
apply (div_le_iff₀ hCC).mpr
nlinarith [hh]
calc
_=‖u‖^2/C^2 := by simp only [div_eq_mul_inv,inv_pow]; ring
_≤‖D u.val‖^2 := hb
_≤B u.val u.val := hlo
obtain ⟨p,hp,hfull,huniq⟩ := solve_form_mod_constants B M F oneV hM hF hB hB' hcoer
refine ⟨p,hp,?_,?_⟩
· intro v
exact (centralForm_apply w D T p v).symm.trans ((hfull v).trans (hFa v))
· intro q hq
apply huniq q
intro v
exact (hq v).trans (hFa v).symm
theorem central_variational_exists_graph [CompleteSpace V] {ι : Type*} [Zero ι]
(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)+
(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i p))
(spectralGraphWeight w (T i v)))=
∑ i : Fin 3,s i*spectralGraphMean w (T i v)) ∧
(∀ q : V,(∀ v : V,inner ℝ (D q) (D v)+
(∑ i : Fin 3,inner ℝ (spectralGraphWeight w (T i q))
(spectralGraphWeight w (T i v)))=
∑ i : Fin 3,s i*spectralGraphMean w (T i v)) → q-p=M q • oneV) := by
let B := centralForm w D T
let F : V →L[ℝ] ℝ := ∑ i : Fin 3,s i • (spectralGraphMean w).comp (T i)
have hFa (v : V) : F v=∑ i : Fin 3,s i*spectralGraphMean w (T i v) := by
simp [F,smul_eq_mul]
have hF : F oneV=0 := by rw [hFa]; simpa only [havg,mul_one] using hs
have hB (u : V) : B u oneV=0 := by
simp only [B,centralForm_apply,hD,inner_zero_right,hT,Finset.sum_const_zero,add_zero]
have hB' (v : V) : B oneV v=0 := by
simp only [B,centralForm_apply,hD,inner_zero_left,hT,Finset.sum_const_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 hh := hP u
have he : B u.val u.val=‖D u.val‖^2+
∑ i : Fin 3,‖spectralGraphWeight w (T i u.val)‖^2 := by
simp only [B,centralForm_apply,real_inner_self_eq_norm_sq]
rw [←he] at hh
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 (centralForm_apply w D T p v).symm.trans ((hfull v).trans (hFa v))
· intro q hq
apply huniq q
intro v
exact (centralForm_apply w D T q v).trans ((hq v).trans (hFa v).symm)
end ScalarConductivity
end
end OAI
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