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import OAI.Geometry.CAT0Fillings.Minimizers.Geometric
import OAI.Geometry.CAT0Fillings.Minimizers.Nonnegative
namespace OAI
section
open Set Filter MeasureTheory Metric TopologicalSpace
open scoped Topology NNReal ENNReal
namespace CAT0Fillings
open ChartGeometry AnalyticMinimizer MassMeasure Rearrangement RadialSobolev
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X]
theorem extremal_nonnegative_euler {k : ℕ} (hk : 0 < k) (hX : IsCAT0 X)
{T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : boundarySucc T = 0)
(hm : 0 < mass T) (hms : mass T < sphereArea (k+2)) (q : ChartGeometry hT.1)
{d r : ℝ} (hd : 0 < d) (hr : 1 < r)
(hfill : ∀ P : Functional X (k+1), IsIntegral (k+1) P → boundarySucc P = 0 →
∃ R : Functional X (k+2), IsIntegral (k+2) R ∧ boundarySucc R = P ∧
mass R ≤ fillingCoefficient (k+1)*(mass P)^(fillingPower (k+1)))
(hext : ∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 →
d*((mass T)^r-(mass B)^r) ≤ fillingVolume (T-B)) :
∃ v : q.Sobolev,
(∀ᵐ x ∂currentMassMeasure hT.1, 0 ≤ (q.inclusion v) x) ∧
criticalNorm q.inclusion (sobolevP (k+2)) v = 1 ∧
0 < energy q.inclusion q.closedGradient (4/((k+2:ℝ)-2)) (k+2:ℝ) v ∧
energy q.inclusion q.closedGradient (4/((k+2:ℝ)-2)) (k+2:ℝ) v <
(k+2:ℝ)*sphereArea (k+2)^(2/(k+2:ℝ)) ∧
(∀ u, energy q.inclusion q.closedGradient (4/((k+2:ℝ)-2)) (k+2:ℝ) v *
(criticalNorm q.inclusion (sobolevP (k+2)) u)^2 ≤
energy q.inclusion q.closedGradient (4/((k+2:ℝ)-2)) (k+2:ℝ) u) ∧
(∀ ψ : q.Sobolev,
(4/((k+2:ℝ)-2))*inner ℝ (q.closedGradient v) (q.closedGradient ψ) +
(k+2:ℝ)*inner ℝ (q.inclusion v) (q.inclusion ψ) =
energy q.inclusion q.closedGradient (4/((k+2:ℝ)-2)) (k+2:ℝ) v *
(∫ x, ((q.inclusion v) x)^(sobolevP (k+2)-1)*(q.inclusion ψ) x ∂currentMassMeasure hT.1)) := by
have hn : (2:ℝ) < k+2 := by exact_mod_cast (show 2 < k+2 by omega)
have hk0 : 0 < (k+2:ℝ)-2 := by linarith
have hA : 0 < 4/((k+2:ℝ)-2) := div_pos (by norm_num) hk0
have hB : 0 < (k+2:ℝ) := by positivity
have hp : 2 < sobolevP (k+2) := by
dsimp [sobolevP]
push_cast
exact (lt_div_iff₀ hk0).mpr (by linarith)
have hsphere : 0 < sphereArea (k+2) := by
unfold sphereArea
exact mul_pos (by positivity) (omega_pos (by omega))
let S := (k+2:ℝ)*sphereArea (k+2)^(2/(k+2:ℝ))
have hS : 0 < S := mul_pos hB (Real.rpow_pos_of_pos hsphere _)
have hmem : ∀ w : q.Sobolev, MemLp (q.inclusion w)
(ENNReal.ofReal (sobolevP (k+2))) (currentMassMeasure hT.1) := by
as_aux_lemma =>
obtain ⟨C,hC,hb⟩ := extremal_closed_sobolev (k := k) (T := T)
hk hX hT hz hm q hd hr hfill hext
(show 0 < S/2 by positivity) (show S/2 < S by linarith)
exact fun w => (hb w).1
obtain ⟨u,hu,hus,hmin⟩ := extremal_critical_minimizer (k := k) (T := T)
hk hX hT hz hm hms q hd hr hfill hext
as_aux_lemma =>
exact q.nonnegative_euler_of_minimum (k := k+1) (T := T)
(A := 4/((k+2:ℝ)-2)) (B := k+2) (p := sobolevP (k+2)) (S := S)
hA.le hB hp hmem u hu hus hmin
end CAT0Fillings
end
section
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
namespace CAT0Fillings.AnalyticMinimizer
variable {α H F : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] [SeparableSpace H]
[NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F]
lemma normalized_scalar {p n Λ σ : ℝ} (hp : 2 < p) (hn : 0 < n) (hΛ : 0 < Λ)
(hσ : 0 < σ) (hsub : Λ < n*σ^((p-2)/p)) :
let c := (Λ/n)^(1/(p-2))
0 < c ∧ c*Λ = n*c^(p-1) ∧ c^2*Λ = n*c^p ∧ 0 < c^p ∧ c^p < σ := by
let c := (Λ/n)^(1/(p-2))
have hc : 0 < c := Real.rpow_pos_of_pos (div_pos hΛ hn) _
have hp0 : 0 < p := by linarith
have hpm : 0 < p-2 := by linarith
have hcp : c^(p-2) = Λ/n := by
dsimp [c]
rw [←Real.rpow_mul (div_nonneg hΛ.le hn.le)]
rw [one_div_mul_cancel hpm.ne',Real.rpow_one]
have hcpm : c^(p-1) = c^(p-2)*c := by
calc
c^(p-1) = c^((p-2)+1) := by congr 1; ring
_ = c^(p-2)*c^((1:ℝ)) := Real.rpow_add hc _ _
_ = c^(p-2)*c := by rw [Real.rpow_one]
have hcpa : c^p = c^(p-2)*c^2 := by
rw [←Real.rpow_two,←Real.rpow_add hc]
congr 1
ring
have he1 : c*Λ = n*c^(p-1) := by rw [hcpm,hcp]; field_simp
have he2 : c^2*Λ = n*c^p := by rw [hcpa,hcp]; field_simp
have hlt : c^(p-2) < σ^((p-2)/p) := by
rw [hcp]
exact (div_lt_iff₀ hn).mpr (by simpa only [mul_comm] using hsub)
have hlt' := Real.rpow_lt_rpow (Real.rpow_nonneg hc.le _) hlt (div_pos hp0 hpm)
rw [←Real.rpow_mul hc.le,←Real.rpow_mul hσ.le] at hlt'
have hp1 : (p-2)*(p/(p-2)) = p := by field_simp
have hp2 : ((p-2)/p)*(p/(p-2)) = 1 := by field_simp
rw [hp1,hp2,Real.rpow_one] at hlt'
exact ⟨hc,he1,he2,Real.rpow_pos_of_pos hc _,hlt'⟩
lemma normalize_euler (I : H →L[ℝ] Lp ℝ 2 μ) (G : H →L[ℝ] F)
{A p n σ : ℝ} (hp : 2 < p) (hn : 0 < n) (hσ : 0 < σ) (u : H)
(hu : criticalNorm I p u = 1) (hupos : ∀ᵐ x ∂μ, 0 ≤ (I u) x)
(hQ : 0 < energy I G A n u) (hsub : energy I G A n u < n*σ^((p-2)/p))
(heuler : ∀ ψ : H, A*inner ℝ (G u) (G ψ)+n*inner ℝ (I u) (I ψ) =
energy I G A n u*(∫ x, (I u x)^(p-1)*(I ψ x) ∂μ)) :
∃ v : H,
(∀ᵐ x ∂μ, 0 ≤ I v x) ∧ 0 < criticalMass I p v ∧ criticalMass I p v < σ ∧
energy I G A n v = n*criticalMass I p v ∧
(∀ ψ : H, A*inner ℝ (G v) (G ψ)+n*inner ℝ (I v) (I ψ) =
n*(∫ x, (I v x)^(p-1)*(I ψ x) ∂μ)) := by
let Λ := energy I G A n u
let c := (Λ/n)^(1/(p-2))
obtain ⟨hc,he1,he2,hcp,hcps⟩ := normalized_scalar hp hn hQ hσ hsub
change 0 < c at hc
change c*Λ = n*c^(p-1) at he1
change c^2*Λ = n*c^p at he2
change 0 < c^p at hcp
change c^p < σ at hcps
have hmass : criticalMass I p (c • u) = c^p := by
rw [criticalMass_eq I (by linarith : 0 < p),criticalNorm_smul,hu,mul_one,abs_of_pos hc]
refine ⟨c • u,?_,by simpa only [hmass] using hcp,by simpa only [hmass] using hcps,?_,?_⟩
· rw [map_smul]
filter_upwards [Lp.coeFn_smul c (I u),hupos] with x hx hp
rw [hx]
exact mul_nonneg hc.le hp
· rw [energy_smul,hmass]
exact he2
· intro ψ
have hi : (∫ x, (I (c • u) x)^(p-1)*(I ψ x) ∂μ) =
c^(p-1)*(∫ x, (I u x)^(p-1)*(I ψ x) ∂μ) := by
rw [←integral_const_mul,map_smul]
apply integral_congr_ae
filter_upwards [Lp.coeFn_smul c (I u),hupos] with x hx hp
rw [hx]
change (c*(I u x))^(p-1)*(I ψ x) = _
rw [Real.mul_rpow hc.le hp]
ring
rw [hi,map_smul,map_smul,real_inner_smul_left,real_inner_smul_left]
calc
A*(c*inner ℝ (G u) (G ψ))+n*(c*inner ℝ (I u) (I ψ)) =
c*(A*inner ℝ (G u) (G ψ)+n*inner ℝ (I u) (I ψ)) := by ring
_ = c*Λ*(∫ x, (I u x)^(p-1)*(I ψ x) ∂μ) := by rw [heuler]; ring
_ = n*(c^(p-1)*(∫ x, (I u x)^(p-1)*(I ψ x) ∂μ)) := by rw [he1]; ring
end CAT0Fillings.AnalyticMinimizer
end
end OAI
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