| import Mathlib |
| 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 |
|
|
