| import OAI.Analysis.AllenCahn.JetProducts |
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| namespace OAI |
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| noncomputable section |
| namespace DeGiorgi |
| open scoped ContDiff |
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| theorem positiveAC_all_jets (n k : ℕ) : |
| JetControl (positiveDefectClass n) (fun _ _ => 1) (fun v => v) k ∧ |
| (∀ i, JetControl (positiveDefectClass n) (fun _ _ => 1) |
| (fun v => coordDeriv (inverseProfile v) i) k) ∧ |
| JetControl (positiveDefectClass n) diffuseDefect diffuseDefect k ∧ |
| ∀ i j, JetControl (positiveDefectClass n) defectRoot (fun v => inverseHessian v i j) k := by |
| induction k with |
| | zero => exact positiveAC_jet_base n |
| | succ k ih => |
| rcases ih with ⟨hJv,hJp,hJD,hJT⟩ |
| have hv (v) (h : v ∈ positiveDefectClass n) := h.1 |
| have hp (i) (v) (h : v ∈ positiveDefectClass n) := coordDeriv_smooth (inverseProfile_smooth h.1 h.2.1) i |
| have hq (v) (h : v ∈ positiveDefectClass n) : ContDiff ℝ ∞ (defectWeight v) := |
| contDiff_const.sub (h.1.pow 2) |
| have hD (v) (h : v ∈ positiveDefectClass n) := diffuseDefect_smooth h.1 h.2.1 |
| have hT (i j) (v) (h : v ∈ positiveDefectClass n) := inverseHessian_smooth h.1 h.2.1 i j |
| have hDn (v) (h : v ∈ positiveDefectClass n) (x) := (h.2.2.2 x).le |
| have hJDu : JetControl (positiveDefectClass n) (fun _ _ => 1) diffuseDefect k := |
| hJD.mono_weight (fun v _ x => diffuseDefect_le_one v x) |
| have hJq : JetControl (positiveDefectClass n) (fun _ _ => 1) defectWeight k := by |
| have HH := hJv.mul_bounded hv hv (fun _ _ _ => zero_le_one) hJv |
| exact (JetControl.const (positiveDefectClass n) 1 k).sub (fun _ _ => contDiff_const) |
| (fun v h => (hv v h).mul (hv v h)) HH |>.congr (fun v _ => by funext x; simp [defectWeight,pow_two]) |
| have hJb (i : Fin n) : JetControl (positiveDefectClass n) (fun _ _ => 1) |
| (fun v x => 2*Real.sqrt 2*v x*coordDeriv (inverseProfile v) i x) k := by |
| have HH := (hJv.const_mul hv (2*Real.sqrt 2)).mul_bounded |
| (fun v h => contDiff_const.mul (hv v h)) (hp i) (fun _ _ _ => zero_le_one) (hJp i) |
| exact HH |
| have hb (v) (_ : v ∈ positiveDefectClass n) (g : Space n → ℝ) (x) : |
| fderiv ℝ g x (diffuseDrift v x) = ∑ i : Fin n, |
| (2*Real.sqrt 2*v x*coordDeriv (inverseProfile v) i x)*coordDeriv g i x := by |
| rw [fderiv_diffuseDrift, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| have hbc (i) (v) (h : v ∈ positiveDefectClass n) : ContDiff ℝ ∞ |
| (fun x => 2*Real.sqrt 2*v x*coordDeriv (inverseProfile v) i x) := |
| (contDiff_const.mul (hv v h)).mul (hp i v h) |
| have hJS : JetControl (positiveDefectClass n) diffuseDefect defectSource k := by |
| have h1 := (JetControl.const (positiveDefectClass n) 1 k).sub (fun _ _ => contDiff_const) hD hJDu |
| have h2 := hJq.mul_bounded hq (fun v h => contDiff_const.sub (hD v h)) |
| (fun _ _ _ => zero_le_one) h1 |
| have h3 := (h2.const_mul (fun v h => (hq v h).mul (contDiff_const.sub (hD v h))) 2).mul_bounded |
| (fun v h => contDiff_const.mul ((hq v h).mul (contDiff_const.sub (hD v h)))) hD hDn hJD |
| have h4 : JetControl (positiveDefectClass n) diffuseDefect |
| (fun v x => ∑ i : Fin n, ∑ j : Fin n, inverseHessian v i j x*inverseHessian v i j x) k := by |
| apply JetControl.sum Finset.univ |
| · intro i _ v h |
| exact ContDiff.sum (fun j _ => (hT i j v h).mul (hT i j v h)) |
| · intro i _ |
| apply JetControl.sum Finset.univ |
| · intro j _ v h |
| exact (hT i j v h).mul (hT i j v h) |
| · intro j _ |
| exact positiveAC_root_product (hT i j) (hT i j) (hJT i j) (hJT i j) |
| have h4s (v) (h : v ∈ positiveDefectClass n) : ContDiff ℝ ∞ |
| (fun x => ∑ i : Fin n, ∑ j : Fin n, inverseHessian v i j x*inverseHessian v i j x) := |
| ContDiff.sum (fun i _ => ContDiff.sum (fun j _ => (hT i j v h).mul (hT i j v h))) |
| exact (h3.sub (fun v h => (contDiff_const.mul ((hq v h).mul |
| (contDiff_const.sub (hD v h)))).mul (hD v h)) |
| (fun v h => contDiff_const.mul (h4s v h)) (h4.const_mul h4s 2)).congr |
| (fun v _ => by funext x; simp only [defectSource,defectPotential,hessianSquare_eq_sum,pow_two]) |
| have hJD' : JetControl (positiveDefectClass n) diffuseDefect diffuseDefect (k+1) := by |
| apply JetControl.bootstrap hD (fun v h => defectSource_smooth h.1 h.2.1) hbc |
| (fun v h => smooth_C2 (hD v h)) (fun _ h => h.2.2.2) hb |
| (fun _ h => diffuseDefect_drift_eq_source h.1 h.2.1 h.2.2.1) |
| (show 0 ≤ 2*Real.sqrt 2 by positivity) (relativeDefectConstant_pos n).le (defectSourceConstant_pos n).le |
| (fun _ h => diffuseDrift_bound (smooth_C2 h.1) h.2.1 h.2.2.1) |
| (fun _ h => diffuseDefect_relative_gradient_bound h.1 h.2.1 h.2.2.1) |
| (fun v h x => ?_) hJb hJD hJS |
| rw [diffuseDefect_drift_eq_source h.1 h.2.1 h.2.2.1] |
| exact (defectSource_bounds h.1 h.2.1 h.2.2.1).1 x |
| have hJT' (i j : Fin n) : JetControl (positiveDefectClass n) defectRoot |
| (fun v => inverseHessian v i j) (k+1) := by |
| let R (v : Space n → ℝ) (x : Space n) := |
| -defectWeight v x*diffuseDefect v x*inverseHessian v i j x + |
| Real.sqrt 2*v x*defectWeight v x*diffuseDefect v x* |
| coordDeriv (inverseProfile v) i x*coordDeriv (inverseProfile v) j x - |
| defectWeight v x*(coordDeriv (inverseProfile v) i x*coordDeriv (diffuseDefect v) j x+ |
| coordDeriv (inverseProfile v) j x*coordDeriv (diffuseDefect v) i x) + |
| 2*Real.sqrt 2*v x*(∑ a : Fin n, inverseHessian v i a x*inverseHessian v a j x) |
| have hdD (a) (v) (h : v ∈ positiveDefectClass n) := coordDeriv_smooth (hD v h) a |
| have hR (v) (h : v ∈ positiveDefectClass n) : ContDiff ℝ ∞ (R v) := |
| ((((hq v h).neg.mul (hD v h)).mul (hT i j v h)).add |
| (((((contDiff_const.mul (hv v h)).mul (hq v h)).mul (hD v h)).mul (hp i v h)).mul (hp j v h))).sub |
| ((hq v h).mul (((hp i v h).mul (hdD j v h)).add ((hp j v h).mul (hdD i v h)))) |>.add |
| ((contDiff_const.mul (hv v h)).mul (ContDiff.sum (fun a _ => (hT i a v h).mul (hT a j v h)))) |
| have hJR : JetControl (positiveDefectClass n) defectRoot R k := by |
| have hnR (v) (_ : v ∈ positiveDefectClass n) (x) := (defectRoot_pos v x).le |
| have h1 := ((hJq.const_mul hq (-1)).mul_bounded |
| (fun v h => contDiff_const.mul (hq v h)) hD (fun _ _ _ => zero_le_one) hJDu).mul_bounded |
| (fun v h => (contDiff_const.mul (hq v h)).mul (hD v h)) (hT i j) hnR (hJT i j) |
| have h2a := ((((hJv.const_mul hv (Real.sqrt 2)).mul_bounded |
| (fun v h => contDiff_const.mul (hv v h)) hq (fun _ _ _ => zero_le_one) hJq).mul_bounded |
| (fun v h => (contDiff_const.mul (hv v h)).mul (hq v h)) (hp i) |
| (fun _ _ _ => zero_le_one) (hJp i)).mul_bounded |
| (fun v h => ((contDiff_const.mul (hv v h)).mul (hq v h)).mul (hp i v h)) (hp j) |
| (fun _ _ _ => zero_le_one) (hJp j)) |
| have h2 := (h2a.mul_bounded |
| (fun v h => (((contDiff_const.mul (hv v h)).mul (hq v h)).mul (hp i v h)).mul (hp j v h)) |
| hD hDn hJD).mono_weight (fun v h x => defect_le_root h x) |
| have h3a := (hJp i).mul_bounded (hp i) (hdD j) hDn (hJD'.deriv hD j) |
| have h3b := (hJp j).mul_bounded (hp j) (hdD i) hDn (hJD'.deriv hD i) |
| have h3c := h3a.add (fun v h => (hp i v h).mul (hdD j v h)) |
| (fun v h => (hp j v h).mul (hdD i v h)) h3b |
| have h3 := (hJq.mul_bounded hq |
| (fun v h => ((hp i v h).mul (hdD j v h)).add ((hp j v h).mul (hdD i v h))) hDn h3c).mono_weight |
| (fun v h x => defect_le_root h x) |
| have h4a : JetControl (positiveDefectClass n) diffuseDefect |
| (fun v x => ∑ a : Fin n, inverseHessian v i a x*inverseHessian v a j x) k := by |
| apply JetControl.sum Finset.univ |
| · intro a _ v h |
| exact (hT i a v h).mul (hT a j v h) |
| · intro a _ |
| exact positiveAC_root_product (hT i a) (hT a j) (hJT i a) (hJT a j) |
| have h4 := ((hJv.const_mul hv (2*Real.sqrt 2)).mul_bounded |
| (fun v h => contDiff_const.mul (hv v h)) |
| (fun v h => ContDiff.sum (fun a _ => (hT i a v h).mul (hT a j v h))) hDn h4a).mono_weight |
| (fun v h x => defect_le_root h x) |
| have hs1 (v) (h : v ∈ positiveDefectClass n) := |
| (((contDiff_const (c := (-1 : ℝ))).mul (hq v h)).mul (hD v h)).mul (hT i j v h) |
| have hs2 (v) (h : v ∈ positiveDefectClass n) := |
| (((((contDiff_const (c := Real.sqrt 2)).mul (hv v h)).mul (hq v h)).mul (hp i v h)).mul (hp j v h)).mul (hD v h) |
| have hs3 (v) (h : v ∈ positiveDefectClass n) := |
| (hq v h).mul (((hp i v h).mul (hdD j v h)).add ((hp j v h).mul (hdD i v h))) |
| have hs4 (v) (h : v ∈ positiveDefectClass n) : ContDiff ℝ ∞ |
| (fun x => 2*Real.sqrt 2*v x*(∑ a : Fin n, inverseHessian v i a x*inverseHessian v a j x)) := |
| ((contDiff_const (c := 2*Real.sqrt 2)).mul (hv v h)).mul (ContDiff.sum (fun a _ => (hT i a v h).mul (hT a j v h))) |
| exact (((h1.add hs1 hs2 h2).sub (fun v h => (hs1 v h).add (hs2 v h)) hs3 h3).add |
| (fun v h => ((hs1 v h).add (hs2 v h)).sub (hs3 v h)) hs4 h4).congr |
| (fun v _ => by funext x; dsimp [R]; ring) |
| exact JetControl.bootstrap (hT i j) hR hbc |
| (fun v h => smooth_C2 (defectRoot_smooth h.1 h.2.1 h.2.2.2)) |
| (fun v _ => defectRoot_pos v) hb |
| (fun v h => inverseHessian_drift_system h.1 h.2.1 h.2.2.1 i j) |
| (show 0 ≤ 2*Real.sqrt 2 by positivity) (relativeDefectConstant_pos n).le |
| (by have := (defectSourceConstant_pos n).le; positivity) |
| (fun _ h => diffuseDrift_bound (smooth_C2 h.1) h.2.1 h.2.2.1) |
| (fun _ h x => (defectRoot_bounds h.1 h.2.1 h.2.2.1 h.2.2.2 x).1) |
| (fun _ h x => (defectRoot_bounds h.1 h.2.1 h.2.2.1 h.2.2.2 x).2) hJb (hJT i j) hJR |
| refine ⟨?_, ?_, hJD', hJT'⟩ |
| · apply JetControl.of_derivs hv (positiveAC_jet_base n).1 |
| intro i |
| have HH := (hJq.mul_bounded hq (hp i) (fun _ _ _ => zero_le_one) (hJp i)).const_mul |
| (fun v h => (hq v h).mul (hp i v h)) (1/Real.sqrt 2) |
| exact HH.congr (fun v h => by |
| funext x |
| rw [coordDeriv_profile (h.1.differentiable (by simp)) h.2.1] |
| ring) |
| · intro i |
| apply JetControl.of_derivs (hp i) ((positiveAC_jet_base n).2.1 i) |
| intro j |
| exact (hJT i j).mono_weight (fun v h x => defectRoot_le_one h.2.2.2 x) |
|
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| end DeGiorgi |
| end |
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| end OAI |
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