lean/OAI/Analysis/AllenCahn/NormalMass.lean

import OAI.Analysis.AllenCahn.PositiveWindows

namespace OAI

noncomputable section
namespace DeGiorgi
open Filter Set Topology
open scoped NNReal
open scoped SchwartzMap Laplacian ContDiff
open FourierTransform
open Filter Topology
open scoped ContDiff

lemma SmallDefectConverges.mass_positive {n : ℕ} {v : ℕ → Space n → ℝ}
    {g : SmallDefectIndex n → Space n → ℝ} (hg : SmallDefectConverges v g)
    (hv : ∀ j, v j ∈ positiveDefectClass n) (ν : Space n) {R : ℝ} (hR : 0<R) :
    0 < ∫ y in -R..R, profileWeight (g none (y • (Real.sqrt 2 • ν)))^3*
      g (some none) (y • (Real.sqrt 2 • ν)) := by
  have hc : Continuous (fun y : ℝ => profileWeight (g none (y • (Real.sqrt 2 • ν)))^3*
      g (some none) (y • (Real.sqrt 2 • ν))) :=
    (((profileWeight_contDiff.continuous.comp (hg.1 none).continuous).comp
      (continuous_id.smul continuous_const)).pow 3).mul
      ((hg.1 (some none)).continuous.comp (continuous_id.smul continuous_const))
  have hp (y : ℝ) : 0 < profileWeight (g none (y • (Real.sqrt 2 • ν)))^3*
      g (some none) (y • (Real.sqrt 2 • ν)) :=
    mul_pos (pow_pos (profileWeight_pos _) _) ((hg.positive hv).2 _)
  exact intervalIntegral.integral_pos (by linarith) hc.continuousOn
    (fun y _ => (hp y).le) ⟨0,⟨by linarith,by linarith⟩,hp 0⟩

theorem small_defect_mean_window_normalized {n : ℕ} {ν : Space n} (hν : ‖ν‖=1)
    {c : ℝ} (hc : 3/4<c) :
    ∃ R > (0 : ℝ), ∃ δ ∈ Set.Ioc (0 : ℝ) (1/2),
      ∀ v ∈ positiveDefectClass n, v 0=0 → diffuseDefect v 0<δ →
      gradient (inverseProfile v) 0=Real.sqrt (1-diffuseDefect v 0) • ν →
      |2*Real.sqrt 2*diffuseMeanCurvature v 0| ≤ c*reducedNormalMass v ν R := by
  obtain ⟨c₀,hc₀,hc₀c⟩ := exists_between hc
  obtain ⟨R,hR,hbound⟩ := SmallDefectConverges.positive_window hν hc₀
  refine ⟨R,hR,?_⟩
  by_contra! hbad
  have hseq (j : ℕ) : ∃ v ∈ positiveDefectClass n, v 0=0 ∧
      diffuseDefect v 0 < min (1/2) (1/(j+1 : ℝ)) ∧
      gradient (inverseProfile v) 0=Real.sqrt (1-diffuseDefect v 0) • ν ∧
      c*reducedNormalMass v ν R < |2*Real.sqrt 2*diffuseMeanCurvature v 0| := by
    exact hbad _ ⟨lt_min (by norm_num) (by positivity),min_le_left _ _⟩
  choose v hv h0 he horient hviol using hseq
  have he0 : Tendsto (fun j => diffuseDefect (v j) 0) atTop (𝓝 0) :=
    squeeze_zero (fun j => ((hv j).2.2.2 0).le)
      (fun j => (he j).le.trans (min_le_right _ _))
      tendsto_one_div_add_atTop_nhds_zero_nat
  obtain ⟨a,g,ha,hg⟩ := smallDefect_compactness hv h0
  have hv' (j) := hv (a j)
  have h0' (j) := h0 (a j)
  have he' := he0.comp ha.tendsto_atTop
  have horient' (j) := horient (a j)
  let F := fun x => profileWeight (inner ℝ ν x)*g (some none) x
  let M := ∫ y in -R..R, profileWeight (g none (y • (Real.sqrt 2 • ν)))^3*g (some none) (y • (Real.sqrt 2 • ν))
  have hM : 0<M := hg.mass_positive hv' ν hR
  have hFp : |fderiv ℝ F 0 (Real.sqrt 2 • ν)| ≤ c₀*M := by
    have H := hbound hg hv' h0' he' horient'
    have hmass : (∫ y in -R..R, F (y • (Real.sqrt 2 • ν))*sech y^4)=M := by
      unfold M
      rw [hg.profile_limit hv' h0' he' horient']
      apply intervalIntegral.integral_congr
      intro y hy
      dsimp only [F]
      rw [profileWeight_normal hν]
      ring
    simpa only [← hmass] using H
  have hm := hg.normal_mass_limit hv' ν R
  have hh := ((hg.mean_limit hv' h0' he' hν horient').const_mul (2*Real.sqrt 2)).abs
  have hh' : Tendsto (fun j => |2*Real.sqrt 2*diffuseMeanCurvature (v (a j)) 0| /
      diffuseDefect (v (a j)) 0) atTop (𝓝 |fderiv ℝ F 0 (Real.sqrt 2 • ν)|) := by
    convert hh using 1
    · funext j
      rw [← mul_div_assoc,abs_div,abs_of_pos ((hv' j).2.2.2 0)]
    · congr 2
      rw [map_smul]
      simp only [smul_eq_mul]
      dsimp only [F]
      ring
  have hrev : c*M ≤ |fderiv ℝ F 0 (Real.sqrt 2 • ν)| := by
    apply le_of_tendsto_of_tendsto (hm.const_mul c) hh'
    filter_upwards [] with j
    have H := (div_lt_div_of_pos_right (hviol (a j)) ((hv' j).2.2.2 0)).le
    simpa only [mul_div_assoc] using H
  nlinarith

end DeGiorgi
end

noncomputable section
namespace DeGiorgi
open Filter Set Topology
open scoped NNReal
open scoped SchwartzMap Laplacian ContDiff
open FourierTransform
open Filter Topology
open scoped ContDiff

lemma fderiv_affine_isometry {n : ℕ} {f : Space n → ℝ} (hf : Differentiable ℝ f)
    (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n) (x w : Space n) :
    fderiv ℝ (fun y => f (z+J y)) x w = fderiv ℝ f (z+J x) (J w) := by
  have H := (hf (z+J x)).hasFDerivAt.comp x (J.hasFDerivAt.const_add z)
  change HasFDerivAt (fun y => f (z+J y)) _ x at H
  rw [H.fderiv]
  rfl

lemma gradient_affine_isometry {n : ℕ} {f : Space n → ℝ} (hf : Differentiable ℝ f)
    (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n) (x : Space n) :
    gradient (fun y => f (z+J y)) x = J.symm (gradient f (z+J x)) := by
  apply ext_inner_right ℝ
  intro w
  rw [inner_gradient_left,fderiv_affine_isometry hf,J.symm.inner_map_eq_flip,
    J.symm_symm,inner_gradient_left]

lemma diffuseDefect_affine_isometry {n : ℕ} {v : Space n → ℝ}
    (hv : ContDiff ℝ ∞ v) (hi : ∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1)
    (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n) (x : Space n) :
    diffuseDefect (fun y => v (z+J y)) x = diffuseDefect v (z+J x) := by
  have he : inverseProfile (fun y => v (z+J y)) = fun y => inverseProfile v (z+J y) := rfl
  unfold diffuseDefect
  rw [he,gradient_affine_isometry ((inverseProfile_smooth hv hi).differentiable (by simp)),J.symm.norm_map]

lemma positiveDefectClass_affine_isometry {n : ℕ} {v : Space n → ℝ}
    (hv : v ∈ positiveDefectClass n) (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n) :
    (fun y => v (z+J y)) ∈ positiveDefectClass n := by
  refine ⟨hv.1.comp (contDiff_const.add J.contDiff),fun x => hv.2.1 _,?_,?_⟩
  · intro x
    have ht : ContDiff ℝ 2 (fun t => v (z+t)) :=
      (smooth_C2 hv.1).comp (contDiff_const.add contDiff_id)
    rw [show Laplacian.laplacian (fun y => v (z+J y)) x = Laplacian.laplacian (fun t => v (z+t)) (J x) from
      PoissonSupport.DeGiorgi.laplacian_comp_isometry ht J x,
      PoissonSupport.DeGiorgi.laplacian_translate (smooth_C2 hv.1)]
    exact hv.2.2.1 _
  · intro x
    rw [diffuseDefect_affine_isometry hv.1 hv.2.1]
    exact hv.2.2.2 _

lemma diffuseNormal_affine_isometry {n : ℕ} {v : Space n → ℝ}
    (hv : v ∈ positiveDefectClass n) (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n) (x : Space n) :
    diffuseNormal (fun y => v (z+J y)) x=J.symm (diffuseNormal v (z+J x)) := by
  have he : inverseProfile (fun y => v (z+J y)) = fun y => inverseProfile v (z+J y) := rfl
  unfold diffuseNormal
  rw [he,gradient_affine_isometry ((inverseProfile_smooth hv.1 hv.2.1).differentiable (by simp)),
    J.symm.norm_map, J.symm.map_smul]

lemma diffuseMeanCurvature_affine_isometry_zero {n : ℕ} {v : Space n → ℝ}
    (hv : v ∈ positiveDefectClass n) (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n)
    (h0 : v z=0) (hz : diffuseDefect v z<1) :
    diffuseMeanCurvature (fun y => v (z+J y)) 0=diffuseMeanCurvature v z := by
  have hv' := positiveDefectClass_affine_isometry hv z J
  have h0' : (fun y => v (z+J y)) 0=0 := by simpa using h0
  have hD : diffuseDefect (fun y => v (z+J y)) 0=diffuseDefect v z := by
    simpa using diffuseDefect_affine_isometry hv.1 hv.2.1 z J 0
  rw [diffuseMeanCurvature_zero hv' h0' (hD ▸ hz),diffuseMeanCurvature_zero hv h0 hz,hD]
  congr 1
  rw [show diffuseDefect (fun y => v (z+J y)) = fun y => diffuseDefect v (z+J y) from
    funext (diffuseDefect_affine_isometry hv.1 hv.2.1 z J),
    fderiv_affine_isometry ((diffuseDefect_smooth hv.1 hv.2.1).differentiable (by simp)),
    diffuseNormal_affine_isometry hv,J.apply_symm_apply]
  simp

lemma gradient_eq_normal {n : ℕ} (v : Space n → ℝ) (z : Space n) :
    gradient (inverseProfile v) z=Real.sqrt (1-diffuseDefect v z) • diffuseNormal v z := by
  simp only [diffuseDefect,sub_sub_cancel,Real.sqrt_sq (norm_nonneg _),diffuseNormal,smul_smul]
  by_cases h : ‖gradient (inverseProfile v) z‖=0
  · rw [h,zero_mul,zero_smul]
    exact norm_eq_zero.mp h
  · rw [mul_inv_cancel₀ h,one_smul]

noncomputable def normalDefectMass {n : ℕ} (v : Space n → ℝ) (z : Space n) (P : ℝ) : ℝ :=
  ∫ s in -P..P, defectWeight v (z+s • diffuseNormal v z)^3 * diffuseDefect v (z+s • diffuseNormal v z)

lemma normal_mass_affine_isometry {n : ℕ} {v : Space n → ℝ}
    (hv : v ∈ positiveDefectClass n) (z : Space n) (J : Space n ≃ₗᵢ[ℝ] Space n)
    (ν : Space n) (hJ : J ν=diffuseNormal v z) (R : ℝ) :
    normalDefectMass v z (R*Real.sqrt 2) =
      Real.sqrt 2*reducedNormalMass (fun x => v (z+J x)) ν R := by
  let f := fun s : ℝ => defectWeight v (z+s • diffuseNormal v z)^3 * diffuseDefect v (z+s • diffuseNormal v z)
  have he (y : ℝ) : defectWeight (fun x => v (z+J x)) (y • (Real.sqrt 2 • ν))^3 *
    diffuseDefect (fun x => v (z+J x)) (y • (Real.sqrt 2 • ν))=f (y*Real.sqrt 2) := by
    rw [diffuseDefect_affine_isometry hv.1 hv.2.1]
    simp only [defectWeight,J.map_smul,hJ,smul_smul,f]
  have H := intervalIntegral.integral_comp_mul_right (a := -R) (b := R) f
    (Real.sqrt_ne_zero'.mpr (by norm_num) : Real.sqrt 2≠0)
  unfold reducedNormalMass
  simp_rw [he]
  rw [H]
  simp only [smul_eq_mul,neg_mul]
  change normalDefectMass v z (R*Real.sqrt 2) =
    Real.sqrt 2*((Real.sqrt 2)⁻¹ * normalDefectMass v z (R*Real.sqrt 2))
  rw [← mul_assoc,mul_inv_cancel₀ (Real.sqrt_ne_zero'.mpr (by norm_num)),one_mul]

theorem diffuse_mean_window_positive {n : ℕ} (hn : 0<n) {c : ℝ}
    (hc : 1/(2*Real.sqrt 2)<c) :
    ∃ P > (0 : ℝ), ∃ δ ∈ Set.Ioc (0 : ℝ) (1/2),
      ∀ v ∈ positiveDefectClass n, ∀ z : Space n,
      v z=0 → diffuseDefect v z<δ →
      |diffuseMeanCurvature v z| ≤ (c/(4*Real.sqrt 2/3))*normalDefectMass v z P := by
  let ν : Space n := EuclideanSpace.basisFun (Fin n) ℝ ⟨0,hn⟩
  have hν : ‖ν‖=1 := by simp [ν]
  let c' := (3*Real.sqrt 2/2)*c
  have hs : 0<Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)
  have hs2 : Real.sqrt 2^2=2 := Real.sq_sqrt (by norm_num)
  have hc' : 3/4<c' := by
    have HH := (div_lt_iff₀ (show 0<2*Real.sqrt 2 by positivity)).mp hc
    dsimp only [c']
    nlinarith
  obtain ⟨R,hR,δ,hδ,hbound⟩ := small_defect_mean_window_normalized hν hc'
  refine ⟨R*Real.sqrt 2,mul_pos hR hs,δ,hδ,?_⟩
  intro v hv z h0 hz
  have hz1 : diffuseDefect v z<1 := lt_of_lt_of_le hz (hδ.2.trans (by norm_num))
  let J : Space n ≃ₗᵢ[ℝ] Space n := (Submodule.span ℝ {ν-diffuseNormal v z})ᗮ.reflection
  have hJ : J ν=diffuseNormal v z := Submodule.reflection_sub (hν.trans (diffuseNormal_unit hz1).symm)
  let w := fun x => v (z+J x)
  have hw : w ∈ positiveDefectClass n := positiveDefectClass_affine_isometry hv z J
  have hw0 : w 0=0 := by simpa [w] using h0
  have hD : diffuseDefect w 0=diffuseDefect v z := by
    simpa [w] using diffuseDefect_affine_isometry hv.1 hv.2.1 z J 0
  have horient : gradient (inverseProfile w) 0=Real.sqrt (1-diffuseDefect w 0) • ν := by
    change gradient (fun x => inverseProfile v (z+J x)) 0=_
    rw [gradient_affine_isometry ((inverseProfile_smooth hv.1 hv.2.1).differentiable (by simp)),
      J.map_zero,add_zero,gradient_eq_normal v z,J.symm.map_smul,← hJ,J.symm_apply_apply,hD]
  have H := hbound w hw hw0 (hD ▸ hz) horient
  have hmean : diffuseMeanCurvature w 0=diffuseMeanCurvature v z :=
    diffuseMeanCurvature_affine_isometry_zero hv z J h0 hz1
  rw [hmean,abs_mul,abs_of_pos (show 0<2*Real.sqrt 2 by positivity)] at H
  rw [normal_mass_affine_isometry hv z J ν hJ R]
  have hcoef : c/(4*Real.sqrt 2/3)*Real.sqrt 2=c'/(2*Real.sqrt 2) := by
    dsimp only [c']
    field_simp [hs.ne']
    nlinarith
  rw [← mul_assoc,hcoef]
  rw [div_mul_eq_mul_div]
  apply (le_div_iff₀ (show 0<2*Real.sqrt 2 by positivity)).mpr
  simpa only [_root_.mul_comm] using H

lemma normalDefectMass_mono {n : ℕ} {v : Space n → ℝ}
    (hv : v ∈ positiveDefectClass n) (z : Space n) {P P' : ℝ} (hP : 0≤P) (hPP' : P≤P') :
    normalDefectMass v z P≤normalDefectMass v z P' := by
  have hf : Continuous (fun s : ℝ => defectWeight v (z+s • diffuseNormal v z)^3 *
      diffuseDefect v (z+s • diffuseNormal v z)) :=
    (((contDiff_const.sub (hv.1.pow 2)).continuous.comp
      (continuous_const.add (continuous_id.smul continuous_const))).pow 3).mul
      ((diffuseDefect_smooth hv.1 hv.2.1).continuous.comp
        (continuous_const.add (continuous_id.smul continuous_const)))
  apply intervalIntegral.integral_mono_interval (neg_le_neg hPP') (by linarith) hPP' _ (hf.intervalIntegrable _ _)
  exact MeasureTheory.ae_of_all _ (fun s => mul_nonneg
    (pow_nonneg (by dsimp [defectWeight]; nlinarith [(hv.2.1 (z+s • diffuseNormal v z)).1,
      (hv.2.1 (z+s • diffuseNormal v z)).2]) _) (hv.2.2.2 _).le)

end DeGiorgi
end

end OAI