| import OAI.Analysis.AllenCahn.PositiveWindows |
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| namespace OAI |
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| noncomputable section |
| namespace DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
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| 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⟩ |
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| 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 |
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| end DeGiorgi |
| end |
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| noncomputable section |
| namespace DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
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| 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 |
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| 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] |
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| 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] |
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| 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 _ |
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| 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] |
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| 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 |
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| 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] |
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| 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) |
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| 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] |
|
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| 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 |
|
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| 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) |
|
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| end DeGiorgi |
| end |
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| end OAI |
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