| import OAI.Analysis.AllenCahn.EllipticCompactness |
| import OAI.Analysis.AllenCahn.ExtremeRays |
|
|
| 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 |
| open MeasureTheory intervalIntegral Set Filter Topology |
|
|
| lemma sech_neg (y : ℝ) : sech (-y) = sech y := by simp [sech] |
|
|
| lemma exp_two_sech_four (y : ℝ) : |
| Real.exp (2*y)*sech y^4 = (1+Real.tanh y)^2*sech y^2 := by |
| rw [show 2*y = y+y by ring, Real.exp_add, ←Real.cosh_add_sinh y, |
| Real.tanh_eq_sinh_div_cosh y] |
| dsimp [sech] |
| field_simp |
|
|
| lemma exp_two_sech_four_primitive (y : ℝ) : |
| HasDerivAt (fun z => Real.tanh z + Real.tanh z^2 + Real.tanh z^3/3) |
| (Real.exp (2*y)*sech y^4) y := by |
| convert ((tanh_hasDerivAt y).add ((tanh_hasDerivAt y).pow 2)).add |
| (((tanh_hasDerivAt y).pow 3).div_const 3) using 1 |
| first | rfl | (rw [exp_two_sech_four]; ring) |
|
|
| lemma integral_exp_two_sech_four (R : ℝ) : |
| (∫ y in -R..R, Real.exp (2*y)*sech y^4) = |
| 2*Real.tanh R + (2/3)*Real.tanh R^3 := by |
| have hi : IntervalIntegrable (fun y => Real.exp (2*y)*sech y^4) volume (-R) R := |
| ((Real.continuous_exp.comp (continuous_const.mul continuous_id)).mul |
| (contDiff_sech.continuous.pow 4)).intervalIntegrable _ _ |
| rw [intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (fun y _ => exp_two_sech_four_primitive y) hi] |
| rw [Real.tanh_neg] |
| ring |
|
|
| lemma moment_exp_two_sech_four (R : ℝ) : |
| (∫ y in -R..R, y*Real.exp (2*y)*sech y^4) = |
| 2*Real.tanh R - 2*R*sech R^2 := by |
| let k := fun y : ℝ => y*Real.exp (2*y)*sech y^4 |
| have hk : Continuous k := (continuous_id.mul (Real.continuous_exp.comp |
| (continuous_const.mul continuous_id))).mul (contDiff_sech.continuous.pow 4) |
| have href : (∫ y in -R..R, k (-y)) = ∫ y in -R..R, k y := by |
| simpa only [neg_neg] using (intervalIntegral.integral_comp_neg (f := k) (a := -R) (b := R)) |
| have hki : IntervalIntegrable k volume (-R) R := hk.intervalIntegrable _ _ |
| have hkni : IntervalIntegrable (fun y => k (-y)) volume (-R) R := (hk.comp continuous_neg).intervalIntegrable _ _ |
| have hsum : (∫ y in -R..R, k y + k (-y)) = 2*(∫ y in -R..R, k y) := by |
| rw [intervalIntegral.integral_add hki hkni, href] |
| ring |
| have he (y) : k y + k (-y) = 4*y*Real.tanh y*sech y^2 := by |
| dsimp [k] |
| rw [mul_assoc y, exp_two_sech_four, mul_assoc (-y), exp_two_sech_four, |
| Real.tanh_neg, sech_neg] |
| ring |
| have hd (y) : HasDerivAt (fun z => 2*(Real.tanh z-z*sech z^2)) |
| (4*y*Real.tanh y*sech y^2) y := by |
| convert ((tanh_hasDerivAt y).sub ((hasDerivAt_id y).mul |
| ((sech_hasDerivAt' y).pow 2))).const_mul 2 using 1 <;> first | rfl | (try simp only [Pi.pow_apply, id_eq]; ring) |
| have hcalc : (∫ y in -R..R, 4*y*Real.tanh y*sech y^2) = |
| 4*Real.tanh R-4*R*sech R^2 := by |
| have hi : IntervalIntegrable (fun y => 4*y*Real.tanh y*sech y^2) volume (-R) R := |
| (((continuous_const.mul continuous_id).mul contDiff_tanh.continuous).mul |
| (contDiff_sech.continuous.pow 2)).intervalIntegrable _ _ |
| rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (fun y _ => hd y) hi] |
| rw [Real.tanh_neg, sech_neg] |
| ring |
| have : (∫ y in -R..R, k y+k (-y)) = ∫ y in -R..R, 4*y*Real.tanh y*sech y^2 := by |
| apply intervalIntegral.integral_congr |
| intro y _; exact he y |
| rw [hsum, hcalc] at this |
| dsimp [k] at this |
| linarith |
| |
| lemma exp_tangent_two (a y : ℝ) : |
| Real.exp (2*y)*(1+(a-2)*y) ≤ Real.exp (a*y) := by |
| have H := mul_le_mul_of_nonneg_left (Real.add_one_le_exp ((a-2)*y)) (Real.exp_pos (2*y)).le |
| rw [←Real.exp_add] at H |
| convert H using 1 <;> congr 1 <;> ring |
| |
| lemma finite_mass_tangent {R a : ℝ} (hR : 0 ≤ R) (ha : a ≤ 2) : |
| (3/4)*(∫ y in -R..R, Real.exp (a*y)*sech y^4) ≥ |
| (3/2)*a-3+(3/4)*(2*Real.tanh R+(2/3)*Real.tanh R^3) := by |
| have hk0 : Continuous (fun y : ℝ => Real.exp (2*y)*sech y^4) := |
| (Real.continuous_exp.comp (continuous_const.mul continuous_id)).mul (contDiff_sech.continuous.pow 4) |
| have hk1 : Continuous (fun y : ℝ => y*Real.exp (2*y)*sech y^4) := |
| (continuous_id.mul (Real.continuous_exp.comp (continuous_const.mul continuous_id))).mul (contDiff_sech.continuous.pow 4) |
| have hka : Continuous (fun y : ℝ => Real.exp (a*y)*sech y^4) := |
| (Real.continuous_exp.comp (continuous_const.mul continuous_id)).mul (contDiff_sech.continuous.pow 4) |
| have hi0 : IntervalIntegrable (fun y => Real.exp (2*y)*sech y^4) volume (-R) R := hk0.intervalIntegrable _ _ |
| have hi1 : IntervalIntegrable (fun y => (a-2)*(y*Real.exp (2*y)*sech y^4)) volume (-R) R := (continuous_const.mul hk1).intervalIntegrable _ _ |
| have hia : IntervalIntegrable (fun y => Real.exp (a*y)*sech y^4) volume (-R) R := hka.intervalIntegrable _ _ |
| have H : (∫ y in -R..R, Real.exp (2*y)*sech y^4 + |
| (a-2)*(y*Real.exp (2*y)*sech y^4)) ≤ ∫ y in -R..R, Real.exp (a*y)*sech y^4 := by |
| apply intervalIntegral.integral_mono_on (by linarith : -R ≤ R) |
| (hi0.add hi1) hia |
| intro y _ |
| have HH := mul_le_mul_of_nonneg_right (exp_tangent_two a y) (by positivity : 0 ≤ sech y^4) |
| calc Real.exp (2*y)*sech y^4+(a-2)*(y*Real.exp (2*y)*sech y^4) = |
| Real.exp (2*y)*(1+(a-2)*y)*sech y^4 := by ring |
| _ ≤ Real.exp (a*y)*sech y^4 := HH |
| rw [intervalIntegral.integral_add hi0 hi1, |
| intervalIntegral.integral_const_mul, integral_exp_two_sech_four, moment_exp_two_sech_four] at H |
| have hm : 2*Real.tanh R - 2*R*sech R^2 ≤ 2 := by |
| nlinarith [Real.tanh_lt_one R, mul_nonneg hR (sq_nonneg (sech R))] |
| nlinarith [mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ha) (sub_nonneg.mpr hm)] |
| |
| lemma jost_weighted_primitive (a y : ℝ) : |
| HasDerivAt (fun z => Real.exp (a*z)*sech z^4) |
| (4*(jost a y*sech y^4) - 3*a*(Real.exp (a*y)*sech y^4)) y := by |
| convert (((hasDerivAt_id y).const_mul a).exp.mul ((sech_hasDerivAt' y).pow 4)) using 1 <;> first | rfl | (dsimp [jost]; ring) |
|
|
| lemma finite_jost_mass_lower {R a : ℝ} (hR : 0 ≤ R) (ha : 0 ≤ a) : |
| (3*a/4)*(∫ y in -R..R, Real.exp (a*y)*sech y^4) ≤ |
| ∫ y in -R..R, jost a y*sech y^4 := by |
| have hk : Continuous (fun y : ℝ => Real.exp (a*y)*sech y^4) := |
| (Real.continuous_exp.comp (continuous_const.mul continuous_id)).mul (contDiff_sech.continuous.pow 4) |
| have hj : Continuous (fun y : ℝ => jost a y*sech y^4) := |
| (contDiff_jost a).continuous.mul (contDiff_sech.continuous.pow 4) |
| have hij : IntervalIntegrable (fun y => 4*(jost a y*sech y^4)) volume (-R) R := (continuous_const.mul hj).intervalIntegrable _ _ |
| have hik : IntervalIntegrable (fun y => 3*a*(Real.exp (a*y)*sech y^4)) volume (-R) R := (continuous_const.mul hk).intervalIntegrable _ _ |
| have H := intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (a := -R) (b := R) (fun y _ => jost_weighted_primitive a y) (hij.sub hik) |
| rw [intervalIntegral.integral_sub hij hik, |
| intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, sech_neg] at H |
| have hh : Real.exp (a*(-R)) ≤ Real.exp (a*R) := by |
| apply Real.exp_le_exp.mpr |
| nlinarith |
| nlinarith [mul_le_mul_of_nonneg_right hh (by positivity : 0 ≤ sech R^4)] |
|
|
| lemma reflected_weighted_mass (R a : ℝ) : |
| (∫ y in -R..R, jostReflected a y*sech y^4) = |
| ∫ y in -R..R, jost a y*sech y^4 := by |
| have H := intervalIntegral.integral_comp_neg |
| (f := fun y => jost a y*sech y^4) (a := -R) (b := R) |
| simpa only [jost_reflection, sech_neg, neg_neg] using H |
|
|
| lemma tanh_exp_two (y : ℝ) : |
| Real.tanh y = (Real.exp (2*y)-1)/(Real.exp (2*y)+1) := by |
| have he : Real.exp (2*y) = Real.exp y ^ 2 := by |
| rw [show 2*y=y+y by ring, Real.exp_add, pow_two] |
| rw [Real.tanh_eq, he, Real.exp_neg] |
| apply (div_eq_div_iff (by positivity : Real.exp y+(Real.exp y)⁻¹ ≠ 0) |
| (by positivity : Real.exp y^2+1 ≠ 0)).mpr |
| field_simp |
|
|
| lemma tanh_nonneg_of_nonneg {R : ℝ} (hR : 0 ≤ R) : 0 ≤ Real.tanh R := by |
| rw [tanh_exp_two] |
| apply div_nonneg |
| · linarith [Real.add_one_le_exp (2*R)] |
| · positivity |
|
|
| lemma tanh_tail_bound {R : ℝ} (hR : 0 < R) : 1-Real.tanh R ≤ 1/R := by |
| have hh : 1-Real.tanh R = 2/(Real.exp (2*R)+1) := by |
| rw [tanh_exp_two] |
| field_simp |
| ring |
| rw [hh] |
| apply (div_le_div_iff₀ (by positivity) hR).mpr |
| linarith [Real.add_one_le_exp (2*R)] |
|
|
| lemma exp_two_mass_deficit {R : ℝ} (hR : 0 < R) : |
| 2-(3/4)*(2*Real.tanh R+(2/3)*Real.tanh R^3) ≤ 3/R := by |
| have ht0 := tanh_nonneg_of_nonneg hR.le |
| have htt := Real.tanh_lt_one R |
| have hE : 2-(3/4)*(2*Real.tanh R+(2/3)*Real.tanh R^3) ≤ 3*(1-Real.tanh R) := by |
| nlinarith [mul_nonneg (sq_nonneg (1-Real.tanh R)) (by linarith : 0 ≤ Real.tanh R+2)] |
| have H := mul_le_mul_of_nonneg_left (tanh_tail_bound hR) (by norm_num : (0 : ℝ) ≤ 3) |
| have he : 3 * (1 / R) = 3 / R := by ring |
| rw [he] at H |
| exact hE.trans H |
|
|
| lemma jost_finite_window {c : ℝ} (hc : 3/4 < c) : |
| ∃ R > (0 : ℝ), ∀ a ∈ Icc (1 : ℝ) 2, |
| a^2-1 ≤ c*(∫ y in -R..R, jost a y*sech y^4) := by |
| have hc0 : 0 < c := by linarith |
| let δ := 1-(3/4)/c |
| have hδ : 0 < δ := by dsimp [δ]; exact sub_pos.mpr ((div_lt_one hc0).mpr hc) |
| have hδ1 : δ < 1 := by dsimp [δ]; have := div_pos (by norm_num : (0:ℝ)<3/4) hc0; linarith |
| have hcd : c*(1-δ) = 3/4 := by dsimp [δ]; field_simp; ring |
| let R := 6/δ |
| have hR : 0 < R := div_pos (by norm_num) hδ |
| have hERR : 2-(3/4)*(2*Real.tanh R+(2/3)*Real.tanh R^3) ≤ δ/2 := by |
| have H := exp_two_mass_deficit hR |
| have hh : 3/R = δ/2 := by dsimp [R]; field_simp; ring |
| rwa [hh] at H |
| refine ⟨R, hR, ?_⟩ |
| intro a ha |
| have ha0 : 0 ≤ a := by linarith [ha.1] |
| let M := (3/4)*(∫ y in -R..R, Real.exp (a*y)*sech y^4) |
| have hM : (1-δ)*((3/2)*a-1) ≤ M := by |
| have H := finite_mass_tangent hR.le ha.2 |
| dsimp [M] |
| nlinarith [mul_nonneg hδ.le (by linarith [ha.1] : 0 ≤ (3/2)*a-3/2)] |
| have hJ : (1-δ)*(a*((3/2)*a-1)) ≤ ∫ y in -R..R, jost a y*sech y^4 := by |
| have H := mul_le_mul_of_nonneg_left hM ha0 |
| have HH := finite_jost_mass_lower hR.le ha0 |
| dsimp [M] at H |
| nlinarith |
| have hpoly : (4/3)*(a^2-1) ≤ a*((3/2)*a-1) := by |
| nlinarith [mul_nonneg_of_nonpos_of_nonpos (by linarith [ha.2] : a-2 ≤ 0) |
| (by linarith [ha.2] : a-4 ≤ 0)] |
| calc a^2-1 = c*((1-δ)*((4/3)*(a^2-1))) := by rw [←mul_assoc, hcd]; ring |
| _ ≤ c*((1-δ)*(a*((3/2)*a-1))) := |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hpoly (by linarith)) hc0.le |
| _ ≤ c*(∫ y in -R..R, jost a y*sech y^4) := mul_le_mul_of_nonneg_left hJ hc0.le |
|
|
| lemma positive_sech_ode_finite_window {c : ℝ} (hc : 3/4 < c) : |
| ∃ R > (0 : ℝ), ∀ (f : ℝ → ℝ) (lam : ℝ), ContDiff ℝ 2 f → |
| (∀ y, 0 < f y) → lam ≤ 4 → |
| (∀ y, deriv (deriv f) y = (lam-2*sech y^2)*f y) → |
| |deriv f 0| ≤ c*(∫ y in -R..R, f y*sech y^4) := by |
| obtain ⟨R, hR, hbound⟩ := jost_finite_window hc |
| have hc0 : 0 ≤ c := by linarith |
| refine ⟨R, hR, ?_⟩ |
| intro f lam hf hp hlam hode |
| have hl := positive_sech_ode_spectral hf hp hode |
| let a := Real.sqrt lam |
| have ha0 : 0 ≤ a := Real.sqrt_nonneg _ |
| have hasq : a^2 = lam := Real.sq_sqrt (by linarith) |
| have ha1 : 1 ≤ a := by nlinarith |
| have ha2 : a ≤ 2 := by nlinarith |
| have hmass : 0 ≤ ∫ y in -R..R, f y*sech y^4 := by |
| apply intervalIntegral.integral_nonneg (by linarith : -R ≤ R) |
| intro y _; exact mul_nonneg (hp y).le (pow_nonneg (le_of_lt (sech_pos y)) 4) |
| rcases eq_or_lt_of_le ha1 with he | he |
| · have hla : lam = 1 := by rw [←hasq, ←he]; norm_num |
| have hrepr := positive_sech_ode_endpoint hf hp (by simpa only [hla] using hode) |
| have hfeq : f = fun y => f 0*sech y := funext hrepr |
| have hd : deriv f 0 = 0 := by |
| have HH := (sech_hasDerivAt 0).const_mul (f 0) |
| have HH' : HasDerivAt f 0 0 := by |
| simpa only [←hfeq, Real.sinh_zero, zero_div, mul_zero, neg_zero] using HH |
| exact HH'.deriv |
| rw [hd, abs_zero] |
| exact mul_nonneg hc0 hmass |
| · obtain ⟨A, B, hA, hB, hAB, hrepr⟩ := positive_sech_ode_classification he hf hp |
| (fun y => by rw [hasq]; exact hode y) |
| have hfeq : f = fun y => A*jost a y+B*jostReflected a y := funext hrepr |
| have hd : deriv f 0 = (A-B)*(a^2-1) := by |
| have HH := ((jost_hasDerivAt a 0).const_mul A).add |
| (((contDiff_jostReflected a).differentiable (by simp) 0).hasDerivAt.const_mul B) |
| have HH' : HasDerivAt (fun y => A*jost a y+B*jostReflected a y) |
| ((A-B)*(a^2-1)) 0 := by |
| have hv : A*(Real.exp (a*0)*(a*(a-Real.tanh 0)-sech 0^2)) + |
| B*deriv (jostReflected a) 0 = (A-B)*(a^2-1) := by |
| rw [jostReflected_deriv_at_zero] |
| simp only [mul_zero, Real.exp_zero, Real.tanh_zero, sech, Real.cosh_zero, inv_one] |
| ring |
| rw [hv] at HH |
| exact HH |
| rw [hfeq] |
| exact HH'.deriv |
| have hjI : IntervalIntegrable (fun y => A*(jost a y*sech y^4)) volume (-R) R := |
| (continuous_const.mul ((contDiff_jost a).continuous.mul (contDiff_sech.continuous.pow 4))).intervalIntegrable _ _ |
| have hkI : IntervalIntegrable (fun y => B*(jostReflected a y*sech y^4)) volume (-R) R := |
| (continuous_const.mul ((contDiff_jostReflected a).continuous.mul (contDiff_sech.continuous.pow 4))).intervalIntegrable _ _ |
| have hmassEq : (∫ y in -R..R, f y*sech y^4) = |
| (A+B)*(∫ y in -R..R, jost a y*sech y^4) := by |
| have heq : (fun y => f y*sech y^4) = fun y => |
| A*(jost a y*sech y^4)+B*(jostReflected a y*sech y^4) := by |
| funext y; rw [hrepr]; ring |
| rw [heq, intervalIntegral.integral_add hjI hkI, |
| intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul, reflected_weighted_mass] |
| ring |
| have hsq : 0 ≤ a^2-1 := by nlinarith |
| have habs : |A-B| ≤ A+B := by |
| rw [abs_le]; constructor <;> linarith |
| rw [hd, abs_mul, abs_of_nonneg hsq, hmassEq] |
| calc |A-B| * (a^2-1) ≤ (A+B)*(a^2-1) := mul_le_mul_of_nonneg_right habs hsq |
| _ ≤ (A+B)*(c*(∫ y in -R..R, jost a y*sech y^4)) := |
| mul_le_mul_of_nonneg_left (hbound a ⟨ha1, ha2⟩) (add_nonneg hA hB) |
| _ = c*((A+B)*(∫ y in -R..R, jost a y*sech y^4)) := by ring |
|
|
| 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 |
| open DeGiorgiCompactness (JetSpace CoherentJets pdeJetCone jetDirectional jetSegmentMass) |
|
|
| lemma jetCone_extreme_base {n : ℕ} {P : Space n → ℝ} {g : JetSpace n} |
| (hg : g ∈ (pdeJetCone P).extremePoints ℝ) : |
| (g [] : Space n → ℝ) ∈ (positiveLinearCone P).extremePoints ℝ := by |
| refine ⟨jetCone_base_mem hg.1, ?_⟩ |
| intro f hf h hh hseg |
| rcases hseg with ⟨a,b,ha,hb,hab,he⟩ |
| let F := smoothJet f hf.1 |
| let H := smoothJet h hh.1 |
| have hF : F ∈ pdeJetCone P := smoothJet_mem hf |
| have hH : H ∈ pdeJetCone P := smoothJet_mem hh |
| have heq : a • F+b • H = g := by |
| apply DeGiorgiCompactness.coherentJets_eq_of_base |
| (DeGiorgiCompactness.coherentJets_add_smul hF.1 hH.1 a b) hg.1.1 |
| ext x |
| exact congrFun he x |
| have hhseg : g ∈ openSegment ℝ F H := ⟨a,b,ha,hb,hab,heq⟩ |
| have hFG := hg.2 hF hH hhseg |
| funext x |
| exact congrArg (fun q : JetSpace n => q [] x) hFG |
|
|
| lemma jetDirectional_eq_fderiv {n : ℕ} {g : JetSpace n} (hg : CoherentJets g) |
| (ξ : Space n) : jetDirectional ξ g = fderiv ℝ (g [] : Space n → ℝ) 0 ξ := by |
| unfold jetDirectional |
| have he : (fun i => g [i] 0) = fun i => DeGiorgiCompactness.coordDeriv (g []) i 0 := |
| funext (fun i => ((hg []).2 i 0).symm) |
| rw [he, DeGiorgiCompactness.coordinateDual_fderiv] |
|
|
| lemma deriv_normal_line {n : ℕ} {f : Space n → ℝ} (hf : Differentiable ℝ f) |
| (ξ : Space n) : deriv (fun t : ℝ => f (t • ξ)) 0 = fderiv ℝ f 0 ξ := by |
| have hd : HasDerivAt (fun t : ℝ => t • ξ) ξ 0 := by |
| simpa only [id_eq, one_smul] using ((hasDerivAt_id (0:ℝ)).smul_const ξ) |
| have H := ((hf (0 • ξ)).hasFDerivAt).comp_hasDerivAt 0 hd |
| simpa only [id_eq, one_smul, zero_smul, Function.comp_def] using H.deriv |
|
|
| lemma conePotential_extreme_window {c : ℝ} (hc : 3/4 < c) : |
| ∃ R > (0 : ℝ), ∀ {n : ℕ} {ν : Space n}, ‖ν‖=1 → |
| ∀ g ∈ (pdeJetCone (conePotential ν)).extremePoints ℝ, |
| |jetDirectional (Real.sqrt 2 • ν) g| ≤ |
| c*jetSegmentMass (Real.sqrt 2 • ν) (fun y => sech y^4) R g := by |
| obtain ⟨R,hR,hbound⟩ := positive_sech_ode_finite_window hc |
| refine ⟨R,hR, ?_⟩ |
| intro n ν hν g hg |
| have hf := jetCone_extreme_base hg |
| obtain ⟨b,F,hb,hFs,hFp,hF0,hrep,hode⟩ := conePotential_extreme_separated hν hf |
| have hνν : inner ℝ ν ν = 1 := by rw [real_inner_self_eq_norm_sq, hν]; norm_num |
| have he : F = fun y : ℝ => g [] (y • (Real.sqrt 2 • ν)) := by |
| funext y |
| have H := hrep (y • (Real.sqrt 2 • ν)) |
| rw [real_inner_smul_right, real_inner_smul_right, hb, mul_zero, mul_zero, |
| Real.exp_zero, one_mul, real_inner_smul_right, real_inner_smul_right, hνν, |
| mul_one] at H |
| have harg : y*Real.sqrt 2/Real.sqrt 2 = y := by field_simp |
| rw [harg] at H |
| exact H.symm |
| have HH := hbound F (4-2*‖b‖^2) (hFs.of_le (WithTop.coe_le_coe.mpr le_top)) hFp (by nlinarith [sq_nonneg ‖b‖]) hode |
| rw [he, deriv_normal_line ((jetCone_base_mem hg.1).1.differentiable (by simp))] at HH |
| rw [jetDirectional_eq_fderiv hg.1.1] |
| exact HH |
|
|
| end DeGiorgi |
| end |
|
|
| end OAI |
|
|
