| import OAI.Analysis.BernoulliCone.UpperSupport |
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| namespace OAI |
|
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| section |
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff Laplacian |
| namespace Bernoulli.DeSilvaJerison |
| namespace CapAlgebra |
| lemma V_bounds {q b:ℝ} (hq:0≤q) (hq1:q≤1) (hb:2242/10000≤b) (hb1:b≤2243/10000) : |
| 0<V q b ∧ V q b≤1 := by |
| have hqq : q^2≤q := by nlinarith |
| have he : V q b=1-(1-q)*(b-3/200*(q+1)) := by unfold V; ring |
| have hl : 0≤b*q := mul_nonneg (by linarith) hq |
| constructor |
| · unfold V |
| nlinarith |
| · rw [he] |
| have hh : 0≤(1-q)*(b-3/200*(q+1)) := mul_nonneg (by linarith) (by linarith) |
| linarith |
| lemma hasDerivAt_V (q b:ℝ) : HasDerivAt (fun q=>V q b) (D q b/2) q := by |
| convert! (((hasDerivAt_const q (203/200-b)).add ((hasDerivAt_id q).const_mul b)).sub |
| (((hasDerivAt_id q).pow 2).const_mul (3/200))) using 1 |
| dsimp [V,D] |
| ring |
| lemma hasDerivAt_D (q b:ℝ) : HasDerivAt (fun q=>D q b) (-3/50) q := by |
| convert! (hasDerivAt_const q (2*b)).sub ((hasDerivAt_id q).const_mul (3/50)) using 1 |
| ring |
| lemma hasDerivAt_E (q b:ℝ) : HasDerivAt (fun q=>E q b) (-9/50) q := by |
| convert! (hasDerivAt_const q (2*b)).sub ((hasDerivAt_id q).const_mul (9/50)) using 1 |
| ring |
| lemma hasDerivAt_H (q b:ℝ) : HasDerivAt (fun q=>H q b) (9/16*D q b*E q b) q := by |
| convert! (hasDerivAt_const q (1:ℝ)).add |
| ((((hasDerivAt_id q).const_mul (9/16)).mul ((hasDerivAt_D q b).pow 2))) using 1 |
| dsimp [D,E] |
| ring |
| lemma V_contDiff (b:ℝ) {n:WithTop ℕ∞} : ContDiff ℝ n (fun q=>V q b) := by |
| exact (contDiff_const.add (contDiff_const.mul contDiff_id)).sub (contDiff_const.mul (contDiff_id.pow 2)) |
| lemma H_contDiff (b:ℝ) {n:WithTop ℕ∞} : ContDiff ℝ n (fun q=>H q b) := by |
| exact contDiff_const.add ((contDiff_const.mul contDiff_id).mul |
| ((contDiff_const.sub (contDiff_const.mul contDiff_id)).pow 2)) |
| /-- The cap prefactor in squared transverse coordinates. -/ |
| def Y (q b C:ℝ) :ℝ := C*V q b/Real.sqrt (H q b) |
| def Y1 (q b C:ℝ) :ℝ := |
| C*(D q b/2/Real.sqrt (H q b)-V q b*(9/16*D q b*E q b)/(2*(Real.sqrt (H q b))^3)) |
| def Y2 (q b C:ℝ) :ℝ := |
| C*((-3/100)/Real.sqrt (H q b)-D q b/2*(9/16*D q b*E q b)/(Real.sqrt (H q b))^3- |
| V q b*(9/16*((-3/50)*E q b+D q b*(-9/50)))/(2*(Real.sqrt (H q b))^3)+ |
| 3*V q b*(9/16*D q b*E q b)^2/(4*(Real.sqrt (H q b))^5)) |
| lemma Y_contDiffAt {q b C:ℝ} (hH:0<H q b) {n:WithTop ℕ∞} : ContDiffAt ℝ n (fun q=>Y q b C) q := |
| (contDiffAt_const.mul (V_contDiff b (n:=n)).contDiffAt).div ((H_contDiff b (n:=n)).contDiffAt.sqrt hH.ne') |
| (Real.sqrt_pos.mpr hH).ne' |
| lemma hasDerivAt_Y {q b C:ℝ} (hH:0<H q b) : HasDerivAt (fun q=>Y q b C) (Y1 q b C) q := by |
| have hS:=Real.sqrt_pos.mpr hH |
| have hSsq:=Real.sq_sqrt hH.le |
| convert! ((hasDerivAt_V q b).const_mul C).div ((hasDerivAt_H q b).sqrt hH.ne') hS.ne' using 1 |
| simp only [Y1] |
| field_simp |
| lemma hasDerivAt_Y1 {q b C:ℝ} (hH:0<H q b) : HasDerivAt (fun q=>Y1 q b C) (Y2 q b C) q := by |
| have hS:=Real.sqrt_pos.mpr hH |
| have hs := (hasDerivAt_H q b).sqrt hH.ne' |
| have hd := hasDerivAt_D q b |
| have he := hasDerivAt_E q b |
| have hv := hasDerivAt_V q b |
| convert! (((hd.div_const 2).div hs hS.ne').sub |
| ((hv.mul ((hd.const_mul (9/16)).mul he)).div ((hs.pow 3).const_mul 2) (mul_ne_zero (by norm_num) (pow_ne_zero _ hS.ne')))).const_mul C using 1 |
| dsimp [Y1,Y2] |
| field_simp |
| ring |
| lemma deriv_Y {q b C:ℝ} (hH:0<H q b) : deriv (fun q=>Y q b C) q=Y1 q b C := (hasDerivAt_Y hH).deriv |
| lemma deriv2_Y {q b C:ℝ} (hH:0<H q b) : deriv (deriv (fun q=>Y q b C)) q=Y2 q b C := by |
| have hh : deriv (fun q=>Y q b C)=ᶠ[𝓝 q] fun q=>Y1 q b C := by |
| filter_upwards [(H_contDiff b (n:=∞)).continuous.continuousAt.eventually (Ioi_mem_nhds hH)] with t ht |
| exact deriv_Y ht |
| rw [hh.deriv_eq] |
| exact (hasDerivAt_Y1 hH).deriv |
| lemma hasDerivAt_invV {q b:ℝ} (hv:V q b≠0) : |
| HasDerivAt (fun q=>(V q b)⁻¹) (-D q b/(2*(V q b)^2)) q := by |
| convert! (hasDerivAt_V q b).inv hv using 1 |
| ring |
| lemma deriv2_invV {q b:ℝ} (hv:V q b≠0) : |
| deriv (deriv (fun q=>(V q b)⁻¹)) q= (D q b)^2/(2*(V q b)^3)+3/(100*(V q b)^2) := by |
| have hh : deriv (fun q=>(V q b)⁻¹)=ᶠ[𝓝 q] fun q=>-D q b/(2*(V q b)^2) := by |
| filter_upwards [((V_contDiff b (n:=∞)).continuous.continuousAt.eventually_ne hv)] with t ht |
| exact (hasDerivAt_invV ht).deriv |
| rw [hh.deriv_eq] |
| have hd := ((hasDerivAt_D q b).neg.div (((hasDerivAt_V q b).pow 2).const_mul 2) (by positivity : 2*(V q b)^2≠0)).deriv |
| change deriv (fun q=>-D q b/(2*(V q b)^2)) q = _ at hd |
| rw [hd] |
| simp only [Pi.pow_apply,Pi.neg_apply] |
| field_simp |
| ring |
| lemma lap_Y {q b C:ℝ} (hV:V q b≠0) (hH:0<H q b) : |
| 4*q*Y2 q b C+12*Y1 q b C = Y q b C/V q b*(NN q b/(H q b)^2) := by |
| have hS := (Real.sqrt_pos.mpr hH).ne' |
| have hSq := Real.sq_sqrt hH.le |
| have hE : E q b=D q b-3/25*q := by unfold E D; ring |
| unfold Y Y1 Y2 NN MN |
| generalize hvv : V q b=v at * |
| generalize hdd : D q b=z at * |
| generalize hee : E q b=e at * |
| generalize hhh : H q b=h at * |
| have hdef : h=1+9/16*q*z^2 := by rw [←hhh, H, hdd] |
| field_simp [hS,hV,hH.ne'] |
| ring_nf |
| simp only [show Real.sqrt h^4=h^2 by calc |
| _ = (Real.sqrt h^2)^2 := by ring |
| _ = _ := by rw [hSq], hSq] |
| rw [hE,hdef] |
| ring |
| |
| lemma invV_mixed {q b C:ℝ} (hV:V q b≠0) (hH:0<H q b) : |
| Y q b C*(4*q*((D q b)^2/(2*(V q b)^3)+3/(100*(V q b)^2))+ |
| 12*(-D q b/(2*(V q b)^2)))+8*q*Y1 q b C*(-D q b/(2*(V q b)^2)) = |
| -Y q b C/(V q b)^2*(MN q b/H q b) := by |
| have hS := (Real.sqrt_pos.mpr hH).ne' |
| have hSq := Real.sq_sqrt hH.le |
| have hE : E q b=D q b-3/25*q := by unfold E D; ring |
| unfold Y Y1 MN |
| generalize hvv : V q b=v at * |
| generalize hdd : D q b=z at * |
| generalize hee : E q b=e at * |
| generalize hhh : H q b=h at * |
| have hdef : h=1+9/16*q*z^2 := by rw [←hhh, H, hdd] |
| field_simp [hS,hV,hH.ne'] |
| ring_nf |
| rw [hSq,hE,hdef] |
| ring |
| |
| end CapAlgebra |
| |
| lemma capH_contDiff : ContDiff ℝ ∞ capH := by |
| have hr : ContDiff ℝ ∞ capRadius := by |
| apply ContDiff.sqrt |
| · exact contDiff_const.add (contDiff_id.pow 2) |
| · intro x; positivity |
| have ht : ContDiff ℝ ∞ capAngle := contDiff_id.div hr (fun x=>(capRadius_pos x).ne') |
| have hf : ContDiff ℝ ∞ (fun s=>f (capAngle s)) := by |
| apply contDiff_iff_contDiffAt.mpr |
| intro s |
| exact (contDiffAt_f (capAngle_mem s)).comp s ht.contDiffAt |
| exact (hr.mul hf).add (contDiff_const.mul (contDiff_const.add (contDiff_const.mul (contDiff_id.pow 2)))) |
|
|
| lemma deriv2_capH (s:ℝ) : deriv (deriv capH) s=-5*(capRadius s)^3+4*capAmplitude := by |
| have hh : deriv capH=capHP := funext fun t=>(hasDerivAt_capH t).deriv |
| rw [hh] |
| exact (hasDerivAt_capHP s).deriv |
|
|
| def capC :ℝ := Real.sqrt (1+9/16*capP^2) |
| def smoothCap (x:Space 7) :ℝ := Cylindrical.lift axis (fun q=>CapAlgebra.Y q capB capC) |
| (fun q=>(CapAlgebra.V q capB)⁻¹) capH x |
|
|
| lemma smoothCap_contDiffAt {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1) : |
| ContDiffAt ℝ ∞ smoothCap x := by |
| have hq:=Cylindrical.radiusSq_nonneg axis x axis_norm |
| have hv := (CapAlgebra.V_bounds hq hx capB_bounds.1 capB_bounds.2).1 |
| exact Cylindrical.contDiffAt_lift (CapAlgebra.Y_contDiffAt (CapAlgebra.H_pos hq)) |
| ((CapAlgebra.V_contDiff capB).contDiffAt.inv hv.ne') capH_contDiff.contDiffAt |
| |
| lemma laplacian_smoothCap {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1) : |
| let q := Cylindrical.radiusSq axis x |
| let V := CapAlgebra.V q capB |
| let ξ := Cylindrical.axis axis x/V |
| Δ smoothCap x = CapAlgebra.Y q capB capC/V^2 * |
| ((-5*(capRadius ξ)^3+4*capAmplitude)*(1+q*ξ^2*(CapAlgebra.D q capB)^2)- |
| ξ*capHP ξ*V*(CapAlgebra.MN q capB/CapAlgebra.H q capB)+ |
| capH ξ*V*(CapAlgebra.NN q capB/(CapAlgebra.H q capB)^2)) := by |
| dsimp only |
| let q := Cylindrical.radiusSq axis x |
| let a := Cylindrical.axis axis x |
| have hq : 0≤q := Cylindrical.radiusSq_nonneg axis x axis_norm |
| have hv := (CapAlgebra.V_bounds hq hx capB_bounds.1 capB_bounds.2).1 |
| have hh := CapAlgebra.H_pos (b:=capB) hq |
| have hc := Cylindrical.laplacian_lift axis_norm |
| (CapAlgebra.Y_contDiffAt (C:=capC) (n:=2) hh) |
| ((CapAlgebra.V_contDiff capB (n:=2)).contDiffAt.inv hv.ne') |
| (capH_contDiff.of_le (by norm_cast : (2:WithTop ℕ∞)≤∞)).contDiffAt |
| change Δ smoothCap x=_ at hc |
| simp only [Space,finrank_euclideanSpace_fin,Nat.cast_ofNat] at hc |
| norm_num only at hc |
| change Δ smoothCap x = |
| CapAlgebra.Y q capB capC * deriv (deriv capH) (a*(CapAlgebra.V q capB)⁻¹) * |
| (((CapAlgebra.V q capB)⁻¹)^2+4*q*a^2*deriv (fun q=>(CapAlgebra.V q capB)⁻¹) q^2)+ |
| deriv capH (a*(CapAlgebra.V q capB)⁻¹)* |
| (a*CapAlgebra.Y q capB capC*(4*q*deriv (deriv (fun q=>(CapAlgebra.V q capB)⁻¹)) q+ |
| 12*deriv (fun q=>(CapAlgebra.V q capB)⁻¹) q)+ |
| 8*q*a*deriv (fun q=>CapAlgebra.Y q capB capC) q*deriv (fun q=>(CapAlgebra.V q capB)⁻¹) q)+ |
| capH (a*(CapAlgebra.V q capB)⁻¹)*(4*q*deriv (deriv (fun q=>CapAlgebra.Y q capB capC)) q+ |
| 12*deriv (fun q=>CapAlgebra.Y q capB capC) q) at hc |
| rw [hc] |
| simp only [CapAlgebra.deriv_Y hh, CapAlgebra.deriv2_Y hh, |
| (CapAlgebra.hasDerivAt_invV hv.ne').deriv, CapAlgebra.deriv2_invV hv.ne', |
| deriv2_capH,(hasDerivAt_capH _).deriv] |
| change _ = CapAlgebra.Y q capB capC/(CapAlgebra.V q capB)^2 * _ |
| have hm:=CapAlgebra.invV_mixed (C:=capC) hv.ne' hh |
| have hn:=CapAlgebra.lap_Y (C:=capC) hv.ne' hh |
| have h1 : ((CapAlgebra.V q capB)⁻¹)^2+4*q*a^2*(-CapAlgebra.D q capB/(2*(CapAlgebra.V q capB)^2))^2 = |
| (1+q*(a/CapAlgebra.V q capB)^2*(CapAlgebra.D q capB)^2)/(CapAlgebra.V q capB)^2 := by |
| field_simp |
| ring |
| have h2 : a*CapAlgebra.Y q capB capC* |
| (4*q*((CapAlgebra.D q capB)^2/(2*(CapAlgebra.V q capB)^3)+3/(100*(CapAlgebra.V q capB)^2))+ |
| 12*(-CapAlgebra.D q capB/(2*(CapAlgebra.V q capB)^2)))+ |
| 8*q*a*CapAlgebra.Y1 q capB capC*(-CapAlgebra.D q capB/(2*(CapAlgebra.V q capB)^2)) = |
| -a*CapAlgebra.Y q capB capC/(CapAlgebra.V q capB)^2*(CapAlgebra.MN q capB/CapAlgebra.H q capB) := by |
| calc |
| _ = a*(CapAlgebra.Y q capB capC* |
| (4*q*((CapAlgebra.D q capB)^2/(2*(CapAlgebra.V q capB)^3)+3/(100*(CapAlgebra.V q capB)^2))+ |
| 12*(-CapAlgebra.D q capB/(2*(CapAlgebra.V q capB)^2)))+ |
| 8*q*CapAlgebra.Y1 q capB capC*(-CapAlgebra.D q capB/(2*(CapAlgebra.V q capB)^2))) := by ring |
| _ = _ := by rw [hm]; ring |
| rw [h1,h2,hn] |
| change _=CapAlgebra.Y q capB capC/(CapAlgebra.V q capB)^2 * |
| ((-5*(capRadius (a/CapAlgebra.V q capB))^3+4*capAmplitude)*(1+q*(a/CapAlgebra.V q capB)^2*(CapAlgebra.D q capB)^2)- |
| (a/CapAlgebra.V q capB)*capHP (a/CapAlgebra.V q capB)*CapAlgebra.V q capB*(CapAlgebra.MN q capB/CapAlgebra.H q capB)+ |
| capH (a/CapAlgebra.V q capB)*CapAlgebra.V q capB*(CapAlgebra.NN q capB/(CapAlgebra.H q capB)^2)) |
| simp only [← div_eq_mul_inv] |
| field_simp |
| ring |
| lemma cap_sHP_nonpos (s:ℝ) : s*capHP s≤0 := by |
| rcases le_total 0 s with h|h |
| · exact mul_nonpos_of_nonneg_of_nonpos h (capHP_nonpos h) |
| · have hh:=mul_nonpos_of_nonneg_of_nonpos (neg_nonneg.mpr h) (capHP_nonpos (neg_nonneg.mpr h)) |
| simpa only [capHP_odd,neg_mul_neg] using hh |
|
|
| /-- The actual smooth cylindrical cap is superharmonic on its positive strip. -/ |
| theorem smoothCap_laplacian_nonpos {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1) |
| (hs:|Cylindrical.axis axis x/CapAlgebra.V (Cylindrical.radiusSq axis x) capB|≤3/4) : |
| Δ smoothCap x≤0 := by |
| let q:=Cylindrical.radiusSq axis x |
| let V:=CapAlgebra.V q capB |
| let s:=Cylindrical.axis axis x/V |
| have hq:0≤q:=Cylindrical.radiusSq_nonneg axis x axis_norm |
| obtain ⟨hv,hv1⟩:=CapAlgebra.V_bounds hq hx capB_bounds.1 capB_bounds.2 |
| obtain ⟨hM,hM1,hN,hN1⟩:=CapAlgebra.coeff_bounds hq hx capB_bounds.1 capB_bounds.2 |
| have hh:=capH_nonneg hs |
| have hph:=capHPP_nonpos s |
| have hsp:=cap_sHP_nonpos s |
| have hK:=cap_K_nonpos hs |
| change -5*capRadius s^3+4*capAmplitude-(673/250)*s*capHP s+(431/200)*capH s≤0 at hK |
| have hvm : V*(CapAlgebra.MN q capB/CapAlgebra.H q capB)≤673/250 := |
| (mul_le_mul_of_nonneg_right hv1 hM).trans (by simpa using hM1) |
| have hvn : V*(CapAlgebra.NN q capB/(CapAlgebra.H q capB)^2)≤431/200 := |
| (mul_le_mul_of_nonneg_right hv1 hN).trans (by simpa using hN1) |
| have h0:=mul_nonpos_of_nonpos_of_nonneg hph |
| (show 0≤q*s^2*(CapAlgebra.D q capB)^2 by positivity) |
| have h1:=mul_le_mul_of_nonneg_left hvm (neg_nonneg.mpr hsp) |
| have h2:=mul_le_mul_of_nonneg_left hvn hh |
| have hbr : (-5*capRadius s^3+4*capAmplitude)*(1+q*s^2*(CapAlgebra.D q capB)^2)- |
| s*capHP s*V*(CapAlgebra.MN q capB/CapAlgebra.H q capB)+ |
| capH s*V*(CapAlgebra.NN q capB/(CapAlgebra.H q capB)^2)≤0 := by |
| nlinarith only [h0,h1,h2,hK] |
| rw [laplacian_smoothCap hx] |
| have hY:0≤CapAlgebra.Y q capB capC := by unfold CapAlgebra.Y capC; positivity |
| exact mul_nonpos_of_nonneg_of_nonpos (div_nonneg hY (sq_nonneg V)) hbr |
|
|
| end Bernoulli.DeSilvaJerison |
|
|
| end |
| end |
|
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| end OAI |
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