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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Analysis.BernoulliCone.CapCalculus
namespace OAI
section
noncomputable section
open Set Filter
open scoped Topology ContDiff Laplacian
namespace Bernoulli.DeSilvaJerison
lemma capH_abs (s:ℝ) : capH |s|=capH s := by
rcases le_total 0 s with h|h
· rw [abs_of_nonneg h]
· rw [abs_of_nonpos h,capH_even]
lemma capHP_sq_abs (s:ℝ) : (capHP |s|)^2=(capHP s)^2 := by
rcases le_total 0 s with h|h
· rw [abs_of_nonneg h]
· rw [abs_of_nonpos h,capHP_odd,neg_sq]
lemma capH_boundary {s:ℝ} (hs:|s|=3/4) : capH s=0 := by
rw [←capH_abs,hs,capH_end]
lemma capHP_sq_boundary {s:ℝ} (hs:|s|=3/4) : (capHP s)^2=(capHP (3/4))^2 := by
rw [←capHP_sq_abs,hs]
lemma capC_pos : 0<capC := by unfold capC; positivity
lemma capC_sq : capC^2=1+9/16*capP^2 := Real.sq_sqrt (by positivity)
lemma cap_boundary_slope_bound : capC^2*(capHP (3/4))^2≤c7 := by
have hp := capP_bounds
have ha := capAmplitude_bounds
have hs : capP^2≤(3886/10000:ℝ)^2 := by nlinarith [hp.1,hp.2]
have hq : 0≤3125/768-capAmplitude/6 := by linarith [ha.2]
have hq1 : (3125/768-capAmplitude/6)^2≤(3125/768-(289237/1000000)/6:ℝ)^2 := by
nlinarith [ha.1]
rw [capC_sq,capHP_end]
have hprod := mul_le_mul (show 1+9/16*capP^2≤1+9/16*(3886/10000:ℝ)^2 by linarith)
hq1 (sq_nonneg _) (by positivity : (0:ℝ)≤1+9/16*(3886/10000:ℝ)^2)
nlinarith only [hprod,c7_lower]
lemma gradient_smoothCap {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1) :
let q := Cylindrical.radiusSq axis x
let V := CapAlgebra.V q capB
let s := Cylindrical.axis axis x/V
gradient smoothCap x =
(2*CapAlgebra.Y1 q capB capC*capH s) • Cylindrical.transverse axis x +
(CapAlgebra.Y q capB capC*capHP s) •
(V⁻¹ • axis+(-Cylindrical.axis axis x*CapAlgebra.D q capB/V^2) • Cylindrical.transverse axis x) := by
dsimp only
let q:=Cylindrical.radiusSq axis x
let a:=Cylindrical.axis axis x
have hq:=Cylindrical.radiusSq_nonneg axis x axis_norm
have hh:0<CapAlgebra.H q capB:=CapAlgebra.H_pos (b:=capB) hq
have hv:0<CapAlgebra.V q capB:= (CapAlgebra.V_bounds hq hx capB_bounds.1 capB_bounds.2).1
have hg:=Cylindrical.gradient_lift (e:=axis) (x:=x)
(CapAlgebra.Y_contDiffAt (C:=capC) (n:=1) hh |>.differentiableAt (by norm_num))
(CapAlgebra.hasDerivAt_invV hv.ne').differentiableAt
(hasDerivAt_capH _).differentiableAt
change gradient smoothCap x= _ at hg
change gradient smoothCap x=
(2*deriv (fun q=>CapAlgebra.Y q capB capC) q*capH (a*(CapAlgebra.V q capB)⁻¹)) • Cylindrical.transverse axis x+
(CapAlgebra.Y q capB capC*deriv capH (a*(CapAlgebra.V q capB)⁻¹)) •
((CapAlgebra.V q capB)⁻¹ • axis+(2*a*deriv (fun q=>(CapAlgebra.V q capB)⁻¹) q) • Cylindrical.transverse axis x) at hg
rw [hg]
simp only [CapAlgebra.deriv_Y hh,(CapAlgebra.hasDerivAt_invV hv.ne').deriv,
(hasDerivAt_capH _).deriv,←div_eq_mul_inv]
have he:2*a*(-CapAlgebra.D q capB/(2*CapAlgebra.V q capB^2)) = -a*CapAlgebra.D q capB/CapAlgebra.V q capB^2 := by ring
rw [he]
/-- Constant free-boundary slope of the actual cylindrical cap, including the
transverse axis. The result is for the genuine Euclidean gradient. -/
theorem smoothCap_boundary_gradient {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1)
(hs:|Cylindrical.axis axis x/CapAlgebra.V (Cylindrical.radiusSq axis x) capB|=3/4) :
‖gradient smoothCap x‖^2=capC^2*(capHP (3/4))^2 := by
let q:=Cylindrical.radiusSq axis x
let a:=Cylindrical.axis axis x
let V:=CapAlgebra.V q capB
let H:=CapAlgebra.H q capB
let s:=a/V
have hq:0≤q:=Cylindrical.radiusSq_nonneg axis x axis_norm
have hv:0<V:= (CapAlgebra.V_bounds hq hx capB_bounds.1 capB_bounds.2).1
have hh:0<H:=CapAlgebra.H_pos hq
have hs2 : s^2=9/16 := by nlinarith only [sq_abs s,hs]
have hz : capH s=0 := capH_boundary hs
have hp : (capHP s)^2=(capHP (3/4))^2 := capHP_sq_boundary hs
have hg:=gradient_smoothCap hx
dsimp only at hg
change gradient smoothCap x=(2*CapAlgebra.Y1 q capB capC*capH s) • Cylindrical.transverse axis x+
(CapAlgebra.Y q capB capC*capHP s) •
(V⁻¹ • axis+(-a*CapAlgebra.D q capB/V^2) • Cylindrical.transverse axis x) at hg
rw [hz] at hg
simp only [mul_zero,zero_smul,zero_add] at hg
have horth:inner ℝ axis (Cylindrical.transverse axis x)=0 := by
rw [real_inner_comm,Cylindrical.transverse_inner axis x axis_norm]
rw [hg,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,norm_add_sq_real]
simp only [norm_smul,Real.norm_eq_abs,mul_pow,sq_abs,axis_norm,mul_one,
real_inner_smul_left,real_inner_smul_right,horth,mul_zero,add_zero,
Cylindrical.transverse_sq axis x axis_norm]
change (CapAlgebra.Y q capB capC)^2*(capHP s)^2*
((V⁻¹)^2+(-a*CapAlgebra.D q capB/V^2)^2*q)=_
have he : ((V⁻¹)^2+(-a*CapAlgebra.D q capB/V^2)^2*q)=H/V^2 := by
have hhdef : H=1+9/16*q*(CapAlgebra.D q capB)^2:=rfl
have ha2 : a^2=9/16*V^2 := by
dsimp only [s] at hs2
field_simp at hs2
nlinarith only [hs2]
field_simp
rw [ha2,hhdef]
ring
rw [he,hp]
unfold CapAlgebra.Y
change (capC*V/Real.sqrt H)^2*(capHP (3/4))^2*(H/V^2)=_
rw [div_pow,mul_pow,Real.sq_sqrt hh.le]
field_simp
theorem smoothCap_boundary_slope {x:Space 7} (hx:Cylindrical.radiusSq axis x≤1)
(hs:|Cylindrical.axis axis x/CapAlgebra.V (Cylindrical.radiusSq axis x) capB|=3/4) :
‖gradient smoothCap x‖^2≤c7 := by
rw [smoothCap_boundary_gradient hx hs]
exact cap_boundary_slope_bound
end Bernoulli.DeSilvaJerison
end
end
end OAI
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