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2fca478 | 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 | import Mathlib
import OAI.Geometry.PrescribedPotential.BernsteinProductMaximum
import OAI.Geometry.PrescribedPotential.BernsteinQuarticAlgebra
namespace OAI
/-! Bernstein Local C2. -/
section
noncomputable section
open Set Filter Topology
open scoped ContDiff
namespace HigherJet
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{ι : Type*} [Fintype ι]
def quarticConstant (C B : ℝ) : ℝ := B*C+C+3*B*C+4*C^2+1
lemma quartic_cutoff_uniform {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
{S T η : E → ℝ} (hS : ContDiffOn ℝ ∞ S U) (hT : ContDiffOn ℝ ∞ T U)
(hη : ContDiffOn ℝ ∞ η U) {C R : ℝ} (hC : 1 ≤ C) (hR : 0 ≤ R)
(hvalues : ∀ x ∈ K, 0 ≤ S x ∧ S x ≤ R ∧ 0 ≤ T x ∧ 0 ≤ η x ∧ η x ≤ 1)
(hinterior : ∀ x ∈ K, η x ≠ 0 → x ∈ interior K)
(v : E → ι → E)
(hineq : ∀ x ∈ K, η x ≠ 0 →
T x-C ≤ frameLaplace (v x) S x ∧
-C*(1+T x*Real.sqrt (T x)) ≤ frameLaplace (v x) T x ∧
-C ≤ frameLaplace (v x) η x ∧
‖frameGradient (v x) η x‖^2 ≤ C*η x ∧
‖frameGradient (v x) S x‖^2 ≤ C*T x) :
∀ z ∈ K, η z = 1 →
T z ≤ (8*C+1+R)*(2*quarticConstant C (8*C+1+R)+2)^2 := by
let A := 8*C+1
let B := A+R
let D := quarticConstant C B
have hA : 1 ≤ A := by dsimp [A]; linarith
have hB : 0 ≤ B := by dsimp [B]; linarith
have hD : 1 ≤ D := by
have h : 0 ≤ B*C+C+3*B*C+4*C^2 := by positivity
dsimp [D,quarticConstant]; linarith
let W : E → ℝ := fun x => η x*(A+S x)*T x
have hW : ContDiffOn ℝ ∞ W U := (hη.mul (contDiffOn_const.add hS)).mul hT
intro z hz hηz
obtain ⟨x,hx,hmax⟩ := hK.exists_isMaxOn ⟨z,hz⟩ (hW.continuousOn.mono hKU)
obtain ⟨hSx,hSRx,hTx,hηx,hη1x⟩ := hvalues x hx
obtain ⟨hSz,_,hTz,_,_⟩ := hvalues z hz
have hL : A*T z ≤ W x := by
have hh := hmax hz
change η z*(A+S z)*T z ≤ W x at hh
rw [hηz,one_mul] at hh
nlinarith only [hh,mul_nonneg hSz hTz]
have hAB : A+S x ≤ B := by dsimp [B]; linarith
have hax : 0 < A+S x := by linarith
by_cases hn : η x=0
· have he : W x=0 := by dsimp [W]; rw [hn]; ring
rw [he] at hL
have : T z ≤ 0 := by nlinarith only [hL,hA,hTz]
exact this.trans (mul_nonneg (by dsimp [A,B] at hB; exact hB) (sq_nonneg _))
have hηpos : 0 < η x := lt_of_le_of_ne hηx (Ne.symm hn)
have hm : IsLocalMax W x := by
filter_upwards [mem_interior_iff_mem_nhds.mp (hinterior x hx hn)] with y hy
exact hmax hy
have hnum := quartic_test_at_max hU hη hS hT (hKU hx) (v x) A hn (ne_of_gt hax) hm
obtain ⟨hLS,hLT,hLη,hGη,hGS⟩ := hineq x hx hn
have hcross : (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) S x))^2 ≤
‖frameGradient (v x) η x‖^2*‖frameGradient (v x) S x‖^2 := by
simpa only [sq_abs,mul_pow] using
(sq_le_sq₀ (abs_nonneg (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) S x)))
(mul_nonneg (norm_nonneg _) (norm_nonneg _))).mpr
(abs_real_inner_le_norm (frameGradient (v x) η x) (frameGradient (v x) S x))
have hb := BernsteinEstimate.quartic_max_reduction hηpos hη1x hTx
(Real.sqrt_nonneg _) (Real.sq_sqrt hTx) hC (by dsimp only [A]; linarith only [hSx]) hAB
hLS hLT hLη (sq_nonneg _) (sq_nonneg _) hGη hGS hcross hnum
have hbound : η x*T x ≤ (2*D+2)^2 :=
BernsteinEstimate.weighted_quartic_bound hηx hη1x hTx hD hb
have hWB : W x ≤ B*(2*D+2)^2 := by
calc
_ = (A+S x)*(η x*T x) := by dsimp [W]; ring
_ ≤ B*(2*D+2)^2 := mul_le_mul hAB hbound (mul_nonneg hηx hTx) hB
change T z ≤ B*(2*D+2)^2
have hAT : T z ≤ A*T z := le_mul_of_one_le_left hTz hA
exact hAT.trans (hL.trans hWB)
end HigherJet
end
end
end OAI
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