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159357f | 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 | import Mathlib
import OAI.Analysis.Conductivity.Branching.CrossingBlock
namespace OAI
noncomputable section
namespace ScalarConductivity
open Real Set Filter Topology MeasureTheory
lemma crossing_uniform_ellipticity {M L : ℝ} (hM0 : 1 ≤ M) (hL : 1 ≤ L)
(hM : ∀ x : ℝ, |deriv smoothTransition x| ≤ M ∧
|deriv (deriv smoothTransition) x| ≤ M)
(hsmall : 40*M*exp (-L) ≤ 1/56) (z x y X Y : ℝ) :
(1/8)*(X^2+Y^2) ≤ crossingXX L z x y*X^2+2*crossingXY L z x y*X*Y+crossingYY L z x y*Y^2 ∧
crossingXX L z x y*X^2+2*crossingXY L z x y*X*Y+crossingYY L z x y*Y^2 ≤ 10*(X^2+Y^2) := by
have hLp : 0 < L := lt_of_lt_of_le (by norm_num) hL
have hε : 0 ≤ 40*M*exp (-L) := by positivity
obtain ⟨hr₁,hr₂⟩ := crossing_residual_bounds hM0 hL hM z
by_cases hz : z ≤ 0
· obtain ⟨hyy,hxy⟩ := crossing_corrections_left hLp hz x y
have hratio := (crossing_mode_ratios (L := L) (z := z)).1 hz
have hα : |crossingResidualSecond L z/(exp (-3*z)*(1:ℝ)^2)| ≤ 40*M*exp (-L) := by
simpa only [one_pow,mul_one] using hr₂
obtain ⟨ha,hb⟩ := crossing_coefficients_small (exp_pos _).ne' (by norm_num : (1:ℝ)≠0)
(by norm_num : (2:ℝ)≠0) hα hratio (by norm_num : |(1:ℝ)/2|≤2) x y
rw [hyy,hxy]
exact crossing_quadratic_ellipticity (a := 9) (b := 1/4)
(by norm_num) (by norm_num) (by norm_num) (by norm_num) hε hsmall ha hb X Y
· have hz' : 0 ≤ z := le_of_not_ge hz
obtain ⟨hxx,hxy⟩ := crossing_corrections_right hLp hz' x y
have hratio := (crossing_mode_ratios (L := L) (z := z)).2 hz'
have heq : crossingResidualFirst L z/(exp (-z)*(2:ℝ)^2) =
(crossingResidualFirst L z/exp (-z))/4 := by ring
have hα : |crossingResidualFirst L z/(exp (-z)*(2:ℝ)^2)| ≤ 40*M*exp (-L) := by
rw [heq,abs_div,abs_of_pos (by norm_num : (0:ℝ)<4)]
exact (div_le_iff₀ (by norm_num : (0:ℝ)<4)).mpr (by linarith)
obtain ⟨ha,hb⟩ := crossing_coefficients_small (exp_pos _).ne' (by norm_num : (2:ℝ)≠0)
(by norm_num : (1:ℝ)≠0) hα hratio (by norm_num : |(2:ℝ)/1|≤2) y x
have hh := crossing_quadratic_ellipticity (a := 1/4) (b := 9)
(by norm_num) (by norm_num) (by norm_num) (by norm_num) hε hsmall ha hb Y X
rw [hxx,hxy]
dsimp only [crossingYY]
constructor <;> nlinarith [hh.1,hh.2]
theorem exists_elliptic_crossing : ∃ L : ℝ, 1 ≤ L ∧ ∀ z x y X Y : ℝ,
(1/8)*(X^2+Y^2) ≤ crossingXX L z x y*X^2+2*crossingXY L z x y*X*Y+crossingYY L z x y*Y^2 ∧
crossingXX L z x y*X^2+2*crossingXY L z x y*X*Y+crossingYY L z x y*Y^2 ≤ 10*(X^2+Y^2) := by
obtain ⟨M,hM0,hM⟩ := exists_transition_derivative_bound
obtain ⟨L,hL,hs⟩ := exists_small_crossing_scale hM0
exact ⟨L,hL,crossing_uniform_ellipticity hM0 hL hM hs⟩
end ScalarConductivity
end
end OAI
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