| 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 |
| |
