| import OAI.Analysis.AllenCahn.ZeroDefect |
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| namespace OAI |
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| noncomputable section |
| namespace DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
| open scoped ENNReal |
| open MeasureTheory Filter Topology |
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| lemma planar_profile_coordinate_derivative {n : ℕ} |
| (e : Space n) (c : ℝ) (j : Fin n) (x : Space n) : |
| fderiv ℝ (fun y : Space n => profile (inner ℝ e y - c)) x |
| (EuclideanSpace.single j 1) = |
| (profileWeight (inner ℝ e x - c) / Real.sqrt 2) * e j := by |
| have hi : HasFDerivAt (fun y : Space n => inner ℝ e y - c) (innerSL ℝ e) x := |
| (innerSL ℝ e).hasFDerivAt.sub_const c |
| have hp := (profile_hasDerivAt (inner ℝ e x - c)).comp_hasFDerivAt x hi |
| simp only [Function.comp_def] at hp |
| rw [hp.fderiv] |
| simp [EuclideanSpace.inner_single_right] |
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| lemma oriented_profile_of_classified {n : ℕ} {u : Space n → ℝ} (j : Fin n) |
| (hi : ∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single j 1)) |
| (hc : Classified u) : |
| ∃ (e : Space n) (c : ℝ), ‖e‖ = 1 ∧ 0 < e j ∧ |
| ∀ x, u x = profile (inner ℝ e x - c) := by |
| rcases hc with h | h | ⟨e, c, he, h⟩ |
| · have hx := (hi 0).2 |
| rw [h 0] at hx |
| exact (lt_irrefl 1 hx).elim |
| · have hx := (hi 0).1 |
| rw [h 0] at hx |
| exact (lt_irrefl (-1 : ℝ) hx).elim |
| · refine ⟨e, c, he, ?_, h⟩ |
| have hp := hmon 0 |
| rw [funext h, planar_profile_coordinate_derivative] at hp |
| exact (mul_pos_iff_of_pos_left |
| (div_pos (profileWeight_pos _) (Real.sqrt_pos.mpr (by norm_num)))).mp hp |
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| lemma monotone_profile_of_defect_zero {u : Space 8 → ℝ} |
| (hu : ContDiff ℝ 2 u) (hi : ∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn u) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single (7 : Fin 8) 1)) |
| {z : Space 8} (hz : diffuseDefect u z = 0) : |
| ∃ (e : Space 8) (c : ℝ), ‖e‖ = 1 ∧ 0 < e 7 ∧ |
| ∀ x, u x = profile (inner ℝ e x - c) := by |
| exact oriented_profile_of_classified 7 hi hmon |
| (classified_of_defect_zero_at hu hi hAC hz) |
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| theorem monotone_profile_or_positive_defect {u : Space 8 → ℝ} |
| (hu : ContDiff ℝ 2 u) (hi : ∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hAC : AllenCahn u) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single (7 : Fin 8) 1)) : |
| (∃ (e : Space 8) (c : ℝ), ‖e‖ = 1 ∧ 0 < e 7 ∧ |
| ∀ x, u x = profile (inner ℝ e x - c)) ∨ |
| (∀ x, 0 < diffuseDefect u x) := by |
| rcases classified_or_positive_defect hu (fun x => ⟨(hi x).1.le, (hi x).2.le⟩) |
| hAC with hc | hp |
| · exact Or.inl (oriented_profile_of_classified 7 hi hmon hc) |
| · exact Or.inr hp.2 |
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| lemma strictMono_on_coordinate_lines {n : ℕ} {u : Space n → ℝ} |
| (hu : ContDiff ℝ 2 u) (j : Fin n) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single j 1)) |
| (x : Space n) : |
| StrictMono (fun t : ℝ => u (x + t • EuclideanSpace.single j 1)) := by |
| apply strictMono_of_deriv_pos |
| intro t |
| rw [deriv_line (hu.differentiable (by norm_num))] |
| exact hmon _ |
|
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| theorem monotone_line_end_limits {u : Space 8 → ℝ} |
| (hu : ContDiff ℝ 2 u) (hi : ∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) |
| (hmon : ∀ x, 0 < fderiv ℝ u x (EuclideanSpace.single (7 : Fin 8) 1)) |
| (x : Space 8) : |
| ∃ a ∈ Set.Icc (-1 : ℝ) 1, ∃ b ∈ Set.Icc (-1 : ℝ) 1, |
| Tendsto (fun t : ℝ => u (x + t • EuclideanSpace.single (7 : Fin 8) 1)) |
| atBot (𝓝 a) ∧ |
| Tendsto (fun t : ℝ => u (x + t • EuclideanSpace.single (7 : Fin 8) 1)) |
| atTop (𝓝 b) ∧ |
| ∀ t : ℝ, a < u (x + t • EuclideanSpace.single (7 : Fin 8) 1) ∧ |
| u (x + t • EuclideanSpace.single (7 : Fin 8) 1) < b := by |
| let f : ℝ → ℝ := fun t => u (x + t • EuclideanSpace.single (7 : Fin 8) 1) |
| have hm : StrictMono f := strictMono_on_coordinate_lines hu 7 hmon x |
| have hb : BddBelow (Set.range f) := ⟨-1, by |
| rintro _ ⟨t, rfl⟩ |
| exact (hi _).1.le⟩ |
| have ha : BddAbove (Set.range f) := ⟨1, by |
| rintro _ ⟨t, rfl⟩ |
| exact (hi _).2.le⟩ |
| refine ⟨⨅ t, f t, ⟨le_ciInf (fun t => (hi _).1.le), ?_⟩, |
| ⨆ t, f t, ⟨?_, ciSup_le (fun t => (hi _).2.le)⟩, |
| tendsto_atBot_ciInf hm.monotone hb, tendsto_atTop_ciSup hm.monotone ha, ?_⟩ |
| · exact (ciInf_le hb (0 : ℝ)).trans (hi _).2.le |
| · exact (hi _).1.le.trans (le_ciSup ha (0 : ℝ)) |
| · intro t |
| exact ⟨(ciInf_le hb (t - 1)).trans_lt (hm (by linarith)), |
| (hm (show t < t + 1 by linarith)).trans_le (le_ciSup ha (t + 1))⟩ |
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| end DeGiorgi |
| end |
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| end OAI |
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