| import OAI.Analysis.AllenCahn.EndpointMaximum |
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| namespace OAI |
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| noncomputable section |
| namespace EndpointSupport.DeGiorgi |
| open Filter Set Topology |
| open scoped NNReal |
| open scoped SchwartzMap Laplacian ContDiff |
| open FourierTransform |
| open Filter Topology |
| open scoped ContDiff |
| open MeasureTheory |
| open scoped ENNReal |
|
|
| private lemma no_tangent_zero {n : ℕ} {w : Space n → ℝ} |
| (hw : ContDiff ℝ 2 w) (hw0 : ∀ x, 0 ≤ w x) |
| (hineq : ∀ x, Laplacian.laplacian w x ≤ 2 * w x) |
| {r : ℝ} (hr : 0 < r) (y z : Space n) (hz : ‖z - y‖ = r) |
| (hwz : w z = 0) (hball : ∀ x, ‖x - y‖ < r → 0 < w x) : False := by |
| obtain ⟨p, hp, hpmin⟩ := (isCompact_closedBall y (r / 2)).exists_isMinOn |
| ⟨y, by simp only [Metric.mem_closedBall, dist_self]; positivity⟩ hw.continuous.continuousOn |
| have hppos : 0 < w p := hball p (by |
| have hnorm : ‖p - y‖ ≤ r / 2 := by simpa [Metric.mem_closedBall, dist_eq_norm] using hp |
| linarith) |
| let a : ℝ := 1 + (2 * (n : ℝ) + 2) / r ^ 2 |
| have hr2 : 0 < r ^ 2 := sq_pos_of_pos hr |
| have ha : 1 ≤ a := by |
| have hq : 0 ≤ (2 * (n : ℝ) + 2) / r ^ 2 := by positivity |
| dsimp [a] |
| linarith |
| have hscale : 2 * (n : ℝ) + 2 ≤ a * r ^ 2 := by |
| dsimp [a] |
| field_simp |
| nlinarith |
| have hcomp := annulus_comparison hw hw0 hineq ha hr hppos.le hscale y |
| (fun x hx => hpmin (by simp [Metric.mem_closedBall, dist_eq_norm, hx])) |
| let F : Space n → ℝ := fun x => w x - w p * barrier a r y x |
| let e : Space n := y - z |
| let f : ℝ → ℝ := fun t => F (z + t • e) |
| have hF : ContDiff ℝ 2 F := hw.sub (contDiff_const.mul (barrier_contDiff a r y)) |
| have hdline : HasDerivAt (fun t : ℝ => z + t • e) e 0 := by |
| simpa using (hasDerivAt_const (0 : ℝ) z).fun_add ((hasDerivAt_id (0 : ℝ)).smul_const e) |
| have hd : HasDerivAt f (fderiv ℝ F z e) 0 := by |
| simpa [f, Function.comp_def] using ((hF.differentiable (by norm_num) (z + (0 : ℝ) • e)).hasFDerivAt.comp_hasDerivAt 0 hdline) |
| have hf0 : f 0 = 0 := by simp [f, F, barrier, gaussian, hz, hwz] |
| have hpos : ∀ᶠ t in 𝓝[>] (0 : ℝ), 0 ≤ f t := by |
| filter_upwards [self_mem_nhdsWithin, |
| nhdsWithin_le_nhds (Iio_mem_nhds (show (0 : ℝ) < 1 / 2 by norm_num))] with t ht ht2 |
| change 0 < t at ht |
| change t < 1 / 2 at ht2 |
| have hnorm : ‖z + t • e - y‖ = (1 - t) * r := by |
| have heq : z + t • e - y = (1 - t) • (z - y) := by dsimp [e]; module |
| rw [heq, norm_smul, Real.norm_eq_abs, abs_of_pos (by linarith : 0 < 1 - t), hz] |
| have hlow : r / 2 ≤ ‖z + t • e - y‖ := by rw [hnorm]; nlinarith |
| have hhigh : ‖z + t • e - y‖ ≤ r := by rw [hnorm]; nlinarith |
| have hh := hcomp (z + t • e) hlow hhigh |
| dsimp [f, F] |
| linarith |
| have hnonneg := nonneg_right_derivative hd hf0 hpos |
| have hmin : IsLocalMin w z := Filter.Eventually.of_forall (fun x => by simpa [hwz] using hw0 x) |
| have hwder := hmin.fderiv_eq_zero |
| have hi : inner ℝ (z - y) e = -r ^ 2 := by |
| have heq : e = -(z - y) := by dsimp [e]; abel |
| rw [heq, inner_neg_right, real_inner_self_eq_norm_sq, hz] |
| have hcalc : fderiv ℝ F z e = |
| -(w p * gaussian a y z * (2 * a * r ^ 2)) := by |
| dsimp only [F] |
| rw [fderiv_fun_sub (hw.differentiable (by norm_num) z) |
| (((barrier_contDiff a r y).differentiable (by norm_num)).const_mul _ z)] |
| simp only [sub_apply, hwder, zero_apply] |
| rw [fderiv_const_mul ((barrier_contDiff a r y).differentiable (by norm_num) z)] |
| simp only [smul_apply, smul_eq_mul] |
| rw [fderiv_barrier, hi] |
| ring |
| have hnegative : fderiv ℝ F z e < 0 := by |
| rw [hcalc] |
| have hg : 0 < gaussian a y z := Real.exp_pos _ |
| have hap : 0 < a := lt_of_lt_of_le (by norm_num) ha |
| exact neg_neg_of_pos (mul_pos (mul_pos hppos hg) (mul_pos (mul_pos (by norm_num) hap) hr2)) |
| exact not_le_of_gt hnegative hnonneg |
|
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| lemma nonnegative_supersolution_eq_zero {n : ℕ} {w : Space n → ℝ} |
| (hw : ContDiff ℝ 2 w) (hw0 : ∀ x, 0 ≤ w x) |
| (hineq : ∀ x, Laplacian.laplacian w x ≤ 2 * w x) |
| {z : Space n} (hwz : w z = 0) : ∀ x, w x = 0 := by |
| intro y |
| by_contra hy |
| have hypos : 0 < w y := lt_of_le_of_ne (hw0 y) (Ne.symm hy) |
| let Z : Set (Space n) := {x | w x = 0} |
| have hZ : IsClosed Z := isClosed_eq hw.continuous continuous_const |
| obtain ⟨p, hp, hdist⟩ := hZ.exists_infDist_eq_dist ⟨z, hwz⟩ y |
| have hne : y ≠ p := by |
| intro heq |
| have hpzero : w p = 0 := hp |
| rw [heq, hpzero] at hypos |
| exact lt_irrefl _ hypos |
| have hr : 0 < dist y p := dist_pos.mpr hne |
| have hball : ∀ x, ‖x - y‖ < dist y p → 0 < w x := by |
| intro x hx |
| by_contra hnot |
| have hwx : w x = 0 := le_antisymm (le_of_not_gt hnot) (hw0 x) |
| have hd := Metric.infDist_le_dist_of_mem (x := y) (show x ∈ Z from hwx) |
| rw [hdist] at hd |
| have he : dist y x = ‖x - y‖ := by rw [dist_comm, dist_eq_norm] |
| rw [he] at hd |
| exact not_le_of_gt hx hd |
| exact no_tangent_zero hw hw0 hineq hr y p (by rw [← dist_eq_norm, dist_comm]) hp hball |
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| lemma constant_one_of_attains_one {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hbounded : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) {z : Space n} (hz : v z = 1) : ∀ x, v x = 1 := by |
| let w : Space n → ℝ := fun x => 1 - v x |
| have hw : ContDiff ℝ 2 w := contDiff_const.sub hv |
| have hw0 : ∀ x, 0 ≤ w x := fun x => sub_nonneg.mpr (hbounded x).2 |
| have hineq : ∀ x, Laplacian.laplacian w x ≤ 2 * w x := by |
| intro x |
| have hlap : Laplacian.laplacian w x = -(v x ^ 3 - v x) := by |
| change Laplacian.laplacian ((fun _ : Space n => (1 : ℝ)) - v) x = _ |
| rw [contDiff_const.contDiffAt.laplacian_sub hv.contDiffAt, |
| InnerProductSpace.laplacian_const] |
| simpa using congrArg Neg.neg (hAC x) |
| rw [hlap] |
| have hp : 0 ≤ (v x - 1) ^ 2 * (v x + 2) := |
| mul_nonneg (sq_nonneg _) (by linarith [(hbounded x).1]) |
| dsimp [w] |
| nlinarith |
| have H := nonnegative_supersolution_eq_zero hw hw0 hineq (z := z) (by simp [w, hz]) |
| intro x |
| have hh := H x |
| dsimp [w] at hh |
| linarith |
|
|
| lemma allenCahn_neg {n : ℕ} {v : Space n → ℝ} (hAC : AllenCahn v) : |
| AllenCahn (-v) := by |
| intro x |
| rw [InnerProductSpace.laplacian_neg] |
| change -(Laplacian.laplacian v x) = (-v x) ^ 3 - (-v x) |
| rw [hAC x] |
| ring |
|
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| lemma constant_neg_one_of_attains_neg_one {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hbounded : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) {z : Space n} (hz : v z = -1) : ∀ x, v x = -1 := by |
| have hnb : ∀ x, (-v) x ∈ Set.Icc (-1 : ℝ) 1 := by |
| intro x |
| constructor <;> dsimp only [Pi.neg_apply] <;> |
| linarith [(hbounded x).1, (hbounded x).2] |
| have H := constant_one_of_attains_one hv.neg hnb (allenCahn_neg hAC) |
| (z := z) (by simp [hz]) |
| intro x |
| have hh := H x |
| change -v x = 1 at hh |
| linarith |
|
|
| lemma endpoint_separation {n : ℕ} {v : Space n → ℝ} |
| (hv : ContDiff ℝ 2 v) (hbounded : ∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) |
| (hAC : AllenCahn v) : |
| (∀ x, v x = 1) ∨ (∀ x, v x = -1) ∨ (∀ x, v x ∈ Set.Ioo (-1 : ℝ) 1) := by |
| by_cases h1 : ∃ x, v x = 1 |
| · obtain ⟨x, hx⟩ := h1 |
| exact Or.inl (constant_one_of_attains_one hv hbounded hAC hx) |
| by_cases hneg1 : ∃ x, v x = -1 |
| · obtain ⟨x, hx⟩ := hneg1 |
| exact Or.inr (Or.inl (constant_neg_one_of_attains_neg_one hv hbounded hAC hx)) |
| refine Or.inr (Or.inr (fun x => ⟨?_, ?_⟩)) |
| · exact lt_of_le_of_ne (hbounded x).1 (fun h => hneg1 ⟨x, h.symm⟩) |
| · exact lt_of_le_of_ne (hbounded x).2 (fun h => h1 ⟨x, h⟩) |
|
|
| end EndpointSupport.DeGiorgi |
| end |
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| end OAI |
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