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namespace OAI
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
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
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
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
end OAI
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