| import OAI.Analysis.AllenCahn.JetProducts |
|
|
| 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 |
|
|
| abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n) |
|
|
| def potential (t : ℝ) : ℝ := (1 - t ^ 2) ^ 2 / 4 |
|
|
| def profile (s : ℝ) : ℝ := Real.tanh (s / Real.sqrt 2) |
|
|
| def surfaceTension : ℝ := 2 * Real.sqrt 2 / 3 |
|
|
| def AllenCahn {n : ℕ} (v : Space n → ℝ) : Prop := |
| ∀ x, Laplacian.laplacian v x = v x ^ 3 - v x |
|
|
| def SecondVariation {n : ℕ} (v φ : Space n → ℝ) : ℝ := |
| ∫ x, (‖gradient φ x‖ ^ 2 + (3 * v x ^ 2 - 1) * φ x ^ 2) |
|
|
| def Stable {n : ℕ} (v : Space n → ℝ) : Prop := |
| ∀ φ : Space n → ℝ, ContDiff ℝ 1 φ → HasCompactSupport φ → |
| 0 ≤ SecondVariation v φ |
|
|
| def Energy {n : ℕ} (v : Space n → ℝ) (Ω : Set (Space n)) : ℝ := |
| ∫ x in Ω, (‖gradient v x‖ ^ 2 / 2 + potential (v x)) |
|
|
| def unitBallVolume (m : ℕ) : ℝ := |
| volume.real (Metric.ball (0 : Space m) 1) |
|
|
| def density {n : ℕ} (v : Space n → ℝ) (R : ℝ) (y : Space n) : ℝ := |
| Energy v (Metric.ball y R) / (surfaceTension * unitBallVolume (n - 1) * R ^ (n - 1)) |
|
|
| def densityAtInfinity {n : ℕ} (v : Space n → ℝ) : ℝ≥0∞ := |
| ⨆ R : { R : ℝ // 0 < R }, ENNReal.ofReal (density v R.val 0) |
|
|
| def Classified {n : ℕ} (v : Space n → ℝ) : Prop := |
| (∀ x, v x = 1) ∨ (∀ x, v x = -1) ∨ |
| ∃ (a : Space n) (b : ℝ), ‖a‖ = 1 ∧ ∀ x, v x = profile (inner ℝ a x - b) |
|
|
| def StableRigiditySeven : Prop := |
| ∀ v : Space 7 → ℝ, ContDiff ℝ 2 v → |
| (∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) → AllenCahn v → Stable v → Classified v |
|
|
| def MonotoneRigidityEight : Prop := |
| ∀ u : Space 8 → ℝ, ContDiff ℝ 2 u → |
| (∀ x, u x ∈ Set.Ioo (-1 : ℝ) 1) → AllenCahn u → |
| (∀ 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) |
|
|
| def FiniteDensityRigidity : Prop := |
| ∀ (n : ℕ), 4 ≤ n → n ≤ 7 → |
| ∀ v : Space n → ℝ, ContDiff ℝ 2 v → |
| (∀ x, v x ∈ Set.Icc (-1 : ℝ) 1) → AllenCahn v → Stable v → |
| densityAtInfinity v < ⊤ → Classified v |
|
|
| end EndpointSupport.DeGiorgi |
| end |
|
|
| 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 |
|
|
| lemma second_deriv_nonneg_of_local_min {f : ℝ → ℝ} {x : ℝ} |
| (hf : ContinuousAt f x) (hm : IsLocalMin f x) : 0 ≤ deriv (deriv f) x := by |
| by_contra h |
| have hn : deriv (deriv f) x < 0 := lt_of_not_ge h |
| have hmax := isLocalMax_of_deriv_deriv_neg hn hm.deriv_eq_zero hf |
| have he : f =ᶠ[𝓝 x] fun _ => f x := by |
| filter_upwards [hm, hmax] with y hy hz using le_antisymm hz hy |
| have hz := he.deriv.deriv_eq |
| have hz0 : deriv (deriv f) x = 0 := by |
| calc deriv (deriv f) x = deriv (deriv (fun _ : ℝ => f x)) x := hz |
| _ = 0 := by simp |
| linarith |
|
|
| private def basis (n : ℕ) := EuclideanSpace.basisFun (Fin n) ℝ |
|
|
| def coordDeriv {n : ℕ} (f : Space n → ℝ) (i : Fin n) (x : Space n) : ℝ := |
| fderiv ℝ f x (basis n i) |
|
|
| lemma coordDeriv_contDiff {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ 2 f) (i : Fin n) : ContDiff ℝ 1 (coordDeriv f i) := by |
| exact (hf.fderiv_right (m := 1) (by norm_num)).clm_apply contDiff_const |
|
|
| lemma laplacian_eq_sum_coordDeriv {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) |
| (x : Space n) : |
| Laplacian.laplacian f x = ∑ i : Fin n, coordDeriv (coordDeriv f i) i x := by |
| rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_orthonormalBasis f (basis n)] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [iteratedFDeriv_two_apply] |
| change _ = fderiv ℝ (fun y => fderiv ℝ f y (basis n i)) x (basis n i) |
| simp [fderiv_clm_apply |
| ((hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) x) |
| (differentiableAt_const (c := basis n i))] |
|
|
| lemma deriv_line {n : ℕ} {f : Space n → ℝ} (hf : Differentiable ℝ f) |
| (x e : Space n) (t : ℝ) : |
| deriv (fun s : ℝ => f (x + s • e)) t = fderiv ℝ f (x + t • e) e := by |
| have hd : HasDerivAt (fun s : ℝ => x + s • e) e t := by |
| simpa using (hasDerivAt_const t x).fun_add ((hasDerivAt_id t).smul_const e) |
| exact ((hf _).hasFDerivAt.comp_hasDerivAt t hd).deriv |
|
|
| lemma laplacian_nonneg_of_local_min {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ 2 f) {x : Space n} (hm : IsLocalMin f x) : |
| 0 ≤ Laplacian.laplacian f x := by |
| rw [laplacian_eq_sum_coordDeriv hf] |
| apply Finset.sum_nonneg |
| intro i _ |
| let g : ℝ → ℝ := fun s => f (x + s • basis n i) |
| have hg : ContinuousAt g 0 := hf.continuous.continuousAt.comp |
| (continuous_const.add (continuous_id.smul continuous_const)).continuousAt |
| have hmin : IsLocalMin g 0 := by |
| have ht : Tendsto (fun s : ℝ => x + s • basis n i) (𝓝 0) (𝓝 x) := by |
| have hc : Continuous (fun s : ℝ => x + s • basis n i) := |
| continuous_const.add (continuous_id.smul continuous_const) |
| simpa only [ContinuousAt, zero_smul, add_zero] using hc.continuousAt (x := 0) |
| change ∀ᶠ s in 𝓝 (0 : ℝ), g 0 ≤ g s |
| simpa [g] using ht.eventually hm |
| have hder : deriv g = fun s => coordDeriv f i (x + s • basis n i) := by |
| funext s |
| exact deriv_line (hf.differentiable (by norm_num)) x (basis n i) s |
| have hh := second_deriv_nonneg_of_local_min hg hmin |
| rw [hder, deriv_line ((coordDeriv_contDiff hf i).differentiable (by norm_num))] at hh |
| simpa [coordDeriv] using hh |
|
|
| lemma compact_minimum_principle {n : ℕ} {f : Space n → ℝ} |
| (hf : ContDiff ℝ 2 f) {K : Set (Space n)} (hK : IsCompact K) |
| (hboundary : ∀ x ∈ K, x ∉ interior K → 0 ≤ f x) |
| (hineq : ∀ x ∈ interior K, Laplacian.laplacian f x ≤ 2 * f x) : |
| ∀ x ∈ K, 0 ≤ f x := by |
| intro x hx |
| by_contra hn |
| have hneg : f x < 0 := lt_of_not_ge hn |
| obtain ⟨y, hy, hmin⟩ := hK.exists_isMinOn ⟨x, hx⟩ hf.continuous.continuousOn |
| have hyneg : f y < 0 := lt_of_le_of_lt (hmin hx) hneg |
| have hyint : y ∈ interior K := by |
| by_contra hh |
| exact (not_le_of_gt hyneg) (hboundary y hy hh) |
| have hlap := laplacian_nonneg_of_local_min hf |
| (hmin.isLocalMin (mem_interior_iff_mem_nhds.mp hyint)) |
| have hle := hineq y hyint |
| linarith |
|
|
| def gaussian {n : ℕ} (a : ℝ) (y : Space n) (x : Space n) : ℝ := |
| Real.exp (-a * ‖x - y‖ ^ 2) |
|
|
| private lemma gaussian_contDiff {n : ℕ} (a : ℝ) (y : Space n) : |
| ContDiff ℝ 2 (gaussian a y) := by |
| exact ((contDiff_const.mul ((contDiff_norm_sq ℝ).comp |
| (contDiff_id.sub contDiff_const)))).exp |
|
|
| private lemma fderiv_gaussian {n : ℕ} (a : ℝ) (y x e : Space n) : |
| fderiv ℝ (gaussian a y) x e = |
| gaussian a y x * (-2 * a * inner ℝ (x - y) e) := by |
| have h := (((hasFDerivAt_id x).sub (hasFDerivAt_const y x)).norm_sq.const_mul (-a)).exp |
| rw [show fderiv ℝ (gaussian a y) x = _ from h.fderiv] |
| simp [gaussian, inner_sub_left] |
| ring |
|
|
| private lemma second_coord_gaussian {n : ℕ} (a : ℝ) (y x : Space n) (i : Fin n) : |
| coordDeriv (coordDeriv (gaussian a y) i) i x = |
| gaussian a y x * (4 * a ^ 2 * (inner ℝ (x - y) (basis n i)) ^ 2 - 2 * a) := by |
| have hfun : coordDeriv (gaussian a y) i = fun z => |
| gaussian a y z * (-2 * a * inner ℝ (z - y) (basis n i)) := by |
| funext z |
| exact fderiv_gaussian a y z (basis n i) |
| rw [hfun] |
| unfold coordDeriv |
| have hg := (gaussian_contDiff a y).differentiable (by norm_num) |
| have hl : Differentiable ℝ (fun z : Space n => inner ℝ (z - y) (basis n i)) := |
| (differentiable_id.sub (differentiable_const y)).inner ℝ (differentiable_const _) |
| rw [fderiv_fun_mul (hg x) ((hl.const_mul (-2 * a)) x)] |
| simp only [add_apply, smul_apply, smul_eq_mul] |
| rw [fderiv_gaussian] |
| have hi : fderiv ℝ (fun z : Space n => -2 * a * inner ℝ (z - y) (basis n i)) x |
| (basis n i) = -2 * a := by |
| rw [fderiv_const_mul (hl x) (-2 * a)] |
| simp only [smul_apply, smul_eq_mul] |
| have hs : DifferentiableAt ℝ (fun z : Space n => z - y) x := |
| differentiableAt_id.sub (differentiableAt_const y) |
| rw [fderiv_inner_apply ℝ hs (differentiableAt_const (c := basis n i))] |
| simp [basis, fderiv_fun_sub] |
| rw [hi] |
| ring |
|
|
| private lemma laplacian_gaussian {n : ℕ} (a : ℝ) (y x : Space n) : |
| Laplacian.laplacian (gaussian a y) x = |
| gaussian a y x * (4 * a ^ 2 * ‖x - y‖ ^ 2 - 2 * a * n) := by |
| rw [laplacian_eq_sum_coordDeriv (gaussian_contDiff a y)] |
| simp_rw [second_coord_gaussian] |
| rw [← Finset.mul_sum, Finset.sum_sub_distrib, ← Finset.mul_sum, |
| (basis n).sum_sq_inner_left] |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| ring |
|
|
| def barrier {n : ℕ} (a r : ℝ) (y : Space n) (x : Space n) : ℝ := |
| gaussian a y x - Real.exp (-a * r ^ 2) |
|
|
| lemma barrier_contDiff {n : ℕ} (a r : ℝ) (y : Space n) : |
| ContDiff ℝ 2 (barrier a r y) := (gaussian_contDiff a y).sub contDiff_const |
|
|
| private lemma laplacian_barrier {n : ℕ} (a r : ℝ) (y x : Space n) : |
| Laplacian.laplacian (barrier a r y) x = |
| gaussian a y x * (4 * a ^ 2 * ‖x - y‖ ^ 2 - 2 * a * n) := by |
| change Laplacian.laplacian ((gaussian a y) - (fun _ : Space n => Real.exp (-a * r ^ 2))) x = _ |
| rw [(gaussian_contDiff a y).contDiffAt.laplacian_sub contDiff_const.contDiffAt, |
| InnerProductSpace.laplacian_const] |
| simpa using laplacian_gaussian a y x |
|
|
| private lemma barrier_le_one {n : ℕ} {a : ℝ} (ha : 0 ≤ a) (r : ℝ) (y x : Space n) : |
| barrier a r y x ≤ 1 := by |
| have he : gaussian a y x ≤ 1 := by |
| rw [gaussian, Real.exp_le_one_iff] |
| exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr ha) (sq_nonneg _) |
| have hh := Real.exp_pos (-a * r ^ 2) |
| dsimp [barrier] |
| linarith |
|
|
| private lemma barrier_subsolution {n : ℕ} {a r : ℝ} (ha : 1 ≤ a) (hr : 0 < r) |
| (hscale : 2 * (n : ℝ) + 2 ≤ a * r ^ 2) (y x : Space n) |
| (hx : r / 2 ≤ ‖x - y‖) : |
| 2 * barrier a r y x ≤ Laplacian.laplacian (barrier a r y) x := by |
| have han : 0 ≤ a := le_trans (by norm_num) ha |
| have hs : r ^ 2 ≤ 4 * ‖x - y‖ ^ 2 := by nlinarith [norm_nonneg (x - y)] |
| have h1 := mul_le_mul_of_nonneg_left hscale han |
| have h2 := mul_le_mul_of_nonneg_left hs (sq_nonneg a) |
| have hcoeff : 0 ≤ 4 * a ^ 2 * ‖x - y‖ ^ 2 - 2 * a * n - 2 := by |
| nlinarith |
| have hp : 0 ≤ gaussian a y x := le_of_lt (Real.exp_pos _) |
| have hm := mul_nonneg hp hcoeff |
| have he := Real.exp_pos (-a * r ^ 2) |
| rw [laplacian_barrier] |
| dsimp only [barrier] |
| nlinarith |
|
|
| private lemma laplacian_sub_const_mul {n : ℕ} {w b : Space n → ℝ} |
| (hw : ContDiff ℝ 2 w) (hb : ContDiff ℝ 2 b) (ε : ℝ) (x : Space n) : |
| Laplacian.laplacian (fun z => w z - ε * b z) x = |
| Laplacian.laplacian w x - ε * Laplacian.laplacian b x := by |
| change Laplacian.laplacian (w - ε • b) x = _ |
| have hs : ContDiff ℝ 2 (ε • b) := hb.const_smul ε |
| rw [hw.contDiffAt.laplacian_sub hs.contDiffAt, |
| InnerProductSpace.laplacian_smul ε hb.contDiffAt] |
| rfl |
|
|
| lemma annulus_comparison {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) |
| {a r ε : ℝ} (ha : 1 ≤ a) (hr : 0 < r) (hε : 0 ≤ ε) |
| (hscale : 2 * (n : ℝ) + 2 ≤ a * r ^ 2) (y : Space n) |
| (hinner : ∀ x, ‖x - y‖ = r / 2 → ε ≤ w x) : |
| ∀ x, r / 2 ≤ ‖x - y‖ → ‖x - y‖ ≤ r → ε * barrier a r y x ≤ w x := by |
| let K : Set (Space n) := {x | r / 2 ≤ ‖x - y‖ ∧ ‖x - y‖ ≤ r} |
| have hn : Continuous (fun x : Space n => ‖x - y‖) := |
| (continuous_id.sub continuous_const).norm |
| have hclosed : IsClosed K := |
| (isClosed_le continuous_const hn).inter (isClosed_le hn continuous_const) |
| have hK : IsCompact K := (isCompact_closedBall y r).of_isClosed_subset hclosed (by |
| intro x hx |
| simpa only [Metric.mem_closedBall, dist_eq_norm] using hx.2) |
| have hF : ContDiff ℝ 2 (fun x => w x - ε * barrier a r y x) := |
| hw.sub (contDiff_const.mul (barrier_contDiff a r y)) |
| have hboundary : ∀ x ∈ K, x ∉ interior K → 0 ≤ w x - ε * barrier a r y x := by |
| intro x hx hxout |
| by_cases hilow : ‖x - y‖ = r / 2 |
| · have hb := mul_le_mul_of_nonneg_left (barrier_le_one (le_trans (by norm_num) ha) r y x) hε |
| have hi := hinner x hilow |
| linarith |
| by_cases hihigh : ‖x - y‖ = r |
| · simpa [barrier, gaussian, hihigh] using hw0 x |
| exfalso |
| apply hxout |
| rw [mem_interior_iff_mem_nhds] |
| have ho : IsOpen {z : Space n | r / 2 < ‖z - y‖ ∧ ‖z - y‖ < r} := |
| (isOpen_lt continuous_const hn).inter (isOpen_lt hn continuous_const) |
| have hxstrict : x ∈ {z : Space n | r / 2 < ‖z - y‖ ∧ ‖z - y‖ < r} := |
| ⟨lt_of_le_of_ne hx.1 (Ne.symm hilow), lt_of_le_of_ne hx.2 hihigh⟩ |
| exact Filter.mem_of_superset (ho.mem_nhds hxstrict) (fun z hz => ⟨hz.1.le, hz.2.le⟩) |
| have hFineq : ∀ x ∈ interior K, |
| Laplacian.laplacian (fun x => w x - ε * barrier a r y x) x ≤ |
| 2 * (w x - ε * barrier a r y x) := by |
| intro x hx |
| have hxb : x ∈ K := interior_subset hx |
| rw [laplacian_sub_const_mul hw (barrier_contDiff a r y)] |
| have h1 := hineq x |
| have h2 := mul_le_mul_of_nonneg_left (barrier_subsolution ha hr hscale y x hxb.1) hε |
| linarith |
| have H := compact_minimum_principle hF hK hboundary hFineq |
| intro x hxlo hxhi |
| have hh := H x ⟨hxlo, hxhi⟩ |
| linarith |
|
|
| lemma fderiv_barrier {n : ℕ} (a r : ℝ) (y x e : Space n) : |
| fderiv ℝ (barrier a r y) x e = |
| gaussian a y x * (-2 * a * inner ℝ (x - y) e) := by |
| unfold barrier |
| rw [fderiv_fun_sub ((gaussian_contDiff a y).differentiable (by norm_num) x) |
| (differentiableAt_const _)] |
| simpa using fderiv_gaussian a y x e |
|
|
| lemma nonneg_right_derivative {f : ℝ → ℝ} {d : ℝ} |
| (hd : HasDerivAt f d 0) (h0 : f 0 = 0) |
| (hpos : ∀ᶠ t in 𝓝[>] (0 : ℝ), 0 ≤ f t) : 0 ≤ d := by |
| apply ge_of_tendsto hd.tendsto_slope_zero_right |
| filter_upwards [self_mem_nhdsWithin, hpos] with t ht hp |
| simpa [h0] using mul_nonneg (inv_nonneg.mpr (le_of_lt ht)) hp |
|
|
| end EndpointSupport.DeGiorgi |
| end |
|
|
| end OAI |
|
|
