| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.TestAsSmooth |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open MeasureTheory TopologicalSpace Filter Set |
| open |
| variable ℝ |
| [FiniteDimensional ℝ |
| variable |
|
|
| lemma φ τ ℝ |
| Integrable τφ |
| convert φ τ |
| (testMultiply φ τ |
| ext |
| simp |
| rw τφ τφ τφ |
| ring |
|
|
| lemma φ τ ℝ |
| testPair φ τ |
| (testMultiply φ τ |
| ∫ x, Real.exp τφ |
| apply |
| filter_upwards |
| simp |
| rw τφ τφ τφ |
| ring |
|
|
| lemma φ τ ℝτ ≤ τ |
| (u |
| Real.exp τ∫ | φ ≤ |
| testPair φ τ |
| (testMultiply φ τ |
| have | φ |
| isOpen_lt φ | |
| rw ← ← |
| apply |
| (exponential_square_integrable φ τ |
| intro |
| dsimp |
| by_cases φ |
| · rw ∈ | φ |
| exact |
| (Real.exp_le_exp.mpr |
| · rw ∉ | φ |
| exact |
|
|
| lemma φ τ ℝτ ≤ τ |
| (r ∀ φ → |
| testPair φ τ |
| (testMultiply φ τ ≤ |
| Real.exp τ |
| rw ← |
| apply φ τ |
| ((testPair_integrable |
| intro |
| by_cases φ |
| · simp |
| · exact |
| (Real.exp_le_exp.mpr |
| (mul_self_nonneg |
|
|
| theorem φ |
| (C ℝ ≤ |
| (hest ∀ τ ≥ |
| testPair φ τ φ τ ≤ |
| C φ τ φ τ |
| (hr ∀ φ → ∀ φ → |
| intro |
| obtain ⟨⟩ |
| let | φ |
| let ∫ |
| have ≤ |
| have ∀ᶠ τ ℝ ≤ |
| C τ |
| filter_upwards τ τ |
| have τ |
| have τ |
| have φ τ |
| have φ τ |
| have τ |
| change τ ≤ τ |
| apply ₀ τ |
| refine |
| rw τττ |
| rw ← ; congr 1; ring] |
| ring |
| have hlim : Tendsto (fun τ : ℝ => C*testPair K r r*Real.exp ((2*(c-b))*τ)) atTop (𝓝 0) := by |
| have he : Tendsto (fun τ : ℝ => Real.exp ((2*(c-b))*τ)) atTop (𝓝 0) := |
| Real.tendsto_exp_atBot.comp ((tendsto_const_mul_atBot_of_neg (by linarith : 2*(c-b) < 0)).mpr tendsto_id) |
| simpa only [mul_zero] using he.const_mul (C*testPair K r r) |
| have hJ0 : J = 0 := le_antisymm (le_of_tendsto_of_tendsto tendsto_const_nhds hlim hbound) hJ |
| have hae : (fun y => u y*u y) =ᵐ[(volume : Measure E).restrict S] 0 := |
| (integral_eq_zero_iff_of_nonneg (fun y => mul_self_nonneg _) (testPair_integrable K u u).restrict).mp hJ0 |
| have hop : IsOpen S := isOpen_lt continuous_const (SmoothScalar.contDiff φ).continuous |
| have heq := (volume : Measure E).eqOn_open_of_ae_eq hae hop (u.contDiff.continuous.mul u.contDiff.continuous).continuousOn |
| continuous_const.continuousOn |
| have hh := heq hbx |
| simpa using (mul_self_eq_zero.mp hh) |
|
|
|
|
| end |
|
|
| open MeasureTheory TopologicalSpace Filter Set |
| open |
| variable ℝ |
| [FiniteDimensional ℝ |
| variable ι ι ι → |
|
|
| theorem ι → ι → |
| (ha ∀ ψ ₀ |
| (hψ : ψ x₀ = 0) (hpos ₀ |
| (hgrad ψ ₀ ≠ |
| (O ₀ ₀ ∈ |
| (hsol ∀ ∈ |
| (hzero ∀ ∈ ψ → → ℝᶠ𝓝 ₀ |
| let η ψ ₀ |
| have η η ₀ ψ ₀ ψ |
| have η η ₀ ≠ |
| simpa η |
| obtain ⟨⟩ η ₀ η η |
| let φ η |
| have φ₀ φ ₀ |
| change η ₀ /2)*(η x₀*η x₀) = 0 |
| rw [hη]; ring |
| have hsmall : ∀ᶠ x in 𝓝 x₀, -1 < H*η x := |
| ((((continuous_const.mul (SmoothScalar.contDiff η).continuous) : |
| Continuous (fun x => H*η x)).tendsto x₀).eventually_const_lt (by change -1 < H*η x₀; rw [hη]; norm_num)) |
| obtain ⟨R,hR,hRsub⟩ := Metric.mem_nhds_iff.mp |
| (inter_mem hU (inter_mem (hO.mem_nhds hx₀) hsmall)) |
| let r := R/ |
| have |
| let ⟨₀ ₀ ⟩ |
| have ∈ ∈ ∧ ∈ ∧ η |
| apply |
| apply ₀ ≤ |
| dsimp ; linarith |
| let χ : ContDiffBump x₀ := ⟨r/2,r, half_pos hr, by linarith⟩ |
| let w : EllipticTest K := ⟨fun x => χ x * u x, χ.contDiff.mul (SmoothScalar.contDiff u), by |
| intro x hx |
| have hχ : χ x = 0 := χ.zero_of_le_dist (by |
| change ¬ dist x x₀ ≤ r at hx |
| exact (lt_of_not_ge hx).le) |
| simp only [hχ, zero_mul, Pi.zero_apply]⟩ |
| have hw (x : E) : w x = χ x * u x := rfl |
| let c := -r^2/8 |
| have hc : c < 0 := by dsimp [c]; nlinarith [sq_pos_of_pos hr] |
| have hres : ∀ x, c < φ x → testSchrodinger K e a V w x = 0 := by |
| intro x hxphi |
| by_cases hxK : x ∈ K |
| · have hxO := (hK x hxK).2.1 |
| by_cases hxi : x ∈ Metric.ball x₀ (r/2) |
| · have heq : (w : E → ℝ) =ᶠ[𝓝 x] u := by |
| filter_upwards [χ.eventuallyEq_one_of_mem_ball hxi] with y hy |
| simp only [hw, Pi.one_apply] at * |
| rw [hy, one_mul] |
| rw [testSchrodinger_eval_congr K e a V u w x heq] |
| exact hsol x hxO |
| · have hxψ : 0 < ψ x := by |
| by_contra hn |
| have hgap := tangentPhase_annular_gap ψ x₀ x r H hr (le_of_not_gt hn) |
| (le_of_not_gt hxi) (hK x hxK).2.2 |
| change φ x ≤ c at hgap |
| exact not_lt_of_ge hgap hxphi |
| have heq : (w : E → ℝ) =ᶠ[𝓝 x] 0 := by |
| filter_upwards [hO.mem_nhds hxO, |
| (isOpen_lt continuous_const (SmoothScalar.contDiff ψ).continuous).mem_nhds hxψ] with y hyO hyψ |
| simp only [hw, hzero y hyO hyψ, mul_zero, Pi.zero_apply] |
| exact testSchrodinger_zero_of_eventually_zero K e a V w x heq |
| · exact (testSchrodinger K e a V w).zero_on_compl hxK |
| have hupper := exponential_support_dominance K φ w (testSchrodinger K e a V w) 8 T c |
| (by norm_num) (fun τ hτ => hcar K (fun x hx => (hK x hx).1) τ hτ w) hres |
| filter_upwards [(isOpen_lt continuous_const (SmoothScalar.contDiff φ).continuous).mem_nhds |
| (show c < φ x₀ by rw [hφ₀]; exact hc), Metric.ball_mem_nhds x₀ (half_pos hr)] with x hxφ hxb |
| have hχ : χ x = 1 := χ.one_of_mem_closedBall (Metric.ball_subset_closedBall hxb) |
| have hh := hupper x hxφ |
| simpa only [hw, hχ, one_mul, Pi.zero_apply] using hh |
|
|
|
|
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
