| import OAI.Analysis.BernoulliCone.RadialShapeTest |
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| namespace OAI |
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| section |
| noncomputable section |
| open MeasureTheory Set Filter Metric |
| open scoped Topology NNReal ENNReal ContDiff |
| namespace Bernoulli.Shape |
| variable {d : ℕ} |
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| lemma small_parameter_bounds {φ : Space d → ℝ} {C : ℝ≥0} |
| (hφ : ContDiff ℝ ∞ φ) (hLip : LipschitzWith C φ) |
| (e : Space d) {t : ℝ} (ht : |t| < (4*((C:ℝ)*‖e‖+1))⁻¹) : |
| ‖t‖₊*C*‖e‖₊ < 1 ∧ ∀ x, (1:ℝ)/2 ≤ 1+t*fderiv ℝ φ x e := by |
| have hC : 0 ≤ (C:ℝ)*‖e‖ := mul_nonneg C.coe_nonneg (norm_nonneg _) |
| have hp : 0 < 4*((C:ℝ)*‖e‖+1) := by positivity |
| have hh : |t| *(4*((C:ℝ)*‖e‖+1)) < 1 := (lt_div_iff₀ hp).mp (by simpa only [one_div] using ht) |
| have hprod : |t| *(C:ℝ)*‖e‖ < (1:ℝ)/4 := by nlinarith [abs_nonneg t] |
| constructor |
| · change ‖t‖*(C:ℝ)*‖e‖ < 1 |
| rw [Real.norm_eq_abs] |
| linarith |
| · intro x |
| have hb := (fderiv ℝ φ x).le_opNorm e |
| have hn := ((hφ.differentiable (by simp)) x).hasFDerivAt.le_of_lipschitz hLip |
| have hb' : |fderiv ℝ φ x e| ≤ (C:ℝ)*‖e‖ := by |
| rw [← Real.norm_eq_abs] |
| exact hb.trans (mul_le_mul_of_nonneg_right hn (norm_nonneg _)) |
| have hh' : |t*fderiv ℝ φ x e| < (1:ℝ)/4 := by |
| rw [abs_mul] |
| exact lt_of_le_of_lt (mul_le_mul_of_nonneg_left hb' (abs_nonneg _)) (by simpa only [mul_assoc] using hprod) |
| have hneg := neg_abs_le (t*fderiv ℝ φ x e) |
| linarith |
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| /-- Domain comparisons for the *actual* Bernoulli energy force the unit slope |
| of a minimizing halfspace. No boundary condition is assumed. -/ |
| theorem positiveLinear_minimizer_unit {e : Space d} (he : e ≠ 0) |
| (hm : GlobalMinimizer (positiveLinear e)) : ‖e‖=1 := by |
| let φ : Space d → ℝ := HarmonicKernel.potential |
| have hφ : Test (ball (0:Space d) 2) φ := potential_test |
| obtain ⟨C,hLip⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hφ.2.1 hφ.1 (by simp) |
| let I : ℝ := ∫ x in ball (0:Space d) 2, firstTerm φ e x |
| have hI : I ≠ 0 := firstTerm_potential_integral_ne_zero he |
| let B : ℝ := (2*‖e‖^4*(C:ℝ)^2)*(volume (ball (0:Space d) 2)).toReal |
| have hε : 0 < (4*((C:ℝ)*‖e‖+1))⁻¹ := by positivity |
| have hz : (1-‖e‖^2)*I/2 = 0 := by |
| apply linear_coefficient_eq_zero _ B hε |
| intro t ht |
| obtain ⟨hK,hdet⟩ := small_parameter_bounds hφ.1 hLip e ht |
| let hf := displacement_lipschitz hLip e t |
| let g : Space d ≃ₜ Space d := perturbationHomeomorph hf hK |
| have hg : ∀ x, g x=deform φ e t x := fun _ => rfl |
| let v := fun y => positiveLinear e (g.symm y) |
| have hvLip := (positiveLinear_lipschitz e).comp (perturbationHomeomorph_symm_lipschitz hf hK) |
| have hsup : tsupport (fun x => v x-positiveLinear e x) ⊆ tsupport φ := |
| pullback_diff_support (u:=positiveLinear e) e t g hg |
| have hc : HasCompactSupport (fun x => v x-positiveLinear e x) := |
| hφ.2.1.of_isClosed_subset (isClosed_tsupport _) hsup |
| have hcB := hsup.trans hφ.2.2 |
| have hcomp := hm.compare_lipschitz (positiveLinear_lipschitz e) hvLip 0 (by norm_num : 0 < (2:ℝ)) |
| (Eventually.of_forall (fun _ => le_max_right _ _)) hc hcB |
| have hupper := energy_pullback_upper hφ.1 hLip he t hdet g hg 0 2 (by |
| intro x hx |
| by_contra hh |
| exact hx (hφ.2.2 (subset_tsupport φ hh))) |
| change energy (ball (0:Space d) 2) v (gradient v) ≤ _ at hupper |
| have hboth := hcomp.trans hupper |
| dsimp [B,I] |
| nlinarith |
| have hz' : (1-‖e‖^2)*I=0 := (div_eq_zero_iff).mp hz |>.resolve_right (by norm_num) |
| have hn : 1-‖e‖^2=0 := (mul_eq_zero.mp hz').resolve_right hI |
| nlinarith [norm_nonneg e] |
| end Bernoulli.Shape |
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| end |
| end |
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| end OAI |
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