| import OAI.Analysis.BernoulliCone.HalfspaceGroundstate |
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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 |
| open scoped _root_.Bernoulli |
| variable {d : ℕ} {D : Set (Space d)} |
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| lemma dominatedVariation_of_comparison [IsFiniteMeasure (volume.restrict D)] |
| {u v : Space d → ℝ} {G V : Space d → Space d} |
| (hu : H1 D u G) (hv : H10 D v V) |
| (hn : ∀ᵐ x ∂volume.restrict D, 0≤u x) |
| {C : ℝ} (hC : 0<C) (hbound : ∀ᵐ x ∂volume.restrict D, |v x| ≤ C*u x) |
| (hmin : ∀ (w : Space d → ℝ) (W : Space d → Space d), |
| H1 D w W → (∀ᵐ x ∂volume.restrict D, 0≤w x) → |
| H10 D (fun x=>w x-u x) (fun x=>W x-G x) → energy D u G≤energy D w W) : |
| (∫ x in D, inner ℝ (G x) (V x))=0 := by |
| apply linear_coefficient_eq_zero _ (∫ x in D, ‖V x‖^2) (inv_pos.mpr hC) |
| intro t ht |
| have htC : |t| * C<1 := by |
| calc |t| * C < C⁻¹*C := mul_lt_mul_of_pos_right ht hC |
| _ = 1 := inv_mul_cancel₀ hC.ne' |
| have hcontrol : ∀ᵐ x ∂volume.restrict D, |
| 0≤u x+t*v x ∧ ((0<u x+t*v x) ↔ 0<u x) := by |
| filter_upwards [hn,hbound] with x hx hbx |
| have hb : |t*v x| ≤ (|t| * C)*u x := by |
| rw [abs_mul,mul_assoc] |
| exact mul_le_mul_of_nonneg_left hbx (abs_nonneg t) |
| by_cases hup : 0<u x |
| · have hp : 0<u x+t*v x := by |
| have := neg_abs_le (t*v x) |
| nlinarith |
| exact ⟨hp.le,iff_of_true hp hup⟩ |
| · have hz : u x=0 := le_antisymm (not_lt.mp hup) hx |
| have hvz : v x=0 := abs_eq_zero.mp (le_antisymm (by simpa only [hz,mul_zero] using hbx) (abs_nonneg _)) |
| simp only [hz,hvz,mul_zero,add_zero,le_refl,lt_self_iff_false,iff_self,and_self] |
| have hc := hmin (fun x=>u x+t*v x) (fun x=>G x+t • V x) |
| (hu.add (hv.1.const_smul t)) (hcontrol.mono fun _ hx=>hx.1) |
| (by simpa only [add_sub_cancel_left] using hv.const_smul t) |
| rw [energy_affine hu.1 hu.2.1 hv.1.2.1 t (hcontrol.mono fun _ hx=>hx.2)] at hc |
| linarith |
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| lemma GlobalMinimizer.dominatedVariation {u v : Space d → ℝ} {L : ℝ≥0} |
| (hm : GlobalMinimizer u) (hL : LipschitzWith L u) |
| {a : Space d} {R C : ℝ} (hR : 0<R) |
| {V : Space d → Space d} (hv : H10 (ball a R) v V) |
| (hC : 0<C) (hbound : ∀ᵐ x ∂volume.restrict (ball a R), |v x| ≤ C*u x) : |
| (∫ x in ball a R, inner ℝ (gradient u x) (V x))=0 := by |
| let : IsFiniteMeasure (volume.restrict (ball a R)) := |
| isFiniteMeasure_restrict.mpr measure_ball_lt_top.ne |
| obtain ⟨_,G,hG,hn,hmin⟩ := hm |
| have hu := hG a R hR (subset_univ _) |
| have he := hu.gradient_unique isOpen_ball (LipschitzSobolev.h1_ball hL a R) |
| have hh := dominatedVariation_of_comparison hu hv |
| (ae_restrict_of_ae (by simpa only [Measure.restrict_univ] using hn)) hC hbound |
| (hmin a R hR (subset_univ _)) |
| rw [←hh] |
| apply integral_congr_ae |
| filter_upwards [he] with x hx |
| rw [hx] |
| end Bernoulli |
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| end |
| end |
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| end OAI |
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