| import OAI.Analysis.BernoulliCone.TraceBound |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Filter InnerProductSpace MeasureTheory |
| open scoped Topology ContDiff NNReal |
| namespace Bernoulli.DeSilvaJerison |
| open Foliation |
|
|
| def shiftU (ε:ℝ) (x:Space 7):ℝ:=U x+ε |
| def ceilingGraph (ε:ℝ) (v:Space 7→ℝ) (x:Space 7):ℝ:=max (shiftU ε x) (v x) |
| def comparisonR (ε:ℝ) (v:Space 7→ℝ) (x:Space 7):ℝ:= |
| if U x+ε<v x then ‖gradient v x‖^2+‖upperFlux (x,U x+ε)‖^2- |
| 2*inner ℝ (upperFlux (x,U x+ε)) (gradient U x) else 0 |
|
|
| lemma shiftU_lipschitz {K:ℝ≥0} (hK:LipschitzWith K U) (ε:ℝ) : |
| LipschitzWith K (shiftU ε):=by |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| simpa only [shiftU,dist_add_right] using hK.dist_le_mul x y |
| lemma ceilingGraph_lipschitz {K L:ℝ≥0} {v:Space 7→ℝ} |
| (hK:LipschitzWith K U) (hL:LipschitzWith L v) (ε:ℝ) : |
| LipschitzWith (max K L) (ceilingGraph ε v):=(shiftU_lipschitz hK ε).max hL |
| lemma ceiling_eq_shift_outside {v:Space 7→ℝ} {ε:ℝ} (hε:0≤ε) {x:Space 7} |
| (hx:x∉tsupport (fun y=>v y-U y)) : ceilingGraph ε v x=shiftU ε x := by |
| have hv:v x=U x:=sub_eq_zero.mp (image_eq_zero_of_notMem_tsupport (f:=fun y=>v y-U y) hx) |
| exact max_eq_left (by dsimp [shiftU]; rw [hv]; linarith) |
| lemma ceiling_sub_compact {v:Space 7→ℝ} {ε:ℝ} (hε:0≤ε) |
| (hc:HasCompactSupport (fun y=>v y-U y)) : |
| HasCompactSupport (fun x=>ceilingGraph ε v x-shiftU ε x) := by |
| apply HasCompactSupport.intro hc |
| intro x hx |
| rw [ceiling_eq_shift_outside hε hx,sub_self] |
| lemma comparison_primitive_support {v:Space 7→ℝ} {ε:ℝ} (hε:0≤ε) : |
| tsupport (verticalPotential upperFlux (shiftU ε) (ceilingGraph ε v))⊆tsupport (fun y=>v y-U y) := by |
| apply closure_minimal _ (isClosed_tsupport _) |
| intro x hx |
| by_contra hn |
| have he:=ceiling_eq_shift_outside hε hn |
| exact hx (by simp [verticalPotential,he]) |
| lemma comparison_primitive_compact {v:Space 7→ℝ} {ε:ℝ} (hε:0≤ε) |
| (hc:HasCompactSupport (fun y=>v y-U y)) : |
| HasCompactSupport (verticalPotential upperFlux (shiftU ε) (ceilingGraph ε v)) := |
| hc.of_isClosed_subset (isClosed_tsupport _) (comparison_primitive_support hε) |
|
|
| lemma shiftU_gradient {ε:ℝ} {x:Space 7} (hx:DifferentiableAt ℝ U x) : |
| gradient (shiftU ε) x=gradient U x := by |
| have hd:HasFDerivAt (shiftU ε) (fderiv ℝ U x) x:=hx.hasFDerivAt.add_const ε |
| unfold gradient |
| rw [hd.fderiv] |
| lemma ceilingGraph_gradient {ε:ℝ} {v:Space 7→ℝ} {x:Space 7} |
| (hv:ContinuousAt v x) (hlt:U x+ε<v x) : |
| gradient (ceilingGraph ε v) x=gradient v x := by |
| have he:ceilingGraph ε v =ᶠ[𝓝 x] v:=by |
| filter_upwards [(U_continuous.continuousAt.add_const ε).eventually_lt hv hlt] with y hy |
| exact max_eq_right hy.le |
| unfold gradient |
| rw [he.fderiv_eq] |
|
|
| lemma comparisonR_ae_bound {v:Space 7→ℝ} {L:ℝ≥0} (hv:LipschitzWith L v) |
| {ε:ℝ} (hε:0<ε) (hc:HasCompactSupport (fun y=>v y-U y)) : |
| ∀ᵐ x:Space 7 ∂volume, |
| divergence (verticalPotential upperFlux (shiftU ε) (ceilingGraph ε v)) x≤comparisonR ε v x := by |
| obtain ⟨K,hK⟩:=exists_lipschitz_U |
| have ha:=shiftU_lipschitz hK ε |
| have hb:=ceilingGraph_lipschitz hK hv ε |
| obtain ⟨M,hM⟩:=clamped_seam_lipschitz ha hb (fun x=>by dsimp [shiftU]; linarith) |
| (ceiling_sub_compact hε.le hc) |
| filter_upwards [radiusSq_pos_ae,hK.ae_differentiableAt,ha.ae_differentiableAt, |
| hb.ae_differentiableAt,hM.ae_differentiableAt] with x hqx hdu hda hdb hdw |
| have h:=upperFlux_primitive_bound ha.continuous.continuousAt hb.continuous.continuousAt |
| (fun y=>le_max_left _ _) (by dsimp [shiftU]; linarith) hqx hda hdb hdw |
| rw [shiftU_gradient hdu] at h |
| by_cases hlt:U x+ε<v x |
| · have hh:shiftU ε x<ceilingGraph ε v x:=(lt_max_iff.mpr (Or.inr hlt)) |
| rw [ite_eq_left hh,ceilingGraph_gradient hv.continuous.continuousAt hlt] at h |
| simpa only [shiftU,comparisonR,ite_eq_left hlt] using h |
| · have hh:¬shiftU ε x<ceilingGraph ε v x:=by |
| simpa only [ceilingGraph,shiftU,lt_max_iff,lt_self_iff_false,false_or] using hlt |
| simpa only [ite_eq_right hh,comparisonR,ite_eq_right hlt] using h |
|
|
| lemma comparisonR_measurable {v:Space 7→ℝ} (hv:Measurable v) (ε:ℝ): |
| Measurable (comparisonR ε v) := by |
| have hA:Measurable (fun x:Space 7=>upperFlux (x,U x+ε)):= |
| upperFlux_measurable.comp (measurable_id.prodMk (U_continuous.measurable.add_const ε)) |
| exact ((LipschitzSobolev.measurable_gradient.norm.pow_const 2 |>.add (hA.norm.pow_const 2)).sub |
| ((hA.inner LipschitzSobolev.measurable_gradient).const_mul 2)).ite |
| (measurableSet_lt (U_continuous.measurable.add_const ε) hv) measurable_const |
|
|
| lemma comparisonR_bound {v:Space 7→ℝ} {K L:ℝ≥0} (hK:LipschitzWith K U) |
| (hL:LipschitzWith L v) {C:ℝ} (hC:0≤C) (hA:∀p:Space 7×ℝ,‖upperFlux p‖≤C) (ε:ℝ) (x:Space 7): |
| ‖comparisonR ε v x‖≤(L:ℝ)^2+C^2+2*C*K := by |
| unfold comparisonR |
| split_ifs |
| · have h1:‖gradient v x‖^2≤(L:ℝ)^2:=sq_le_sq₀ (norm_nonneg _) L.2 |>.mpr (LipschitzSobolev.norm_gradient_le hL x) |
| have h2:‖upperFlux (x,U x+ε)‖^2≤C^2:=sq_le_sq₀ (norm_nonneg _) hC |>.mpr (hA _) |
| have h3:‖inner ℝ (upperFlux (x,U x+ε)) (gradient U x)‖≤C*K:= |
| (norm_inner_le_norm _ _).trans (mul_le_mul (hA _) (LipschitzSobolev.norm_gradient_le hK x) (norm_nonneg _) hC) |
| calc |
| _ ≤ ‖(‖gradient v x‖^2+‖upperFlux (x,U x+ε)‖^2)‖+‖2*inner ℝ (upperFlux (x,U x+ε)) (gradient U x)‖:=norm_sub_le _ _ |
| _ ≤ (L:ℝ)^2+C^2+2*C*K:=by |
| rw [Real.norm_eq_abs,abs_of_nonneg (add_nonneg (sq_nonneg _) (sq_nonneg _)),norm_mul] |
| norm_num only [Real.norm_ofNat] |
| nlinarith |
| · simp only [norm_zero]; positivity |
|
|
| /-- The genuine compact graph primitive yields an integrated comparison at |
| positive height. Hypotheses concern a competitor, never desired minimality. -/ |
| theorem comparisonR_integral_nonneg {v:Space 7→ℝ} {L:ℝ≥0} (hv:LipschitzWith L v) |
| {ε:ℝ} (hε:0<ε) (hc:HasCompactSupport (fun y=>v y-U y)) |
| {c:Space 7} {r:ℝ} (hS:tsupport (fun y=>v y-U y)⊆Metric.ball c r) : |
| 0≤∫x in Metric.ball c r,comparisonR ε v x := by |
| let P:=verticalPotential upperFlux (shiftU ε) (ceilingGraph ε v) |
| obtain ⟨K,hK⟩:=exists_lipschitz_U |
| have hPc:HasCompactSupport P:=comparison_primitive_compact hε.le hc |
| have hPl:LocallyLipschitz P:=upperFlux_primitive_locallyLipschitz |
| (shiftU_lipschitz hK ε).locallyLipschitz (ceilingGraph_lipschitz hK hv ε).locallyLipschitz |
| (fun x=>le_max_left _ _) (fun x=>by dsimp [shiftU]; linarith) |
| obtain ⟨M,hM⟩:=locallyLipschitz_compact_lipschitz hPl hPc |
| obtain ⟨C,hC,hA⟩:=upperFlux_uniform_bound |
| have hR:IntegrableOn (comparisonR ε v) (Metric.ball c r):= |
| (integrableOn_const (μ:=volume) (C:=(L:ℝ)^2+C^2+2*C*K) (measure_ball_lt_top.ne)).mono' |
| (comparisonR_measurable hv.continuous.measurable ε).aestronglyMeasurable.restrict |
| (Eventually.of_forall (comparisonR_bound hK hv hC.le hA ε)) |
| have hz:(∫x in Metric.ball c r,divergence P x)=0:=by |
| rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx=> |
| divergence_zero_outside_support (fun hp=>hx (hS (comparison_primitive_support hε.le hp))))] |
| exact integral_divergence_compact hM hPc |
| rw [←hz] |
| exact integral_mono_ae (integrable_divergence_compact hM hPc).integrableOn hR |
| ((comparisonR_ae_bound hv hε hc).filter_mono (ae_mono Measure.restrict_le_self)) |
| end Bernoulli.DeSilvaJerison |
| |
| end |
| end |
| |
| end OAI |
| |
