| import OAI.Analysis.BernoulliCone.FluxSeam |
| |
| namespace OAI |
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| universe uE uG |
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| section |
| noncomputable section |
| open MeasureTheory Set Filter InnerProductSpace |
| open scoped Topology ContDiff NNReal |
| namespace Bernoulli.Foliation |
| variable {E : Type uE} {G : Type uG} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] |
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| omit [NormedSpace ℝ G] in |
| lemma locallyLipschitz_compact_lipschitz [NormedSpace ℝ G] {F:E→G} (hl:LocallyLipschitz F) |
| (hc:HasCompactSupport F) : ∃K:ℝ≥0,LipschitzWith K F := by |
| obtain ⟨R,hR⟩:=hc.isBounded.subset_ball (0:E) |
| obtain ⟨K,hK⟩:=hl.locallyLipschitzOn.exists_lipschitzOnWith_of_compact |
| (isCompact_closedBall (0:E) (R+1)) |
| obtain ⟨M,hM⟩:=hc.exists_bound_of_continuousOn hl.continuous.continuousOn |
| let C:ℝ≥0:=⟨max (K:ℝ) (max M 0),le_max_of_le_right (le_max_right _ _)⟩ |
| have hKC:(K:ℝ)≤C:=le_max_left _ _ |
| have hMC:M≤(C:ℝ):=(le_max_left M 0).trans (le_max_right _ _) |
| have hbound (x y:E) (hx:x∈tsupport F) (hy:y∉Metric.closedBall (0:E) (R+1)): |
| dist (F x) (F y)≤(C:ℝ)*dist x y := by |
| have hFx:dist x 0<R:=hR hx |
| have hFy:R+1<dist y 0:=by simpa only [Metric.mem_closedBall,not_le] using hy |
| have hdy:1≤dist x y:=by have hh:=dist_triangle y x 0 |
| have hyS:y∉tsupport F:=by |
| intro hys |
| have hyR:=hR hys |
| change dist y 0<R at hyR |
| linarith |
| rw [image_eq_zero_of_notMem_tsupport hyS,dist_zero_right] |
| exact (hM x hx).trans (hMC.trans (le_mul_of_one_le_right C.2 hdy)) |
| refine ⟨C,LipschitzWith.of_dist_le_mul fun x y=>?_⟩ |
| by_cases hx:x∈tsupport F |
| · by_cases hy:y∈Metric.closedBall (0:E) (R+1) |
| · have hxB:x∈Metric.closedBall (0:E) (R+1):=by |
| have hh:=hR hx |
| change dist x 0<R at hh |
| change dist x 0≤R+1 |
| linarith |
| exact (hK.dist_le_mul x hxB y hy).trans (mul_le_mul_of_nonneg_right hKC dist_nonneg) |
| · exact hbound x y hx hy |
| · by_cases hy:y∈tsupport F |
| · by_cases hxB:x∈Metric.closedBall (0:E) (R+1) |
| · have hyB:y∈Metric.closedBall (0:E) (R+1):=by |
| have hh:=hR hy |
| change dist y 0<R at hh |
| change dist y 0≤R+1 |
| linarith |
| exact (hK.dist_le_mul x hxB y hyB).trans (mul_le_mul_of_nonneg_right hKC dist_nonneg) |
| · simpa only [dist_comm] using hbound y x hy hxB |
| · rw [image_eq_zero_of_notMem_tsupport hx,image_eq_zero_of_notMem_tsupport hy,dist_self] |
| positivity |
|
|
| lemma integral_directional_compact {d:ℕ} {f:Space d→ℝ} {K:ℝ≥0} |
| (hf:LipschitzWith K f) (hc:HasCompactSupport f) (v:Space d) : |
| (∫x,fderiv ℝ f x v)=0 := by |
| obtain ⟨φ,hφ,hφc,hφS,hφ1⟩:=LocalExtension.exists_cutoff hc isOpen_univ (subset_univ _) |
| have hφT:Test univ φ:=⟨hφ,hφc,hφS⟩ |
| have hh:=LipschitzSobolev.weakGradient hf univ φ hφT v |
| simp only [Measure.restrict_univ,inner_gradient_left] at hh |
| have hr:(fun x:Space d=>fderiv ℝ f x v * φ x)=(fun x=>fderiv ℝ f x v):=by |
| funext x |
| by_cases hx:x∈tsupport f |
| · have he:φ x=1:=(show φ =ᶠ[𝓝 x] (fun _=>1) from hφ1.filter_mono (nhds_le_nhdsSet hx)).self_of_nhds |
| rw [he,mul_one] |
| · rw [fderiv_of_notMem_tsupport ℝ hx,zero_apply,zero_mul] |
| have he:(fun x:Space d=>f x*(fderiv ℝ φ x) v)=0:=by |
| funext x |
| by_cases hx:x∈tsupport f |
| · have heq:φ =ᶠ[𝓝 x] (fun _=>1):=hφ1.filter_mono (nhds_le_nhdsSet hx) |
| simp only [heq.fderiv_eq,fderiv_fun_const,zero_apply,mul_zero,Pi.zero_apply] |
| · rw [image_eq_zero_of_notMem_tsupport hx,zero_mul] |
| rfl |
| rw [hr,he] at hh |
| simpa only [integral_zero,Pi.zero_def,zero_eq_neg] using hh |
| lemma integrable_directional_compact {d:ℕ} {f:Space d→ℝ} {K:ℝ≥0} |
| (hf:LipschitzWith K f) (hc:HasCompactSupport f) (v:Space d) : |
| Integrable (fun x=>fderiv ℝ f x v) := by |
| have hs:Function.support (fun x=>fderiv ℝ f x v)⊆tsupport f:= |
| (subset_tsupport _).trans (tsupport_fderiv_apply_subset ℝ v) |
| apply (integrableOn_iff_integrable_of_support_subset hs).mp |
| apply (integrableOn_const (μ:=volume) (C:=((K:ℝ)*‖v‖)) hc.measure_ne_top).mono' |
| (((continuous_id.clm_apply continuous_const : Continuous (fun L:Space d→L[ℝ]ℝ=>L v)).measurable.comp (measurable_fderiv ℝ f)).aestronglyMeasurable.restrict) |
| filter_upwards [] with x |
| exact ((fderiv ℝ f x).le_opNorm v).trans |
| (mul_le_mul_of_nonneg_right (norm_fderiv_le_of_lipschitz ℝ hf) (norm_nonneg v)) |
| |
| lemma integrable_divergence_compact {d:ℕ} {F:Space d→Space d} {K:ℝ≥0} |
| (hF:LipschitzWith K F) (hc:HasCompactSupport F) : Integrable (divergence F) := by |
| let e:=stdOrthonormalBasis ℝ (Space d) |
| let f:=fun i x=>inner ℝ (e i) (F x) |
| have hf i:LipschitzWith (‖innerSL ℝ (e i)‖₊*K) (f i):= |
| (innerSL ℝ (e i)).lipschitzWith.comp hF |
| have hfc i:HasCompactSupport (f i):=by |
| apply HasCompactSupport.intro hc |
| intro x hx |
| simp only [f,image_eq_zero_of_notMem_tsupport hx,inner_zero_right] |
| have he:divergence F =ᵐ[volume] (fun x=>∑i,fderiv ℝ (f i) x (e i)):=by |
| filter_upwards [hF.ae_differentiableAt] with x hx |
| unfold divergence |
| apply Finset.sum_congr rfl |
| intro i hi |
| have hd:HasFDerivAt (f i) ((innerSL ℝ (e i)) ∘L fderiv ℝ F x) x:= |
| (innerSL ℝ (e i)).hasFDerivAt.comp x hx.hasFDerivAt |
| rw [hd.fderiv] |
| rfl |
| exact (integrable_finsetSum _ (fun i hi=>integrable_directional_compact (hf i) (hfc i) (e i))).congr he.symm |
| |
| lemma divergence_zero_outside_support {d:ℕ} {F:Space d→Space d} {x:Space d} |
| (hx:x∉tsupport F) : divergence F x=0 := by |
| unfold divergence |
| simp only [fderiv_of_notMem_tsupport ℝ hx,zero_apply,inner_zero_right, |
| Finset.sum_const_zero] |
| |
| /-- The divergence of a genuine compactly supported Lipschitz vector field |
| has integral zero, with the a.e. derivative justified by Rademacher. -/ |
| theorem integral_divergence_compact {d:ℕ} {F:Space d→Space d} {K:ℝ≥0} |
| (hF:LipschitzWith K F) (hc:HasCompactSupport F) : |
| (∫x,divergence F x)=0 := by |
| let e:=stdOrthonormalBasis ℝ (Space d) |
| let f:=fun i x=>inner ℝ (e i) (F x) |
| have hf i:LipschitzWith (‖innerSL ℝ (e i)‖₊*K) (f i):= |
| (innerSL ℝ (e i)).lipschitzWith.comp hF |
| have hfc i:HasCompactSupport (f i):=by |
| apply HasCompactSupport.intro hc |
| intro x hx |
| simp only [f,image_eq_zero_of_notMem_tsupport hx,inner_zero_right] |
| have he:divergence F =ᵐ[volume] (fun x=>∑i,fderiv ℝ (f i) x (e i)):=by |
| filter_upwards [hF.ae_differentiableAt] with x hx |
| unfold divergence |
| apply Finset.sum_congr rfl |
| intro i hi |
| have hd:HasFDerivAt (f i) ((innerSL ℝ (e i)) ∘L fderiv ℝ F x) x:= |
| (innerSL ℝ (e i)).hasFDerivAt.comp x hx.hasFDerivAt |
| rw [hd.fderiv] |
| rfl |
| rw [integral_congr_ae he,integral_finsetSum _ (fun i hi=>integrable_directional_compact (hf i) (hfc i) (e i))] |
| simp only [integral_directional_compact (hf _) (hfc _),Finset.sum_const_zero] |
| end Bernoulli.Foliation |
| |
| end |
| end |
| |
| end OAI |
| |
