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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import Mathlib
import OAI.Geometry.SmoothYau.Geometry.CoefficientGradientApply
import OAI.Geometry.SmoothYau.Geometry.RoundMetricExists
namespace OAI
noncomputable section
open Set Filter Function
open scoped Topology ContDiff Manifold SchwartzMap
open Set Filter Manifold Bundle MeasureTheory NNReal
open scoped Topology ContDiff ENNReal
open Set Filter Topology NNReal
open Set Filter Module
open scoped Topology
namespace YauCounterexamples
open Set Filter MeasureTheory Manifold Function
open scoped Topology ContDiff
variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
[TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M] [T2Space M]
omit [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M] [T2Space M] in
lemma integrableOn_compact_tsupport {O : Set E} (hO : IsOpen O) {f : E → ℝ}
(hf : ContinuousOn f O) (hc : HasCompactSupport f) (hs : tsupport f ⊆ O) :
IntegrableOn f O := by
apply IntegrableOn.of_inter_support hO.measurableSet
exact ((hf.mono hs).integrableOn_compact hc).mono_set (fun _ hx => subset_tsupport f hx.2)
omit [T2Space M] in
theorem metricChartIntegral_green (g : SmoothMetric E M) (p : M) {u v : M → ℝ}
(hu : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) ∞ u)
(hv : ContMDiff 𝓘(ℝ, E) 𝓘(ℝ, ℝ) ∞ v)
(huc : HasCompactSupport u) (hus : tsupport u ⊆ (chartAt E p).source) :
metricChartIntegral g p (fun x => u x * laplaceBeltrami g v x) =
-metricChartIntegral g p (coordinateGradientPair g u v) := by
let f := realChartLocalize (E := E) p u
let O := (chartAt E p).target
let b := Module.finBasis ℝ E
have hf : ContDiff ℝ ∞ f := contDiff_realChartLocalize p u huc hus hu
have hfc : HasCompactSupport f := hasCompactSupport_realChartLocalize p u huc hus
have hfs : tsupport f ⊆ O := tsupport_realChartLocalize_subset p u huc hus
have hflux (i : CoordIndex E) : ContDiffOn ℝ ∞ (fun y => metricFlux g v p y i) O :=
fun y hy => (contDiffAt_metricFlux hv g p hy i).contDiffWithinAt
have hdf (i : CoordIndex E) : ContDiff ℝ ∞ (fun y => fderiv ℝ f y (b i)) :=
(hf.fderiv_right (m := ∞) (by simp)).clm_apply contDiff_const
have hFi (i : CoordIndex E) : IntegrableOn (fun y => f y *
fderiv ℝ (fun z => metricFlux g v p z i) y (b i)) O := by
apply integrableOn_compact_tsupport (chartAt E p).open_target
· apply hf.continuous.continuousOn.mul
intro y hy
exact (((contDiffAt_metricFlux hv g p hy i).fderiv_right (m := ∞) (by simp)).clm_apply
contDiffAt_const).continuousAt.continuousWithinAt
· exact hfc.mul_right
· exact tsupport_mul_subset_left.trans hfs
have hGi (i : CoordIndex E) : IntegrableOn (fun y => fderiv ℝ f y (b i) * metricFlux g v p y i) O := by
apply integrableOn_compact_tsupport (chartAt E p).open_target
· exact (hdf i).continuous.continuousOn.mul (hflux i).continuousOn
· exact (hfc.fderiv_apply ℝ (b i)).mul_right
· exact tsupport_mul_subset_left.trans ((tsupport_fderiv_apply_subset ℝ (b i)).trans hfs)
have hleft : metricChartIntegral g p (fun x => u x * laplaceBeltrami g v x) =
∫ y in O, ∑ i, f y * fderiv ℝ (fun z => metricFlux g v p z i) y (b i) := by
apply setIntegral_congr_fun (chartAt E p).open_target.measurableSet
intro y hy
dsimp only
rw [laplaceBeltrami_inChart hv g p _ ((chartAt E p).map_target hy),
(chartAt E p).right_inv hy]
simp only [localLaplacian, ← Finset.mul_sum]
rw [show f y = u ((chartAt E p).symm y) from realChartLocalize_eq p u hy]
have hd : Real.sqrt (metricCoefficients g p y).det ≠ 0 :=
(Real.sqrt_pos.2 (metricCoefficients_posDef g p hy).det_pos).ne'
dsimp only [b]
field_simp
have hright : metricChartIntegral g p (coordinateGradientPair g u v) =
∫ y in O, ∑ i, fderiv ℝ f y (b i) * metricFlux g v p y i := by
apply setIntegral_congr_fun (chartAt E p).open_target.measurableSet
intro y hy
dsimp only
rw [coordinateGradientPair_inChart hu hv g p _ ((chartAt E p).map_target hy),
(chartAt E p).right_inv hy]
dsimp only [f]
rw [realChartLocalize_fderiv p u hy]
simp only [localGradientPair, metricFlux, Matrix.mulVec, dotProduct, Finset.mul_sum, b]
apply Finset.sum_congr rfl
intro i _
ring_nf
rw [hleft, hright, integral_finsetSum _ (fun i _ => hFi i),
integral_finsetSum _ (fun i _ => hGi i), ← Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro i _
exact local_integration_by_parts (chartAt E p).open_target hf hfc hfs (hflux i) (b i)
end YauCounterexamples
end
end OAI
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