| 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 |
| |
