| import Mathlib |
| import OAI.Geometry.SmoothYau.Limits.C1ExtensionNearCompact |
|
|
| 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 : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| lemma local_integration_by_parts_C1 {O : Set E} (hO : IsOpen O) |
| {f F : E → ℝ} (hf : ContDiff ℝ 1 f) (hfc : HasCompactSupport f) |
| (hfs : tsupport f ⊆ O) (hF : ContDiffOn ℝ 1 F O) (v : E) : |
| (∫ x in O, f x * fderiv ℝ F x v) = -(∫ x in O, fderiv ℝ f x v * F x) := by |
| obtain ⟨H, hH, he⟩ := C1_extension_near_compact hfc hO hfs hF |
| have he' (x : E) (hx : x ∈ tsupport f) : F =ᶠ[𝓝 x] H := |
| he.filter_mono (nhds_le_nhdsSet hx) |
| have hdf := (hf.fderiv_right (m := 0) (by simp)).clm_apply (contDiff_const (c := v)) |
| have hdH := (hH.fderiv_right (m := 0) (by simp)).clm_apply (contDiff_const (c := v)) |
| have h1 (x : E) : f x * fderiv ℝ F x v = f x * fderiv ℝ H x v := by |
| by_cases hx : x ∈ tsupport f |
| · rw [(he' x hx).fderiv_eq] |
| · simp only [image_eq_zero_of_notMem_tsupport hx, zero_mul] |
| have h2 (x : E) : fderiv ℝ f x v * F x = fderiv ℝ f x v * H x := by |
| by_cases hx : x ∈ tsupport f |
| · rw [(he' x hx).eq_of_nhds] |
| · rw [fderiv_of_notMem_tsupport ℝ hx] |
| simp |
| simp_rw [h1, h2] |
| rw [setIntegral_eq_integral_of_forall_compl_eq_zero |
| (fun x hx => by rw [image_eq_zero_of_notMem_tsupport (mt (@hfs x) hx), zero_mul]), |
| setIntegral_eq_integral_of_forall_compl_eq_zero |
| (fun x hx => by rw [fderiv_of_notMem_tsupport ℝ (mt (@hfs x) hx)]; simp)] |
| exact integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable |
| (hdf.continuous.mul hH.continuous |>.integrable_of_hasCompactSupport |
| ((hfc.fderiv_apply ℝ v).mul_right)) |
| (hf.continuous.mul hdH.continuous |>.integrable_of_hasCompactSupport hfc.mul_right) |
| (hf.continuous.mul hH.continuous |>.integrable_of_hasCompactSupport hfc.mul_right) |
| (fun x _ => (hf.differentiable (by simp)).differentiableAt) |
| (fun x _ => (hH.differentiable (by simp)).differentiableAt) |
| |
| end YauCounterexamples |
| |
| end |
| |
| end OAI |
| |
