| import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep |
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| namespace OAI |
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|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal ContDiff InnerProductSpace |
| namespace RPISingularHarnack |
| variable {n : ℕ} |
|
|
| theorem gradient_tsupport_subset (f : Euc n → ℝ) : tsupport (gradient f) ⊆ tsupport f := by |
| apply closure_minimal _ (isClosed_tsupport _) |
| intro x hx |
| by_contra h |
| have hd := fderiv_of_notMem_tsupport ℝ h |
| have hg : gradient f x=0 := by simp only [gradient,hd,map_zero] |
| exact hx hg |
|
|
| theorem gradient_norm_eq_fderiv (f : Euc n → ℝ) (x : Euc n) : ‖gradient f x‖=‖fderiv ℝ f x‖ := |
| (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map _ |
|
|
| theorem integral_inner_gradient_eq_setIntegral {G : Euc n → Euc n} {φ : Euc n → ℝ} |
| {U : Set (Euc n)} (hU : MeasurableSet U) (hs : tsupport φ ⊆ U) : |
| (∫ x, inner ℝ (G x) (gradient φ x)) = ∫ x in U, inner ℝ (G x) (gradient φ x) := by |
| rw [←integral_indicator hU] |
| congr 1; funext x |
| by_cases hx : x∈U |
| · simp [hx] |
| · have hg : gradient φ x=0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hs (gradient_tsupport_subset φ h))) |
| simp [hx,hg] |
|
|
| theorem mollified_gradient_pairing_tendsto {G : Euc n → Euc n} (hG : Continuous G) |
| (p : Euc n) (r : ℝ) |
| {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ) |
| : Tendsto (fun j ↦ ∫ x in Metric.ball p r, inner ℝ (G x) (gradient (RPI.Boundary.c1Mollify φ j) x)) atTop |
| (𝓝 (∫ x in Metric.ball p r, inner ℝ (G x) (gradient φ x))) := by |
| let φj := RPI.Boundary.c1Mollify φ |
| obtain ⟨M,hM,hb⟩ := RPI.Boundary.c1Mollify_uniform_bound hφ hcφ |
| obtain ⟨C,hC⟩ := (isCompact_closedBall p r).exists_bound_of_continuousOn hG.continuousOn |
| have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by |
| rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩ |
| apply tendsto_integral_of_dominated_convergence (fun _ ↦ |C| * M) |
| · intro j |
| exact (hG.inner (gradient_continuous_C1 ((RPI.Boundary.c1Mollify_contDiff hφ.continuous j).of_le (by simp)))).aestronglyMeasurable |
| · exact integrable_const _ |
| · intro j |
| filter_upwards [ae_restrict_mem measurableSet_ball] with x hx |
| calc |
| ‖inner ℝ (G x) (gradient (φj j) x)‖ ≤ ‖G x‖*‖gradient (φj j) x‖ := norm_inner_le_norm _ _ |
| _ ≤ |C| * M := mul_le_mul ((hC x (Metric.ball_subset_closedBall hx)).trans (le_abs_self C)) |
| (by rw [gradient_norm_eq_fderiv]; exact (hb j x).2) (norm_nonneg _) (abs_nonneg _) |
| · apply ae_of_all |
| intro x |
| have hg : Tendsto (fun j ↦ gradient (φj j) x) atTop (𝓝 (gradient φ x)) := by |
| exact (InnerProductSpace.toDual ℝ (Euc n)).symm.continuous.tendsto (fderiv ℝ φ x) |>.comp |
| (RPI.Boundary.c1Mollify_fderiv_tendsto hφ hcφ x) |
| exact (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ G x) atTop (𝓝 (G x))).inner hg |
|
|
| theorem continuous_div_free_C1_test {G : Euc n → Euc n} (hG : Continuous G) |
| (p : Euc n) (r : ℝ) |
| (h : ∀ φ : SmoothLocalTest (Metric.ball p r), (∫ x, inner ℝ (G x) (gradient φ.val x))=0) |
| {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ) |
| (hsφ : tsupport φ ⊆ Metric.ball p r) : |
| (∫ x, inner ℝ (G x) (gradient φ x))=0 := by |
| let φj := RPI.Boundary.c1Mollify φ |
| have hs := RPI.Boundary.c1Mollify_eventually_tsupport_subset hcφ Metric.isOpen_ball hsφ |
| have hlim := mollified_gradient_pairing_tendsto hG p r hφ hcφ |
| have hz : ∀ᶠ j in atTop, (∫ x in Metric.ball p r, inner ℝ (G x) (gradient (φj j) x))=0 := by |
| filter_upwards [hs] with j hj |
| rw [←integral_inner_gradient_eq_setIntegral measurableSet_ball hj] |
| exact h ⟨φj j,RPI.Boundary.c1Mollify_contDiff hφ.continuous j, |
| RPI.Boundary.c1Mollify_hasCompactSupport hcφ j,hj⟩ |
| rw [integral_inner_gradient_eq_setIntegral measurableSet_ball hsφ] |
| exact tendsto_nhds_unique hlim (tendsto_const_nhds.congr' (hz.mono (fun _ hj ↦ hj.symm))) |
| |
| theorem WeakLinearEquation.test_C1 {A : Euc n → Euc n →L[ℝ] Euc n} |
| {f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f) |
| (hF : Continuous F) {p : Euc n} {r : ℝ} (h : WeakLinearEquation A f F (Metric.ball p r)) |
| {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ) |
| (hsφ : tsupport φ ⊆ Metric.ball p r) : |
| (∫ x, inner ℝ (A x (gradient f x)-F x) (gradient φ x))=0 := by |
| apply continuous_div_free_C1_test ((hA.clm_apply (gradient_continuous_C1 hf)).sub hF) p r _ hφ hcφ hsφ |
| intro ψ |
| change (∫ x, inner ℝ (A x (gradient f x)-F x) (gradient ψ.val x))=0 |
| simp_rw [inner_sub_left] |
| erw [integral_sub (inner_gradient_pairing_integrable (hA.clm_apply (gradient_continuous_C1 hf)) ψ) |
| (inner_gradient_pairing_integrable hF ψ),h ψ,sub_self] |
| |
| end RPISingularHarnack |
| |
| end |
| |
| end OAI |
| |
