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2fca478 | 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 | import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
namespace OAI
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
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