| import OAI.Geometry.RCD.Coordinates.RegularDirichletCore |
| import OAI.Geometry.RCD.Sobolev.WeakLaplacianProduct |
| import OAI.Geometry.RCD.Sobolev.GoodLaplacianCutoff |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| A bounded Lipschitz function with weak L² Laplacian is multiplied by an |
| actual good cutoff. On its plateau both the function and source agree |
| with the originals. No constant-source or source-gradient premise is used. |
| -/ |
|
|
| open Set Filter Metric MeasureTheory |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] (μ : Measure X) [IsFiniteMeasureOnCompacts μ] (hQ : QuadraticCheegerEnergy μ) |
|
|
| theorem bounded_regular_localization (u : SobolevSpace μ hQ) (d : Lp ℝ 2 μ) |
| (hd : IsWeakLaplacian μ hQ u d) (v : X → ℝ) {L : ℝ≥0} |
| (hv : LipschitzWith L v) (hu : SobolevSpace.toLp μ hQ u =ᵐ[μ] v) |
| (A : ℝ≥0) (hA : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ hQ u x|≤A) |
| {p : X} {r C : ℝ} (χ : GoodLaplacianCutoff μ hQ p r C) : |
| ∃ (F : compactLaplacianCore μ hQ univ) (q : Lp ℝ 2 μ), |
| IsWeakLaplacian μ hQ F.val q ∧ |
| SobolevSpace.toLp μ hQ F.val =ᵐ[μ.restrict (ball p r)] SobolevSpace.toLp μ hQ u ∧ |
| q =ᵐ[μ.restrict (ball p r)] d := by |
| have hχb : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ hQ χ.sobolev x|≤(1:ℝ≥0) := by |
| filter_upwards [χ.representative] with x hx |
| rw [hx,abs_of_nonneg (χ.bounds x).1] |
| exact (χ.bounds x).2 |
| have hχg := SobolevSpace.minimalSlope_le_lipschitz μ hQ χ.sobolev χ.lipschitz χ.representative |
| have hΓ : ∀ᵐ x ∂μ, localGamma μ hQ χ.sobolev χ.sobolev x≤((C/r).toNNReal:ℝ)^2 := by |
| filter_upwards [localGamma_self μ hQ χ.sobolev,hχg, |
| (Lp.coeFn_nonneg _).mpr (SobolevSpace.minimalSlope_nonneg μ hQ χ.sobolev)] with x hx hg hn |
| rw [hx] |
| exact pow_le_pow_left₀ hn hg 2 |
| let F := SobolevSpace.product μ hQ u χ.sobolev A 1 |
| obtain ⟨q,hq,hqr⟩ := hd.product μ hQ u χ.sobolev d χ.laplacian χ.weak_laplacian |
| A 1 (C/r).toNNReal hA hχb hΓ |
| have hFr : SobolevSpace.toLp μ hQ F =ᵐ[μ] fun x => χ.weight x*v x := by |
| filter_upwards [SobolevSpace.product_coe μ hQ u χ.sobolev A 1 hA hχb,hu,χ.representative] with x hx hy hz |
| change SobolevSpace.toLp μ hQ (SobolevSpace.product μ hQ u χ.sobolev A 1) x = _ |
| rw [hx,hy,hz,mul_comm] |
| obtain ⟨B,hB⟩ := lipschitz_mul_compact_cutoff χ.lipschitz hv (χ.hasCompactSupport μ hQ) |
| (fun x => by rw [abs_of_nonneg (χ.bounds x).1]; exact (χ.bounds x).2) |
| have hFc : F ∈ compactLaplacianCore μ hQ univ := |
| ⟨⟨_,B,hB,(χ.hasCompactSupport μ hQ).mul_right,subset_univ _,hFr⟩,q,hq⟩ |
| refine ⟨⟨F,hFc⟩,q,hq,?_,?_⟩ |
| · apply (ae_restrict_iff' measurableSet_ball).mpr |
| filter_upwards [hFr,hu] with x hx hy hball |
| rw [hx,χ.one x hball,one_mul,hy] |
| · apply (ae_restrict_iff' measurableSet_ball).mpr |
| filter_upwards [hqr,χ.representative,χ.laplacian_locality, |
| localGamma_zero_on_level μ hQ χ.sobolev u 1, |
| localGamma_comm μ hQ u χ.sobolev] with x hx hχ hΔ hz hc hball |
| have hχ1 := χ.one x hball |
| have hΓ0 : localGamma μ hQ u χ.sobolev x=0 := hc.trans (hz (hχ.trans hχ1)) |
| rw [hx,hχ,hχ1,hΔ (Or.inl (by simpa only [mem_ball,dist_comm] using hball)),hΓ0] |
| ring |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
