| import OAI.Geometry.RCD.Sobolev.SobolevPointedTestRecovery |
| import OAI.Analysis.RCDHeat.HopfTransport |
|
|
| namespace OAI |
|
|
| /-! # Compact Hopf |
|
|
| open Set Metric Filter MeasureTheory |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
|
|
| theorem compact_hopf_dirichlet (μ : Measure X) [IsFiniteMeasureOnCompacts μ] |
| (hQ : QuadraticCheegerEnergy μ) (hgeo : IsGeodesicSpace X) |
| {p : X} {r b S t : ℝ} (hr : 0<r) (ht : 0<t) {M : ℝ≥0} |
| (hroom : b+2*t*(M:ℝ)<S) {F : X → ℝ} |
| (hF : LipschitzWith ⟨(M:ℝ)/r,div_nonneg M.coe_nonneg hr.le⟩ F) |
| (hs : tsupport F⊆closedBall p (b*r)) : |
| ∃ w : SobolevSpace.dirichlet μ hQ (ball p (S*r)), |
| (SobolevSpace.toLp μ hQ w.val =ᵐ[μ] hopfLax F (t*r^2)) ∧ |
| ∀ᵐ x ∂μ, SobolevSpace.minimalSlope μ hQ w.val x≤ |
| (localSlope (hopfLax F (t*r^2)) x).toReal := by |
| have hT : 0<t*r^2 := mul_pos ht (sq_pos_of_pos hr) |
| have hFL := hopfLax_lipschitz hF hgeo hT |
| have hzero (x : X) (hx : b*r<dist p x) : F x=0 := by |
| by_contra hne |
| have hh := hs (subset_tsupport F hne) |
| rw [mem_closedBall,dist_comm] at hh |
| linarith |
| have hqs : tsupport (hopfLax F (t*r^2))⊆closedBall p ((b+2*t*(M:ℝ))*r) := by |
| apply closure_minimal _ isClosed_closedBall |
| intro x hx |
| by_contra hnot |
| have hd : (b+2*t*(M:ℝ))*r<dist p x := by |
| simpa only [mem_closedBall,dist_comm,not_le] using hnot |
| apply hx |
| apply hopfLax_zero_outside hF p (b*r) hzero hT |
| change b*r+2*(t*r^2)*((M:ℝ)/r)<dist p x |
| have heq : b*r+2*(t*r^2)*((M:ℝ)/r)=(b+2*t*(M:ℝ))*r := by |
| field_simp |
| rwa [heq] |
| have hqc : HasCompactSupport (hopfLax F (t*r^2)) := |
| (isCompact_closedBall _ _).of_isClosed_subset (isClosed_tsupport _) hqs |
| let w0 := SobolevSpace.ofLipschitzCompact μ hQ hFL hqc |
| have hrep : SobolevSpace.toLp μ hQ w0 =ᵐ[μ] hopfLax F (t*r^2) := |
| SobolevSpace.ofLipschitzCompact_coe μ hQ hFL hqc |
| have hdir : w0∈SobolevSpace.dirichlet μ hQ (ball p (S*r)) := |
| (SobolevSpace.dirichletCore μ hQ (ball p (S*r))).le_topologicalClosure |
| ⟨hopfLax F (t*r^2),_,hFL,hqc,hqs.trans (closedBall_subset_ball |
| (mul_lt_mul_of_pos_right hroom hr)),hrep⟩ |
| exact ⟨⟨w0,hdir⟩,hrep,SobolevSpace.ofLipschitzCompact_minimalSlope_le_slope μ hQ hFL hqc⟩ |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
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|
