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e4d9d2b | 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 | import OAI.Geometry.RCD.Sobolev.SobolevPointedTestRecovery
import OAI.Analysis.RCDHeat.HopfTransport
namespace OAI
/-! # Compact Hopf--Lax regularizations in the original Dirichlet space -/
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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