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a97875e | 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 | import OAI.Analysis.CircleDomains.Sobolev.ComplexScalarEnergy
import OAI.Analysis.CircleDomains.Sobolev.LocalComplexWitness
namespace OAI
/-! A compact continuous complex-plane scalar with square-integrable
classical differential on an open set has its actual physical Sobolev
witness. This is used for the compact horizontal priority test. -/
open Set Function Filter MeasureTheory
open scoped Topology ContDiff ENNReal
noncomputable section
namespace CircleDomainRigidity
theorem complex_source_sobolev_witness
{D : Set ℂ} (hD : IsOpen D) {f : ℂ → ℝ}
(hc : Continuous f) (hs : HasCompactSupport f)
(hf : ContDiffOn ℝ ∞ f D)
(hE : MemLp (fderiv ℝ f) 2 (volume.restrict D)) :
Nonempty (DeGiorgi.MemW1pWitness 2 (f ∘ complexPlaneCoordinates)
(complexPlaneCoordinates ⁻¹' D)) := by
let Ω := complexPlaneCoordinates ⁻¹' D
let F := f ∘ complexPlaneCoordinates
have hΩ : IsOpen Ω := hD.preimage complexPlaneCoordinates.continuous
have hF : ContDiffOn ℝ ∞ F Ω :=
hf.comp complexPlaneCoordinates.contDiff.contDiffOn (fun _ hp => hp)
have hmajor : MemLp (fun p => ‖fderiv ℝ f (complexPlaneCoordinates p)‖) 2 (volume.restrict Ω) :=
hE.norm.comp_measurePreserving (complexPlaneCoordinates_measurePreserving.restrict_preimage hD.measurableSet)
have hpartial (i : Fin 2) : MemLp
(fun p => fderiv ℝ F p (EuclideanSpace.single i 1)) 2 (volume.restrict Ω) := by
have hm : AEStronglyMeasurable (fun p => fderiv ℝ F p (EuclideanSpace.single i 1))
(volume.restrict Ω) :=
((hF.continuousOn_fderiv_of_isOpen hΩ (by simp)).clm_apply continuousOn_const).aestronglyMeasurable hΩ.measurableSet
apply hmajor.of_le hm
filter_upwards [ae_restrict_mem hΩ.measurableSet] with p hp
have hd : DifferentiableAt ℝ f (complexPlaneCoordinates p) :=
(hf.contDiffAt (hD.mem_nhds hp)).differentiableAt (by simp)
change ‖fderiv ℝ (f ∘ complexPlaneCoordinates) p (EuclideanSpace.single i 1)‖ ≤ _
rw [fderiv_comp p hd complexPlaneCoordinates.differentiableAt, complexPlaneCoordinates.fderiv]
have haxis : ‖complexPlaneCoordinates (EuclideanSpace.single i 1)‖ = 1 := by
fin_cases i
· change ‖complexPlaneCoordinates (EuclideanSpace.single (0 : Fin 2) 1)‖ = 1
rw [complexPlaneCoordinates_zero, norm_one]
· change ‖complexPlaneCoordinates (EuclideanSpace.single (1 : Fin 2) 1)‖ = 1
rw [complexPlaneCoordinates_one, Complex.norm_I]
simpa only [ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe, haxis, mul_one,
Real.norm_eq_abs, abs_norm] using
(fderiv ℝ f (complexPlaneCoordinates p)).le_opNorm (complexPlaneCoordinates (EuclideanSpace.single i 1))
refine ⟨{
memLp := ((hc.comp complexPlaneCoordinates.continuous).memLp_of_hasCompactSupport
(hs.comp_homeomorph complexPlaneCoordinates.toHomeomorph)).restrict Ω
weakGrad := fun p => WithLp.toLp 2 (fun i => fderiv ℝ F p (EuclideanSpace.single i 1))
weakGrad_component_memLp := hpartial
isWeakGrad := fun i => weakPartial_of_smooth_on hΩ (hF.of_le (by simp)) i }⟩
end CircleDomainRigidity
end
end OAI
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