| 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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|
