File size: 1,892 Bytes
8fadf68 | 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 | import OAI.Geometry.NodalSets.Charts.SphereReferenceMeasurePositive
import OAI.Geometry.NodalSets.Spectral.SphereEigenGlobalRepresentative
namespace OAI
namespace Yau.Target
open MeasureTheory Manifold Set Yau.Geometry
open scoped ContDiff
noncomputable section
attribute [local instance] sphereGlobalRepresentativeMeasurable sphereGlobalRepresentativeBorel
theorem sphere_L2_representative_pairing (d : SphereEnergyData) (f : SphereWeightedL2 d)
(u : Base → ℝ) (ha : u =ᵐ[sphereWeightedMeasure d.density] (f : Base → ℝ)) :
sphereWeightedPairing d.density u u=‖f‖^2 := by
rw [sphereWeightedPairing,sphereWeightedL2_norm_sq_integral]
apply integral_congr_ae
filter_upwards [(sphereWeightedMeasure_ae_iff d _).mp ha] with x hx
rw [hx]
ring
theorem sphere_eigen_global_normalized (d : SphereEnergyData)
(hrho : ∀ p : Base, ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x)))
(mu : ℝ) (hmu : mu ≠ 0) (f : SphereWeightedL2 d)
(heigen : sphereL2Resolvent d f=mu • f) (hn : ‖f‖=1) :
∃ u : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u ∧
u =ᵐ[sphereWeightedMeasure d.density] (f : Base → ℝ) ∧
sphereWeightedPairing d.density u u=1 ∧ u ≠ 0 ∧
∀ p x, x ∈ interior (Yau.realCenteredCube 4 (1/64)) →
Yau.coordDiv (realMatrixFlux (sphereChartPrincipalDensity d p) (u ∘ sphereChartCoordMap p)) x+
sphereEigenForcingCoefficient d p mu x*u (sphereChartCoordMap p x)=0 := by
obtain ⟨u,hu,ha,he⟩ := sphere_eigen_global_representative d hrho mu hmu f heigen
have hnorm : sphereWeightedPairing d.density u u=1 := by
rw [sphere_L2_representative_pairing d f u ha,hn,one_pow]
refine ⟨u,hu,ha,hnorm,?_,he⟩
intro hz
simp only [hz,sphereWeightedPairing,Pi.zero_apply,mul_zero,integral_zero] at hnorm
norm_num at hnorm
end
end Yau.Target
end OAI
|