openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /RadialAngularScalar.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialAngularForms | |
| namespace OAI | |
| /-! Adding a spherical-harmonic eigenvalue to the regular scalar spectral forms. -/ | |
| open Set | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| open ProfileCertificate | |
| noncomputable def radialAngularScalarAction (n : ℕ) (z : ProfileMatchingBall) (η : ℝ) | |
| (q f : ℝ → ℝ) (r : ℝ) : ℝ := | |
| radialScalarAction n z q f r+η*radialAngularDensity n z r*f r | |
| noncomputable def radialAngularScalarForm (n : ℕ) (z : ProfileMatchingBall) (η R : ℝ) | |
| (q f g : ℝ → ℝ) : ℝ := | |
| radialScalarForm n z R q f g+η*radialAngularForm n z R f g | |
| noncomputable def radialAngularScalarVirial (n : ℕ) (z : ProfileMatchingBall) (η R : ℝ) | |
| (q dq f : ℝ → ℝ) : ℝ := | |
| radialScalarVirial n z R q dq f+η*radialAngularVirial n z R f | |
| noncomputable def radialAngularScalarBoundary (n : ℕ) (z : ProfileMatchingBall) (η R s t : ℝ) | |
| (q f g : ℝ → ℝ) : ℝ := | |
| radialSpectralBoundary n z R s t q f g+η*radialAngularBoundary n z R f | |
| theorem radialAngularScalarForm_symm (n : ℕ) (z : ProfileMatchingBall) (η R : ℝ) | |
| (q f g : ℝ → ℝ) : | |
| radialAngularScalarForm n z η R q f g=radialAngularScalarForm n z η R q g f := by | |
| unfold radialAngularScalarForm | |
| rw [radialScalarForm_symm n z R q f g,radialAngularForm_symm n z R f g] | |
| theorem radialAngularScalarAction_continuousOn (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z) | |
| (n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z=0) (η R : ℝ) (q f : ℝ → ℝ) | |
| (hq : ContinuousOn q (Icc 0 R)) (hf : ContDiff ℝ 2 f) : | |
| ContinuousOn (radialAngularScalarAction n z η q f) (Icc 0 R) := by | |
| exact (radialScalarAction_continuousOn n z hX hz R q f hq hf).add | |
| (((radialAngularDensity_continuous n z hX hz).const_mul η).mul hf.continuous).continuousOn | |
| theorem radialAngularScalarForm_spectralTest_boundary (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z) | |
| (n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z=0) (η R s t : ℝ) (hR : 0 ≤ R) | |
| (q dq f g : ℝ → ℝ) (hq : ContinuousOn q (Icc 0 R)) | |
| (hdq : ContinuousOn dq (Icc 0 R)) (hq' : ∀ r ∈ Ioo 0 R, HasDerivAt q (dq r) r) | |
| (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 1 g) : | |
| (∫ r in (0 : ℝ)..R, | |
| radialSpectralTest n z s t f g r*radialAngularScalarAction n z η q f r)= | |
| s*radialAngularScalarForm n z η R q f f+t*radialAngularScalarForm n z η R q f g+ | |
| radialAngularScalarVirial n z η R q dq f+ | |
| radialAngularScalarBoundary n z η R s t q f g := by | |
| have hT := radialSpectralTest_continuousOn n z hX hz R s t f g hf hg | |
| have hAi : IntervalIntegrable (fun r => | |
| radialSpectralTest n z s t f g r*radialScalarAction n z q f r) | |
| MeasureTheory.volume 0 R := | |
| (hT.mul (radialScalarAction_continuousOn n z hX hz R q f hq hf)).intervalIntegrable_of_Icc hR | |
| have hKi : IntervalIntegrable (fun r => | |
| radialSpectralTest n z s t f g r*radialAngularDensity n z r*f r) | |
| MeasureTheory.volume 0 R := | |
| ((hT.mul (radialAngularDensity_continuous n z hX hz).continuousOn).mul | |
| hf.continuous.continuousOn).intervalIntegrable_of_Icc hR | |
| have he : (∫ r in (0 : ℝ)..R, | |
| radialSpectralTest n z s t f g r*radialAngularScalarAction n z η q f r)= | |
| (∫ r in (0 : ℝ)..R, radialSpectralTest n z s t f g r*radialScalarAction n z q f r)+ | |
| η*(∫ r in (0 : ℝ)..R, radialSpectralTest n z s t f g r*radialAngularDensity n z r*f r) := by | |
| calc | |
| _ = ∫ r in (0 : ℝ)..R, | |
| radialSpectralTest n z s t f g r*radialScalarAction n z q f r+ | |
| η*(radialSpectralTest n z s t f g r*radialAngularDensity n z r*f r) := by | |
| apply intervalIntegral.integral_congr | |
| intro r _ | |
| unfold radialAngularScalarAction | |
| ring | |
| _ = _ := by | |
| rw [intervalIntegral.integral_add hAi (hKi.const_mul η),intervalIntegral.integral_const_mul] | |
| rw [he,radialScalarForm_spectralTest_boundary n z hX hz R s t hR q dq f g hq hdq hq' hf hg, | |
| radialAngularForm_spectralTest n z hX hz R s t hR f g (hf.of_le (by norm_num)) hg.continuous] | |
| unfold radialAngularScalarForm radialAngularScalarVirial radialAngularScalarBoundary | |
| ring | |
| end DefocusingNLS | |
| end OAI | |