| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.AngularNormalTrace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped NNReal ENNReal |
|
|
| theorem integral_circle_faces {w : ℝ} (hw : 0<w) {F : UnitAddCircle → ℝ} |
| (hF : Continuous F) : |
| (∫ θ,F θ ∂AddCircle.haarAddCircle)= |
| ∑ j : Fin 4,∫ b in (-1:ℝ)..1, |
| F ((faceRayAngle w j b:ℝ):UnitAddCircle)*faceRayDensity w j b := by |
| have hc : Continuous (fun a : ℝ => F (a:UnitAddCircle)) := |
| hF.comp (AddCircle.continuous_mk' 1) |
| have hh (j : Fin 4) : |
| (∫ b in (-1:ℝ)..1,F ((faceRayAngle w j b:ℝ):UnitAddCircle)*faceRayDensity w j b)= |
| ∫ a in faceRayAngle w j (-1)..faceRayAngle w j 1,F (a:UnitAddCircle) := by |
| exact intervalIntegral.integral_comp_mul_deriv |
| (fun b _ => hasDerivAt_faceRayAngle w j b) (continuous_faceRayDensity w j).continuousOn hc |
| simp_rw [hh] |
| have he := faceRayAngle_endpoints hw |
| rw [AddCircle.integral_haarAddCircle] |
| simp only [inv_one,one_smul] |
| rw [←AddCircle.intervalIntegral_preimage 1 (faceRayAngle w 0 (-1))] |
| rw [Fin.sum_univ_four,←he.2] |
| have h0 := he.1 (0:Fin 3) |
| have h1 := he.1 (1:Fin 3) |
| have h2 := he.1 (2:Fin 3) |
| change faceRayAngle w 0 1=faceRayAngle w 1 (-1) at h0 |
| change faceRayAngle w 1 1=faceRayAngle w 2 (-1) at h1 |
| change faceRayAngle w 2 1=faceRayAngle w 3 (-1) at h2 |
| rw [←h0,←h1,←h2] |
| rw [intervalIntegral.integral_add_adjacent_intervals (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _), |
| intervalIntegral.integral_add_adjacent_intervals (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _), |
| intervalIntegral.integral_add_adjacent_intervals (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _)] |
|
|
| theorem integral_rectangularRay_le {w : ℝ} (hw : 0<w) (hw' : w≤1) |
| {f : (Fin 2 → ℝ) → ℝ} (hf : Continuous f) |
| (hn : ∀ z,0≤f z) : |
| (∫ θ,f (rectangularRay w θ) ∂AddCircle.haarAddCircle)≤ |
| ((1/w)/(2*Real.pi))*∑ j : Fin 4,∫ b in (-1:ℝ)..1, |
| f ![w*squareFace j b 0,squareFace j b 1] := by |
| rw [integral_circle_faces (F:=fun θ => f (rectangularRay w θ)) hw (hf.comp (continuous_rectangularRay hw hw'))] |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro j _ |
| rw [←intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_mono_on (by norm_num) |
| · have ha : Continuous (faceRayAngle w j) := continuous_iff_continuousAt.mpr |
| (fun b => (hasDerivAt_faceRayAngle w j b).continuousAt) |
| exact ((hf.comp (continuous_rectangularRay hw hw')).comp |
| ((AddCircle.continuous_mk' 1).comp ha)).mul |
| (continuous_faceRayDensity w j) |>.intervalIntegrable _ _ |
| · have hc : Continuous (fun b : ℝ => ![w*squareFace j b 0,squareFace j b 1]) := by |
| apply continuous_pi |
| intro i |
| fin_cases i |
| · unfold squareFace; fun_prop |
| · unfold squareFace; fun_prop |
| exact (continuous_const.mul (hf.comp hc)).intervalIntegrable _ _ |
| · intro b hb |
| rw [faceRayAngle_image hw j (abs_le.mpr hb)] |
| exact (mul_le_mul_of_nonneg_left (faceRayDensity_bounds hw hw' j b).2 |
| (hn ![w*squareFace j b 0,squareFace j b 1])).trans_eq (mul_comm _ _) |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
