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a458723 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | 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
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