| import Mathlib |
| import OAI.Analysis.Conductivity.Geometry.BoxIteration |
| import OAI.Analysis.Conductivity.Geometry.TorusFaceIntegration |
| import OAI.Analysis.Conductivity.Fourier.AngularIntegratedTrace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped NNReal ENNReal |
|
|
| lemma sourceExtendedBox_compact (l r : ℝ) : IsCompact (sourceExtendedBox l r) := isCompact_Icc |
|
|
| theorem sourceExtended_integral_bound {g : (Fin 3 → ℝ) → ℝ} |
| (hg : Continuous g) (hn : ∀ y,0≤g y) (i j : Fin 4) {l r : ℝ} |
| (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) : |
| (∫ x in sourceExtendedBox l r,g (sourceCollarPiece i j x))≤ |
| 100*(∫ y in sourceCollarPiece i j '' sourceExtendedBox l r,g y) := by |
| rw [sourceExtended_integral i j hl hr,←integral_const_mul] |
| apply setIntegral_mono_on |
| · exact (hg.comp (sourceCollarPiece_contDiff i j).continuous).continuousOn.integrableOn_compact isCompact_Icc |
| · have hc : Continuous (fun x : Fin 3 → ℝ => abs ((sourceCollarDerivative i j x).det)) := by |
| simp_rw [sourceCollarDerivative_det] |
| unfold sourceCollarRadius squareFace |
| fun_prop |
| exact (continuous_const.mul (hc.mul (hg.comp (sourceCollarPiece_contDiff i j).continuous))).continuousOn.integrableOn_compact isCompact_Icc |
| · exact measurableSet_Icc |
| · intro x hx |
| have hh := sourceExtended_jacobian_lower i j hl hr hx |
| change g (sourceCollarPiece i j x)≤100*(abs ((sourceCollarDerivative i j x).det)*g (sourceCollarPiece i j x)) |
| nlinarith [mul_nonneg (hn (sourceCollarPiece i j x)) (show 0≤abs ((sourceCollarDerivative i j x).det)-1/100 by linarith)] |
|
|
| lemma continuous_collarFace_integral {g : (Fin 3 → ℝ) → ℝ} (hg : Continuous g) (i j : Fin 4) : |
| Continuous (fun t : ℝ => ∫ a in (-1:ℝ)..1,∫ b in (-1:ℝ)..1,g (sourceCollarPiece i j ![t,a,b])) := by |
| apply continuous_parametric_unitInterval (f:=fun t a => ∫ b in (-1:ℝ)..1,g (sourceCollarPiece i j ![t,a,b])) |
| apply continuous_parametric_unitInterval (f:=fun p : ℝ×ℝ => fun b => g (sourceCollarPiece i j ![p.1,p.2,b])) |
| exact hg.comp ((sourceCollarPiece_contDiff i j).continuous.comp (by fun_prop)) |
|
|
| theorem sourceAngular_volume_bound {g : (Fin 3 → ℝ) → ℝ} |
| (hg : Continuous g) (hn : ∀ y,0≤g y) {l r : ℝ} |
| (hlr : l≤r) (hl : -(1:ℝ)/100≤l) (hr : r≤1/100) : |
| (∫ t in l..r,∫ x : UnitAddTorus (Fin 2),g (sourceAngularCollar t x))≤ |
| (100*((1/sourceRadialWidth)/(2*Real.pi))*((1:ℝ)/(2*Real.pi)))* |
| ∑ i : Fin 4,∑ j : Fin 4,∫ y in sourceCollarPiece i j '' sourceExtendedBox l r,g y := by |
| let c : ℝ := ((1/sourceRadialWidth)/(2*Real.pi))*((1:ℝ)/(2*Real.pi)) |
| have hw : 0<sourceRadialWidth := by norm_num [sourceRadialWidth,sourceHole] |
| have hc : 0≤c := by dsimp [c]; positivity |
| have hcont : Continuous (fun t : ℝ => ∫ x : UnitAddTorus (Fin 2),g (sourceAngularCollar t x)) := by |
| have hcs : Continuous (fun p : ℝ×UnitAddTorus (Fin 2) => sourceAngularCollar p.1 p.2) := by |
| simpa only [zero_add] using continuous_sourceAngular 0 |
| have hh := continuous_parametric_integral_of_continuous (μ:=volume) |
| (f:=fun t (x : UnitAddTorus (Fin 2)) => g (sourceAngularCollar t x)) |
| (hg.comp hcs) (s:=univ) isCompact_univ |
| simpa using hh |
| have hsum : Continuous (fun t : ℝ => ∑ i : Fin 4,∑ j : Fin 4, |
| ∫ a in (-1:ℝ)..1,∫ b in (-1:ℝ)..1,g (sourceCollarPiece i j ![t,a,b])) := |
| continuous_finsetSum _ (fun i _ => continuous_finsetSum _ (fun j _ => continuous_collarFace_integral hg i j)) |
| calc |
| _≤∫ t in l..r,c*(∑ i : Fin 4,∑ j : Fin 4, |
| ∫ a in (-1:ℝ)..1,∫ b in (-1:ℝ)..1,g (sourceCollarPiece i j ![t,a,b])) := |
| intervalIntegral.integral_mono_on hlr (hcont.intervalIntegrable l r) |
| ((continuous_const.mul hsum).intervalIntegrable l r) (fun t _ => integral_sourceAngular_le hg hn t) |
| _=c*(∑ i : Fin 4,∑ j : Fin 4,∫ x in sourceExtendedBox l r,g (sourceCollarPiece i j x)) := by |
| rw [intervalIntegral.integral_const_mul] |
| congr 1 |
| rw [intervalIntegral.integral_finsetSum (fun i _ => (continuous_finsetSum _ (fun j _ => continuous_collarFace_integral hg i j)).intervalIntegrable l r)] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [intervalIntegral.integral_finsetSum (fun j _ => (continuous_collarFace_integral hg i j).intervalIntegrable l r)] |
| apply Finset.sum_congr rfl |
| intro j _ |
| exact (integral_sourceExtendedBox (hg.comp (sourceCollarPiece_contDiff i j).continuous) hlr).symm |
| _≤c*(∑ i : Fin 4,∑ j : Fin 4,100*(∫ y in sourceCollarPiece i j '' sourceExtendedBox l r,g y)) := by |
| apply mul_le_mul_of_nonneg_left _ hc |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| exact sourceExtended_integral_bound hg hn i j hl hr |
| _=_ := by simp_rw [←Finset.mul_sum]; dsimp [c]; ring |
|
|
| end ScalarConductivity |
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| end |
|
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| end OAI |
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