| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.IntervalTraceEnergy |
| import OAI.Analysis.Conductivity.Sources.AngularTrace |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
| open Set MeasureTheory Filter Topology UnitAddTorus |
| open scoped NNReal ENNReal |
|
|
| def sourceAngularVelocity (x : UnitAddTorus (Fin 2)) : Fin 3 → ℝ := |
| let z := sourceRayCoordinates x |
| ![-sourceLength*z 2*z 0,-z 2*z 1,-z 3] |
|
|
| lemma sourceAngular_affine (t a : ℝ) (x : UnitAddTorus (Fin 2)) : |
| sourceAngularCollar (t+a) x=sourceAngularCollar t x+a • sourceAngularVelocity x := by |
| ext i |
| fin_cases i < |
|
|
| lemma sourceAngularVelocity_norm (x : UnitAddTorus (Fin 2)) : ‖sourceAngularVelocity x‖≤8 := by |
| have hz := sourceRayCoordinates_mem x |
| have hb (i : Fin 4) : |sourceRayCoordinates x i|≤2 := abs_le.mpr ⟨hz.1 i,hz.2 i⟩ |
| apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ)≤8)).mpr |
| intro i |
| fin_cases i < |
| · have hh := mul_le_mul (hb 2) (hb 0) (abs_nonneg _) (by norm_num : (0:ℝ)≤2) |
| norm_num [sourceLength] at * |
| nlinarith [abs_nonneg (sourceRayCoordinates x 2*sourceRayCoordinates x 0)] |
| · have hh := mul_le_mul (hb 2) (hb 1) (abs_nonneg _) (by norm_num : (0:ℝ)≤2) |
| linarith |
| · linarith [hb 3] |
|
|
| lemma continuous_sourceAngularVelocity : Continuous sourceAngularVelocity := by |
| apply continuous_pi |
| intro i |
| fin_cases i < |
| · exact (continuous_const.mul ((continuous_apply (2 : Fin 4)).comp continuous_sourceRayCoordinates)).mul |
| ((continuous_apply (0 : Fin 4)).comp continuous_sourceRayCoordinates) |
| · exact (((continuous_apply (2 : Fin 4)).comp continuous_sourceRayCoordinates).neg).mul |
| ((continuous_apply (1 : Fin 4)).comp continuous_sourceRayCoordinates) |
| · exact ((continuous_apply (3 : Fin 4)).comp continuous_sourceRayCoordinates).neg |
|
|
| lemma sourceAngular_normal_deriv {f : (Fin 3 → ℝ) → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (t a : ℝ) (x : UnitAddTorus (Fin 2)) : |
| deriv (fun s => f (sourceAngularCollar (t+s) x)) a= |
| fderiv ℝ f (sourceAngularCollar (t+a) x) (sourceAngularVelocity x) := by |
| have hd := (((hf.differentiable (by simp) (sourceAngularCollar t x+a • sourceAngularVelocity x)).hasFDerivAt).comp_hasDerivAt a |
| (((hasDerivAt_id a).smul_const (sourceAngularVelocity x)).const_add (sourceAngularCollar t x))).deriv |
| change deriv (fun s : ℝ => f (sourceAngularCollar t x+s • sourceAngularVelocity x)) a=_ at hd |
| simpa only [sourceAngular_affine,one_smul] using hd |
|
|
| theorem sourceAngular_pointwise_trace {f : (Fin 3 → ℝ) → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (t : ℝ) {η : ℝ} (hη : 0<η) |
| (x : UnitAddTorus (Fin 2)) : |
| (f (sourceAngularCollar t x))^2≤(1+1/η)*(∫ a in (0:ℝ)..η, |
| 64*‖fderiv ℝ f (sourceAngularCollar (t+a) x)‖^2+ |
| 2*(f (sourceAngularCollar (t+a) x))^2) := by |
| have hq : ContDiff ℝ (↑(⊤ : ℕ∞)) (fun a : ℝ => f (sourceAngularCollar (t+a) x)) := by |
| simp_rw [sourceAngular_affine] |
| exact hf.comp (contDiff_const.add (contDiff_id.smul contDiff_const)) |
| have hh := fourier_mode_trace_energy hq hη (m:=1) (by norm_num) |
| simp only [add_zero,one_pow,one_mul] at hh |
| apply hh.trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply intervalIntegral.integral_mono_on hη.le |
| · exact (((hq.continuous_deriv (by simp)).pow 2).add |
| (continuous_const.mul (hq.continuous.pow 2))).intervalIntegrable _ _ |
| · have hc : Continuous (fun a : ℝ => sourceAngularCollar (t+a) x) := by |
| simp_rw [sourceAngular_affine] |
| exact ((continuous_const.mul ((hf.continuous_fderiv (by simp)).comp hc |>.norm.pow 2)).add |
| (continuous_const.mul (hq.continuous.pow 2))).intervalIntegrable _ _ |
| · intro a _ |
| rw [sourceAngular_normal_deriv hf] |
| have hb := (fderiv ℝ f (sourceAngularCollar (t+a) x)).le_opNorm (sourceAngularVelocity x) |
| have hb' := hb.trans (mul_le_mul_of_nonneg_left (sourceAngularVelocity_norm x) (norm_nonneg _)) |
| have hs := mul_self_le_mul_self (norm_nonneg _) hb' |
| simp only [Real.norm_eq_abs,←pow_two,sq_abs] at hs |
| nlinarith |
| |
| lemma rayDirection_coe_angle (a : ℝ) : |
| rayDirection ((a/(2*Real.pi):ℝ):UnitAddCircle)=![Real.cos a,Real.sin a] := by |
| have he : fourier 1 ((a/(2*Real.pi):ℝ):UnitAddCircle)=Complex.exp ((a:ℂ)*Complex.I) := by |
| rw [fourier_coe_apply] |
| congr 1 |
| push_cast |
| have hp : (Real.pi:ℂ)≠0 := Complex.ofReal_ne_zero.mpr Real.pi_pos.ne' |
| field_simp |
| ext i |
| fin_cases i |
| · simp [rayDirection,he] |
| · simp [rayDirection,he] |
| |
| lemma rectangularRay_of_direction {w k : ℝ} (hw : 0<w) (hk : 0<k) |
| (θ : UnitAddCircle) (j : Fin 4) {b : ℝ} (hb : |b|≤1) |
| (he : rayDirection θ=k • ![w*squareFace j b 0,squareFace j b 1]) : |
| rectangularRay w θ=![w*squareFace j b 0,squareFace j b 1] := by |
| have hd : rayDenominator w θ=k := by |
| dsimp [rayDenominator] |
| rw [he] |
| simp only [Pi.smul_apply,smul_eq_mul,Matrix.cons_val_zero,Matrix.cons_val_one, |
| abs_mul,abs_of_pos hk,abs_of_pos hw] |
| have he' : k*(w*|squareFace j b 0|)/w=k*|squareFace j b 0| := by field_simp |
| rw [he',←mul_max_of_nonneg _ _ hk.le,squareFace_max j hb,mul_one] |
| ext i |
| rw [rectangularRay,he,hd] |
| simp only [Pi.smul_apply,smul_eq_mul] |
| exact mul_div_cancel_left₀ _ hk.ne' |
| |
| def faceRayAngle (w : ℝ) (j : Fin 4) (b : ℝ) : ℝ := |
| ![Real.arctan (b/w),Real.pi/2+Real.arctan (w*b), |
| Real.pi+Real.arctan (b/w),3*Real.pi/2+Real.arctan (w*b)] j/(2*Real.pi) |
| |
| lemma faceRayAngle_image {w : ℝ} (hw : 0<w) (j : Fin 4) {b : ℝ} (hb : |b|≤1) : |
| rectangularRay w ((faceRayAngle w j b:ℝ):UnitAddCircle)= |
| ![w*squareFace j b 0,squareFace j b 1] := by |
| have hp (x : ℝ) : 0<Real.sqrt (1+x^2) := Real.sqrt_pos.mpr (by positivity) |
| fin_cases j |
| · apply rectangularRay_of_direction hw (k:=(w*Real.sqrt (1+(b/w)^2))⁻¹) |
| (inv_pos.mpr (mul_pos hw (hp _))) _ 0 hb |
| dsimp only [faceRayAngle] |
| rw [rayDirection_coe_angle] |
| ext i; fin_cases i <;> simp [squareFace,squareFaceBase,squareFaceDirection, |
| Real.cos_arctan,Real.sin_arctan] <;> field_simp |
| · apply rectangularRay_of_direction hw (k:=(Real.sqrt (1+(w*b)^2))⁻¹) |
| (inv_pos.mpr (hp _)) _ 1 hb |
| dsimp only [faceRayAngle] |
| rw [rayDirection_coe_angle] |
| ext i; fin_cases i <;> simp [squareFace,squareFaceBase,squareFaceDirection, |
| Real.cos_add,Real.sin_add,Real.cos_arctan,Real.sin_arctan]; ring |
| · apply rectangularRay_of_direction hw (k:=(w*Real.sqrt (1+(b/w)^2))⁻¹) |
| (inv_pos.mpr (mul_pos hw (hp _))) _ 2 hb |
| dsimp only [faceRayAngle] |
| rw [rayDirection_coe_angle] |
| ext i; fin_cases i <;> simp [squareFace,squareFaceBase,squareFaceDirection, |
| Real.cos_add,Real.sin_add,Real.cos_arctan,Real.sin_arctan] <;> field_simp |
| · apply rectangularRay_of_direction hw (k:=(Real.sqrt (1+(w*b)^2))⁻¹) |
| (inv_pos.mpr (hp _)) _ 3 hb |
| dsimp only [faceRayAngle] |
| rw [rayDirection_coe_angle] |
| have he : 3*Real.pi/2=Real.pi+Real.pi/2 := by ring |
| rw [he] |
| ext i; fin_cases i <;> simp [squareFace,squareFaceBase,squareFaceDirection, |
| Real.cos_add,Real.sin_add,Real.cos_arctan,Real.sin_arctan]; ring |
| |
| lemma faceRayAngle_endpoints {w : ℝ} (hw : 0<w) : |
| (∀ j : Fin 3,faceRayAngle w j.castSucc 1=faceRayAngle w j.succ (-1)) ∧ |
| faceRayAngle w 3 1=faceRayAngle w 0 (-1)+1 := by |
| have hx := Real.arctan_inv_of_pos hw |
| constructor |
| · intro j |
| fin_cases j <;> simp [faceRayAngle,one_div,neg_div,Real.arctan_neg,hx] <;> ring |
| · simp [faceRayAngle,one_div,neg_div,Real.arctan_neg,hx] |
| field_simp |
| ring |
| |
| def faceRayDensity (w : ℝ) (j : Fin 4) (b : ℝ) : ℝ := |
| ![(1/(1+(b/w)^2))/w,w/(1+(w*b)^2), |
| (1/(1+(b/w)^2))/w,w/(1+(w*b)^2)] j/(2*Real.pi) |
| |
| lemma hasDerivAt_faceRayAngle (w : ℝ) (j : Fin 4) (b : ℝ) : |
| HasDerivAt (faceRayAngle w j) (faceRayDensity w j b) b := by |
| have h0 := ((Real.hasDerivAt_arctan (b/w)).comp b ((hasDerivAt_id b).div_const w)).div_const (2*Real.pi) |
| have h1 := (((Real.hasDerivAt_arctan (w*b)).comp b ((hasDerivAt_id b).const_mul w)).const_add (Real.pi/2)).div_const (2*Real.pi) |
| have h2 := (((Real.hasDerivAt_arctan (b/w)).comp b ((hasDerivAt_id b).div_const w)).const_add Real.pi).div_const (2*Real.pi) |
| have h3 := (((Real.hasDerivAt_arctan (w*b)).comp b ((hasDerivAt_id b).const_mul w)).const_add (3*Real.pi/2)).div_const (2*Real.pi) |
| unfold faceRayAngle faceRayDensity |
| fin_cases j |
| · simpa [faceRayAngle,faceRayDensity,div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm] using h0 |
| · simpa [faceRayAngle,faceRayDensity,div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm] using h1 |
| · simpa [faceRayAngle,faceRayDensity,div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm] using h2 |
| · simpa [faceRayAngle,faceRayDensity,div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm] using h3 |
| |
| lemma continuous_faceRayDensity (w : ℝ) (j : Fin 4) : Continuous (faceRayDensity w j) := by |
| have h0 : Continuous (fun b : ℝ => (1/(1+(b/w)^2))/w/(2*Real.pi)) := by |
| have hd : Continuous (fun b : ℝ => 1+(b/w)^2) := by fun_prop |
| exact ((continuous_const.div hd (fun b => ne_of_gt (by positivity))).div_const w).div_const _ |
| have h1 : Continuous (fun b : ℝ => (w/(1+(w*b)^2))/(2*Real.pi)) := by |
| have hd : Continuous (fun b : ℝ => 1+(w*b)^2) := by fun_prop |
| exact (continuous_const.div hd (fun b => ne_of_gt (by positivity))).div_const _ |
| fin_cases j |
| · exact h0 |
| · exact h1 |
| · exact h0 |
| · exact h1 |
| |
| lemma faceRayDensity_bounds {w : ℝ} (hw : 0<w) (hw' : w≤1) (j : Fin 4) (b : ℝ) : |
| 0≤faceRayDensity w j b ∧ faceRayDensity w j b≤(1/w)/(2*Real.pi) := by |
| have hp : 0<2*Real.pi := by positivity |
| have H0 : 0≤(1/(1+(b/w)^2))/w ∧ (1/(1+(b/w)^2))/w≤1/w := by |
| constructor |
| · positivity |
| · exact div_le_div_of_nonneg_right (by apply div_le_one_of_le₀; nlinarith |
| have H1 : 0≤w/(1+(w*b)^2) ∧ w/(1+(w*b)^2)≤1/w := by |
| constructor |
| · positivity |
| · have hh : w/(1+(w*b)^2)≤w := div_le_self hw.le (by nlinarith [sq_nonneg (w*b)]) |
| apply hh.trans |
| apply (le_div_iff₀ hw).mpr |
| nlinarith |
| fin_cases j |
| · exact ⟨div_nonneg H0.1 hp.le,div_le_div_of_nonneg_right H0.2 hp.le⟩ |
| · exact ⟨div_nonneg H1.1 hp.le,div_le_div_of_nonneg_right H1.2 hp.le⟩ |
| · exact ⟨div_nonneg H0.1 hp.le,div_le_div_of_nonneg_right H0.2 hp.le⟩ |
| · exact ⟨div_nonneg H1.1 hp.le,div_le_div_of_nonneg_right H1.2 hp.le⟩ |
|
|
| end ScalarConductivity |
| |
| end |
|
|
| end OAI |
|
|
