| import OAI.MathematicalPhysics.Elasticity.Parameters |
| import OAI.MathematicalPhysics.Elasticity.ODE |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set |
| namespace ElasticityODE |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R] |
| def frameDbar (b : ℂ → C(Time,R)) (p : ℂ) : C(Time,R) := |
| fderiv ℝ (fun p => fundamental (b p)) p 1+ |
| Complex.I • fderiv ℝ (fun p => fundamental (b p)) p Complex.I |
| def coefficientDbar (b : ℂ → C(Time,R)) (p : ℂ) : C(Time,R) := |
| fderiv ℝ b p 1+Complex.I • fderiv ℝ b p Complex.I |
| lemma frameDbar_hasDerivWithinAt (b : ℂ → C(Time,R)) |
| (hb : ContDiff ℝ (⊤ : ℕ∞) b) (p : ℂ) (t : Time) : |
| HasDerivWithinAt (extend (frameDbar b p)) |
| (b p t*frameDbar b p t+coefficientDbar b p t*fundamental (b p) t) Time t := by |
| have h1 := fundamental_variation_hasDerivWithinAt b hb p 1 t |
| have hi := (fundamental_variation_hasDerivWithinAt b hb p Complex.I t).const_smul Complex.I |
| have h := h1.add hi |
| have he : |
| b p t*fderiv ℝ (fun p => fundamental (b p)) p 1 t+fderiv ℝ b p 1 t*fundamental (b p) t+ |
| Complex.I • (b p t*fderiv ℝ (fun p => fundamental (b p)) p Complex.I t+ |
| fderiv ℝ b p Complex.I t*fundamental (b p) t)= |
| b p t*frameDbar b p t+coefficientDbar b p t*fundamental (b p) t := by |
| simp only [frameDbar,coefficientDbar,ContinuousMap.add_apply,ContinuousMap.smul_apply, |
| smul_add,mul_add,add_mul,mul_smul_comm,smul_mul_assoc] |
| abel |
| rw [he] at h |
| exact h.congr_of_mem (fun x hx => by |
| change extend (frameDbar b p) (⟨x,hx⟩ : Time)=_ |
| rw [extend_apply] |
| change _=extend (fderiv ℝ (fun p => fundamental (b p)) p 1) (⟨x,hx⟩ : Time)+ |
| Complex.I • extend (fderiv ℝ (fun p => fundamental (b p)) p Complex.I) (⟨x,hx⟩ : Time) |
| rw [extend_apply,extend_apply] |
| rfl) t.property |
| lemma frameDbar_initial (b : ℂ → C(Time,R)) (hb : ContDiff ℝ (⊤ : ℕ∞) b) (p : ℂ) : |
| frameDbar b p ⟨0,by simp [Time]⟩=0 := by |
| simp only [frameDbar,ContinuousMap.add_apply,ContinuousMap.smul_apply, |
| fundamental_variation_initial b hb,smul_zero,add_zero] |
| |
| /-- The homotopy defect vanishes. This derives the planar frame equation from |
| an actual coefficient homotopy, instead of assuming an Oka trivialization. -/ |
| theorem fundamental_transport (b : ℂ → C(Time,R)) |
| (hb : ContDiff ℝ (⊤ : ℕ∞) b) (p : ℂ) (A : R) |
| (he : ∀ t : Time, coefficientDbar b p t=A+(t:ℝ) • (A*b p t-b p t*A)) (t : Time) : |
| frameDbar b p t=(t:ℝ) • (A*fundamental (b p) t) := by |
| let u : C(Time,R) := frameDbar b p- |
| ⟨fun s => (s:ℝ) • (A*fundamental (b p) s), |
| continuous_subtype_val.smul (continuous_const.mul (fundamental (b p)).continuous)⟩ |
| let v : C(Time,R) := 0 |
| have hu (s : Time) : HasDerivWithinAt (extend u) (b p s*u s) Time s := by |
| have hU := (fundamental_deriv (b p) s).const_mul A |
| have hs : HasDerivWithinAt (fun x : ℝ => x) 1 Time s := (hasDerivAt_id (s:ℝ)).hasDerivWithinAt |
| have hT := hs.smul hU |
| have hD := (frameDbar_hasDerivWithinAt b hb p s).sub hT |
| have ha : b p s*frameDbar b p s+coefficientDbar b p s*fundamental (b p) s- |
| ((s:ℝ) • (A*(b p s*fundamental (b p) s))+(1:ℝ) • (A*extend (fundamental (b p)) s))= |
| b p s*u s := by |
| rw [he,extend_apply] |
| change _=b p s*(frameDbar b p s-(s:ℝ) • (A*fundamental (b p) s)) |
| simp only [one_smul,add_mul,smul_mul_assoc,sub_mul,smul_sub,mul_sub,mul_smul_comm,mul_assoc] |
| abel |
| rw [ha] at hD |
| exact hD.congr_of_mem (fun x hx => by |
| change extend u (⟨x,hx⟩ : Time)=_ |
| rw [extend_apply] |
| change frameDbar b p ⟨x,hx⟩-x • (A*fundamental (b p) ⟨x,hx⟩)=_ |
| rw [← extend_apply (frameDbar b p) ⟨x,hx⟩, |
| ← extend_apply (fundamental (b p)) ⟨x,hx⟩] |
| rfl) s.property |
| have hv (s : Time) : HasDerivWithinAt (extend v) (b p s*v s) Time s := by |
| have hh : HasDerivWithinAt (fun _ : ℝ => (0:R)) 0 Time s := hasDerivWithinAt_const _ _ _ |
| have hz : b p s*v s=0 := mul_zero _ |
| rw [hz] |
| exact hh.congr_of_mem (fun x hx => by |
| change extend v (⟨x,hx⟩ : Time)=0 |
| rw [extend_apply]; rfl) s.property |
| have hzero : u ⟨0,by simp [Time]⟩=v ⟨0,by simp [Time]⟩ := by |
| change frameDbar b p ⟨0,by simp [Time]⟩-(0:ℝ) • _=0 |
| rw [frameDbar_initial b hb,zero_smul,sub_zero] |
| have h := congrArg (fun f : C(Time,R) => f t) |
| (linear_ode_unique (leftCoeff (b p)) u v hu hv hzero) |
| exact sub_eq_zero.mp h |
| end ElasticityODE |
|
|
| end |
| end |
| section |
| noncomputable section |
| open Set Complex Module |
| open scoped SchwartzMap LineDeriv BigOperators |
| namespace ElasticityPlanar |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R] |
| [FiniteDimensional ℂ R] |
| variable {n : Type*} [Fintype n] [DecidableEq n] |
| def coords (q : Basis n ℂ R) : R ≃L[ℂ] (n → ℂ) := q.equivFun.toContinuousLinearEquiv |
| def coord (q : Basis n ℂ R) (i : n) : R →L[ℂ] ℂ := |
| (ContinuousLinearMap.proj i).comp (coords q).toContinuousLinearMap |
| omit [CompleteSpace R] [DecidableEq n] in |
| lemma coords_coord (q : Basis n ℂ R) (x : R) (i : n) : coords q x i=coord q i x := rfl |
| omit [CompleteSpace R] [DecidableEq n] in |
| lemma coord_linear_apply (q : Basis n ℂ R) (L : R →ₗ[ℂ] R) (x : R) (i : n) : |
| coord q i (L x)=∑ j, coord q i (L (q j))*coord q j x := by |
| have h : L x=∑ j, (q.equivFun x j) • L (q j) := by |
| calc |
| L x=L (∑ j, q.equivFun x j • q j) := congrArg L (q.sum_equivFun x).symm |
| _=∑ j, q.equivFun x j • L (q j) := by simp only [map_sum,map_smul] |
| rw [h,map_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [map_smul,smul_eq_mul,mul_comm] |
| rfl |
|
|
| def classicalDbar (f : ℂ → R) (z : ℂ) : R := fderiv ℝ f z 1+I • fderiv ℝ f z I |
| omit [CompleteSpace R] [DecidableEq n] in |
| lemma coord_dbar (q : Basis n ℂ R) (f : ℂ → R) (hf : Differentiable ℝ f) (z : ℂ) (i : n) : |
| coord q i (classicalDbar f z)= |
| fderiv ℝ (fun z => coord q i (f z)) z 1+I*fderiv ℝ (fun z => coord q i (f z)) z I := by |
| have h := ((coord q i).restrictScalars ℝ).hasFDerivAt.comp z (hf z).hasFDerivAt |
| change HasFDerivAt (fun z => coord q i (f z)) _ z at h |
| rw [h.fderiv] |
| simp only [classicalDbar,map_add,map_smul,ContinuousLinearMap.comp_apply, |
| smul_eq_mul] |
| rfl |
|
|
| omit [CompleteSpace R] [DecidableEq n] in |
| /-- The commutator homotopy equation in a genuine finite-dimensional complex |
| Banach algebra, with actual joint smooth representatives. -/ |
| theorem exists_algebra_homotopy {S Ω Ω' : Set ℂ} (hS : IsCompact S) |
| (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω) |
| (q : Basis n ℂ R) (A : ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A) |
| (hAc : HasCompactSupport A) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') : |
| ∃ B : ℝ × ℂ → R, ContDiff ℝ (⊤ : ℕ∞) B ∧ |
| ∀ s z, z∈Ω' → classicalDbar (fun z => B (s,z)) z= |
| A z+s • (A z*B (s,z)-B (s,z)*A z) := by |
| let a : n → n → ℂ → ℂ := fun i j z => coord q i (q j*A z-A z*q j) |
| have ha (i j : n) : ContDiff ℝ (⊤ : ℕ∞) (a i j) := |
| ((coord q i).restrictScalars ℝ).contDiff.comp |
| ((contDiff_const.mul hA).sub (hA.mul contDiff_const)) |
| have hac (i j : n) : HasCompactSupport (a i j) := |
| ((hAc.mul_left.sub hAc.mul_right).comp_left (g := coord q i) (map_zero _)) |
| have hf (i : n) : ContDiff ℝ (⊤ : ℕ∞) (fun z => coord q i (A z)) := |
| ((coord q i).restrictScalars ℝ).contDiff.comp hA |
| have hfc (i : n) : HasCompactSupport (fun z => coord q i (A z)) := |
| hAc.comp_left (g := coord q i) (map_zero _) |
| let f : n → 𝓢(ℂ,ℂ) := fun i => (hfc i).toSchwartzMap (hf i) |
| obtain ⟨v,hv,he⟩ := exists_classical_homotopy hS hΩS hΩ' hsub a ha hac f hg hc hs h1 |
| let B : ℝ × ℂ → R := fun p => (coords q).symm (fun i => v p.1 i p.2) |
| have hL : ContDiff ℝ (⊤ : ℕ∞) (coords q).symm := by |
| let L : (n → ℂ) →L[ℝ] R := (coords q).symm.toContinuousLinearMap.restrictScalars ℝ |
| exact ContinuousLinearMap.contDiff (𝕜 := ℝ) (E := n → ℂ) (F := R) L |
| have hB : ContDiff ℝ (⊤ : ℕ∞) B := hL.comp (contDiff_pi.mpr hv) |
| have hev (s : ℝ) (i : n) : (fun z => coord q i (B (s,z)))=fun z => v s i z := by |
| funext z |
| change (coords q) ((coords q).symm (fun i => v s i z)) i=v s i z |
| rw [ContinuousLinearEquiv.apply_symm_apply] |
| refine ⟨B,hB,fun s z hz => ?_⟩ |
| apply (coords q).injective |
| ext i |
| change coord q i (classicalDbar (fun z => B (s,z)) z)= |
| coord q i (A z+s • (A z*B (s,z)-B (s,z)*A z)) |
| have hBs : Differentiable ℝ (fun z => B (s,z)) := |
| (hB.comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp) |
| have hD := coord_dbar q (fun z => B (s,z)) hBs z i |
| rw [hev] at hD |
| have he' := he s i hz |
| simp only [schwartzDbar,add_apply,smul_apply,SchwartzMap.lineDerivOp_apply_eq_fderiv,smul_eq_mul] at he' |
| rw [← hD] at he' |
| have hcomm : coord q i (B (s,z)*A z-A z*B (s,z))=∑ j, a i j z*v s j z := by |
| let L : R →ₗ[ℂ] R := ((ContinuousLinearMap.mul ℂ R).flip (A z)- |
| ContinuousLinearMap.mul ℂ R (A z)).toLinearMap |
| have hh := coord_linear_apply q L (B (s,z)) i |
| change coord q i (B (s,z)*A z-A z*B (s,z))= |
| ∑ j, a i j z*coord q j (B (s,z)) at hh |
| simpa only [congrFun (hev s _) z] using hh |
| have hs' : coord q i (A z+s • (A z*B (s,z)-B (s,z)*A z))= |
| coord q i (A z)-(s:ℂ)*(∑ j, a i j z*v s j z) := by |
| rw [map_add] |
| have hneg : A z*B (s,z)-B (s,z)*A z=-(B (s,z)*A z-A z*B (s,z)) := by abel |
| rw [hneg] |
| have hl := ((coord q i).restrictScalars ℝ).map_smul s (-(B (s,z)*A z-A z*B (s,z))) |
| change coord q i (s • (-(B (s,z)*A z-A z*B (s,z))))=s • coord q i (-(B (s,z)*A z-A z*B (s,z))) at hl |
| rw [hl,map_neg,hcomm] |
| simp [Complex.real_smul,sub_eq_add_neg] |
| rw [hs'] |
| change _+∑ j, (s:ℂ)*a i j z*v s j z=coord q i (A z) at he' |
| rw [Finset.mul_sum] |
| simp_rw [mul_assoc] at he' ⊢ |
| linear_combination he' |
| end ElasticityPlanar |
| |
| end |
| end |
| section |
| noncomputable section |
| open Set Complex Module |
| open scoped SchwartzMap LineDeriv BigOperators |
| namespace ElasticityPlanar |
| open ElasticityRegularity |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R] |
| [FiniteDimensional ℂ R] |
| variable {n : Type*} [Fintype n] [DecidableEq n] |
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] |
| omit [CompleteSpace R] [DecidableEq n] in |
| /-- The commutator homotopy equation in a genuine finite-dimensional complex |
| Banach algebra, with actual joint smooth representatives. -/ |
| theorem exists_algebra_variable_homotopy {S Ω Ω' K : Set ℂ} (hS : IsCompact S) |
| (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω) |
| (hK : IsCompact K) (q : Basis n ℂ R) (A : P × ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A) |
| (hAc : ∀ p z, z∉K → A (p,z)=0) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') : |
| ∃ B : (ℝ × P) × ℂ → R, ContDiff ℝ (⊤ : ℕ∞) B ∧ |
| ∀ s p z, z∈Ω' → classicalDbar (fun z => B ((s,p),z)) z= |
| A (p,z)+s • (A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z)) := by |
| let A' : (ℝ × P) × ℂ → R := fun q => A (q.1.2,q.2) |
| have hA' : ContDiff ℝ (⊤ : ℕ∞) A' := |
| hA.comp ((contDiff_snd.comp contDiff_fst).prodMk contDiff_snd) |
| let a : n → n → (ℝ × P) × ℂ → ℂ := fun i j w => |
| (w.1.1 : ℂ)*coord q i (q j*A' w-A' w*q j) |
| have ha (i j : n) : ContDiff ℝ (⊤ : ℕ∞) (a i j) := |
| (Complex.ofRealCLM.contDiff.comp (contDiff_fst.comp contDiff_fst)).mul |
| (((coord q i).restrictScalars ℝ).contDiff.comp |
| ((contDiff_const.mul hA').sub (hA'.mul contDiff_const))) |
| have hac (i j : n) (w : ℝ × P) (z : ℂ) (hz : z∉K) : a i j (w,z)=0 := by |
| simp only [a,A',hAc w.2 z hz,mul_zero,zero_mul,sub_self,map_zero] |
| let f : n → (ℝ × P) × ℂ → ℂ := fun i w => coord q i (A' w) |
| have hf (i : n) : ContDiff ℝ (⊤ : ℕ∞) (f i) := |
| ((coord q i).restrictScalars ℝ).contDiff.comp hA' |
| have hfc (i : n) (w : ℝ × P) (z : ℂ) (hz : z∉K) : f i (w,z)=0 := by |
| simp only [f,A',hAc w.2 z hz,map_zero] |
| obtain ⟨v,hv,he⟩ := exists_classical_variable hS hΩS hΩ' hsub hK a ha hac f hf hfc hg hc hs h1 |
| let B : (ℝ × P) × ℂ → R := fun p => (coords q).symm (fun i => v p.1 i p.2) |
| have hL : ContDiff ℝ (⊤ : ℕ∞) (coords q).symm := by |
| let L : (n → ℂ) →L[ℝ] R := (coords q).symm.toContinuousLinearMap.restrictScalars ℝ |
| exact ContinuousLinearMap.contDiff (𝕜 := ℝ) (E := n → ℂ) (F := R) L |
| have hB : ContDiff ℝ (⊤ : ℕ∞) B := hL.comp (contDiff_pi.mpr hv) |
| have hev (w : ℝ × P) (i : n) : (fun z => coord q i (B (w,z)))=fun z => v w i z := by |
| funext z |
| change (coords q) ((coords q).symm (fun i => v w i z)) i=v w i z |
| rw [ContinuousLinearEquiv.apply_symm_apply] |
| refine ⟨B,hB,fun s p z hz => ?_⟩ |
| apply (coords q).injective |
| ext i |
| change coord q i (classicalDbar (fun z => B ((s,p),z)) z)= |
| coord q i (A (p,z)+s • (A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z))) |
| have hBs : Differentiable ℝ (fun z => B ((s,p),z)) := |
| (hB.comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp) |
| have hD := coord_dbar q (fun z => B ((s,p),z)) hBs z i |
| rw [hev] at hD |
| have he' := he (s,p) i hz |
| change schwartzDbar (v (s,p) i) z+∑ j, (s:ℂ)*coord q i (q j*A (p,z)-A (p,z)*q j)*v (s,p) j z=coord q i (A (p,z)) at he' |
| simp only [schwartzDbar,add_apply,smul_apply,SchwartzMap.lineDerivOp_apply_eq_fderiv,smul_eq_mul] at he' |
| rw [← hD] at he' |
| have hcomm : coord q i (B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z))=∑ j, coord q i (q j*A (p,z)-A (p,z)*q j)*v (s,p) j z := by |
| let L : R →ₗ[ℂ] R := ((ContinuousLinearMap.mul ℂ R).flip (A (p,z))- |
| ContinuousLinearMap.mul ℂ R (A (p,z))).toLinearMap |
| have hh := coord_linear_apply q L (B ((s,p),z)) i |
| change coord q i (B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z))= |
| ∑ j, coord q i (q j*A (p,z)-A (p,z)*q j)*coord q j (B ((s,p),z)) at hh |
| simpa only [congrFun (hev (s,p) _) z] using hh |
| have hs' : coord q i (A (p,z)+s • (A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z)))= |
| coord q i (A (p,z))-(s:ℂ)*(∑ j, coord q i (q j*A (p,z)-A (p,z)*q j)*v (s,p) j z) := by |
| rw [map_add] |
| have hneg : A (p,z)*B ((s,p),z)-B ((s,p),z)*A (p,z)=-(B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z)) := by abel |
| rw [hneg] |
| have hl := ((coord q i).restrictScalars ℝ).map_smul s (-(B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z))) |
| change coord q i (s • (-(B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z))))=s • coord q i (-(B ((s,p),z)*A (p,z)-A (p,z)*B ((s,p),z))) at hl |
| rw [hl,map_neg,hcomm] |
| simp [Complex.real_smul,sub_eq_add_neg] |
| rw [hs'] |
| change _+∑ j, (s:ℂ)*coord q i (q j*A (p,z)-A (p,z)*q j)*v (s,p) j z=coord q i (A (p,z)) at he' |
| rw [Finset.mul_sum] |
| simp_rw [mul_assoc] at he' ⊢ |
| linear_combination he' |
| end ElasticityPlanar |
| |
| end |
| end |
| section |
| noncomputable section |
| open Set Complex Module |
| namespace ElasticityPlanar |
| open ElasticityODE ElasticityCompact |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R] |
| omit [CompleteSpace R] in |
| lemma dbar_evaluation (b : ℂ → C(Time,R)) (hb : Differentiable ℝ b) (z : ℂ) (t : Time) : |
| classicalDbar (fun z => b z t) z=coefficientDbar b z t := by |
| have h := (ContinuousMap.evalCLM ℝ t : C(Time,R) →L[ℝ] R).hasFDerivAt.comp z (hb z).hasFDerivAt |
| change HasFDerivAt (fun z => b z t) _ z at h |
| rw [classicalDbar,h.fderiv] |
| rfl |
| lemma dbar_fundamental (b : ℂ → C(Time,R)) (hb : ContDiff ℝ (⊤ : ℕ∞) b) (z : ℂ) (t : Time) : |
| classicalDbar (fun z => fundamental (b z) t) z=frameDbar b z t := |
| dbar_evaluation (fun z => fundamental (b z)) |
| ((fundamental_contDiff b hb).differentiable (by simp)) z t |
| variable [FiniteDimensional ℂ R] {n : Type*} [Fintype n] [DecidableEq n] |
| omit [DecidableEq n] in |
| /-- A global smooth invertible planar frame on the prescribed inner domain. |
| It is built from the proved planar solver and coefficient homotopy flow. -/ |
| theorem exists_planar_frame {S Ω Ω' : Set ℂ} (hS : IsCompact S) |
| (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω) |
| (q : Basis n ℂ R) (A : ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A) |
| (hAc : HasCompactSupport A) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') : |
| ∃ V : ℂ → R, ContDiff ℝ (⊤ : ℕ∞) V ∧ (∀ z, IsUnit (V z)) ∧ |
| ∀ z∈Ω', classicalDbar V z=A z*V z := by |
| obtain ⟨B,hB,he⟩ := exists_algebra_homotopy hS hΩS hΩ' hsub q A hA hAc hg hc hs h1 |
| let f : ℂ × ℝ → R := fun p => B (p.2,p.1) |
| have hf : ContDiff ℝ (⊤ : ℕ∞) f := hB.comp (contDiff_snd.prodMk contDiff_fst) |
| let b : ℂ → C(Time,R) := curryCompact f hf.continuous (fun t : Time => (t:ℝ)) continuous_subtype_val |
| have hb : ContDiff ℝ (⊤ : ℕ∞) b := curryCompact_contDiff f hf _ continuous_subtype_val |
| let t : Time := ⟨1,by simp [Time]⟩ |
| let V : ℂ → R := fun z => fundamental (b z) t |
| have hV : ContDiff ℝ (⊤ : ℕ∞) V := by |
| have hL : ContDiff ℝ (⊤ : ℕ∞) (ContinuousMap.evalCLM ℝ t : C(Time,R) →L[ℝ] R) := |
| ContinuousLinearMap.contDiff (E := C(Time,R)) (F := R) _ |
| exact hL.comp (fundamental_contDiff b hb) |
| refine ⟨V,hV,fun z => fundamental_isUnit (b z) t,fun z hz => ?_⟩ |
| have hcoeff (s : Time) : coefficientDbar b z s=A z+(s:ℝ) • (A z*b z s-b z s*A z) := by |
| rw [← dbar_evaluation b (hb.differentiable (by simp)) z s] |
| exact he (s:ℝ) z hz |
| have ht := fundamental_transport b hb z (A z) hcoeff t |
| rw [← dbar_fundamental b hb] at ht |
| simpa only [t,one_smul] using ht |
| end ElasticityPlanar |
|
|
| end |
| end |
| section |
| noncomputable section |
| open Set Complex Module |
| namespace ElasticityPlanar |
| open ElasticityODE ElasticityCompact |
| variable {R : Type*} [NormedRing R] [NormedAlgebra ℂ R] [CompleteSpace R] |
| [FiniteDimensional ℂ R] {n : Type*} [Fintype n] |
| variable {P : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] |
| /-- A genuine jointly smooth global planar frame with arbitrary transverse |
| parameters. Invertibility holds everywhere, not only at a chosen base point. -/ |
| theorem exists_variable_planar_frame {S Ω Ω' K : Set ℂ} (hS : IsCompact S) |
| (hΩS : Ω ⊆ S) (hΩ' : IsOpen Ω') (hsub : Ω' ⊆ Ω) |
| (hK : IsCompact K) (q : Basis n ℂ R) (A : P × ℂ → R) (hA : ContDiff ℝ (⊤ : ℕ∞) A) |
| (hAc : ∀ p z, z∉K → A (p,z)=0) {g : ℂ → ℂ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hc : HasCompactSupport g) (hs : tsupport g ⊆ Ω) (h1 : Set.EqOn g 1 Ω') : |
| ∃ V : P × ℂ → R, ContDiff ℝ (⊤ : ℕ∞) V ∧ (∀ w, IsUnit (V w)) ∧ |
| ∀ p z, z∈Ω' → classicalDbar (fun z => V (p,z)) z=A (p,z)*V (p,z) := by |
| obtain ⟨B,hB,he⟩ := exists_algebra_variable_homotopy hS hΩS hΩ' hsub hK q A hA hAc hg hc hs h1 |
| let f : (P × ℂ) × ℝ → R := fun w => B ((w.2,w.1.1),w.1.2) |
| have hf : ContDiff ℝ (⊤ : ℕ∞) f := hB.comp |
| ((contDiff_snd.prodMk (contDiff_fst.comp contDiff_fst)).prodMk (contDiff_snd.comp contDiff_fst)) |
| let b : P × ℂ → C(Time,R) := curryCompact f hf.continuous (fun t : Time => (t:ℝ)) continuous_subtype_val |
| have hb : ContDiff ℝ (⊤ : ℕ∞) b := curryCompact_contDiff f hf _ continuous_subtype_val |
| let t : Time := ⟨1,by simp [Time]⟩ |
| let V : P × ℂ → R := fun w => fundamental (b w) t |
| have hV : ContDiff ℝ (⊤ : ℕ∞) V := by |
| have hL : ContDiff ℝ (⊤ : ℕ∞) (ContinuousMap.evalCLM ℝ t : C(Time,R) →L[ℝ] R) := |
| ContinuousLinearMap.contDiff (E := C(Time,R)) (F := R) _ |
| exact hL.comp (fundamental_contDiff b hb) |
| refine ⟨V,hV,fun w => fundamental_isUnit (b w) t,fun p z hz => ?_⟩ |
| let bp : ℂ → C(Time,R) := fun z => b (p,z) |
| have hbp : ContDiff ℝ (⊤ : ℕ∞) bp := hb.comp (contDiff_const.prodMk contDiff_id) |
| have hcoeff (s : Time) : coefficientDbar bp z s=A (p,z)+(s:ℝ) • (A (p,z)*bp z s-bp z s*A (p,z)) := by |
| rw [← dbar_evaluation bp (hbp.differentiable (by simp)) z s] |
| exact he (s:ℝ) p z hz |
| have ht := fundamental_transport bp hbp z (A (p,z)) hcoeff t |
| rw [← dbar_fundamental bp hbp] at ht |
| simpa only [t,one_smul] using ht |
| end ElasticityPlanar |
| |
| end |
| end |
| |
| end OAI |
| |
