| import Mathlib |
| import OAI.Geometry.PrescribedPotential.ChartTransport |
| import OAI.Geometry.PrescribedPotential.CoreBounds |
|
|
| namespace OAI |
|
|
| /-! Chart Transport Higher. -/ |
|
|
| section |
|
|
| |
|
|
| noncomputable section |
| open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv TemperedDistribution |
| open scoped ContDiff SchwartzMap Classical Laplacian |
|
|
| namespace GlobalElliptic |
| open SobolevChart |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] |
| variable (e : OpenPartialHomeomorph E E) (he : ContDiffOn ℝ ∞ e e.source) |
|
|
| include he in |
| lemma transportCoefficient_smooth (v : E) (j : Fin (Module.finrank ℝ E)) : |
| ContDiffOn ℝ ∞ (fun x => ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ)) e.source := by |
| apply Complex.ofRealCLM.contDiff.comp_contDiffOn |
| apply ((EuclideanSpace.proj j).contDiff.comp (stdOrthonormalBasis ℝ E).repr.contDiff).comp_contDiffOn |
| exact (he.fderiv_of_isOpen e.open_source (by simp)).clm_apply contDiffOn_const |
|
|
| def transportCoefficient (κ : ChartCutoff e.source) (v : E) (j : Fin (Module.finrank ℝ E)) : |
| ChartCutoff e.source := κ.mulOn e.open_source (transportCoefficient_smooth e he v j) |
|
|
| @[simp] lemma transportCoefficient_apply (κ : ChartCutoff e.source) (v : E) |
| (j : Fin (Module.finrank ℝ E)) (x : E) : |
| transportCoefficient e he κ v j x = |
| κ x * ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ) := rfl |
|
|
| omit [FiniteDimensional ℝ E] in |
| lemma chartTransport_tsupport (κ : ChartCutoff e.source) (f : 𝓢(E, ℂ)) : |
| tsupport (chartTransport e he κ f : E → ℂ) ⊆ tsupport (κ : E → ℂ) := |
| κ.product_tsupport e.open_source ((f.smooth ⊤).comp_contDiffOn he) |
|
|
| |
|
|
| lemma chartTransport_deriv (κ : ChartCutoff e.source) (v : E) (f : 𝓢(E, ℂ)) : |
| ∂_{v} (chartTransport e he κ f) = chartTransport e he (κ.deriv v) f + |
| ∑ j, chartTransport e he (transportCoefficient e he κ v j) |
| (∂_{stdOrthonormalBasis ℝ E j} f) := by |
| ext x |
| by_cases hx : x ∈ e.source |
| · have hed := (he.contDiffAt (e.open_source.mem_nhds hx)).differentiableAt (by simp) |
| have hfd := f.differentiableAt.comp x hed |
| simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv, chartTransport_apply, |
| _root_.add_apply, _root_.sum_apply, transportCoefficient_apply] |
| change fderiv ℝ (fun y => κ y * f (e y)) x v = |
| fderiv ℝ κ.val x v * f (e x) + _ |
| have hm := fderiv_fun_mul κ.val.differentiableAt hfd |
| have hc := fderiv_comp x f.differentiableAt hed |
| dsimp only [Function.comp_def] at hm hc |
| rw [hm, hc] |
| simp only [_root_.add_apply, _root_.smul_apply, |
| ContinuousLinearMap.comp_apply, smul_eq_mul] |
| have hb := congrArg (fun z => fderiv ℝ f (e x) z) |
| ((stdOrthonormalBasis ℝ E).sum_repr (fderiv ℝ e x v)) |
| simp only [map_sum, map_smul, Complex.real_smul] at hb |
| rw [← hb, Finset.mul_sum] |
| ring_nf |
| · have hk : x ∉ tsupport (κ : E → ℂ) := fun ht => hx (κ.support_sub ht) |
| have ht : x ∉ tsupport (chartTransport e he κ f : E → ℂ) := |
| fun ht => hk (chartTransport_tsupport e he κ f ht) |
| have hder : (∂_{v} (chartTransport e he κ f) : 𝓢(E, ℂ)) x = 0 := by |
| rw [SchwartzMap.lineDerivOp_apply_eq_fderiv, |
| (notMem_tsupport_iff_eventuallyEq.mp ht).fderiv_eq] |
| simp |
| have hk' : κ.deriv v x = 0 := image_eq_zero_of_notMem_tsupport |
| (fun h => hk (κ.deriv_tsupport v h)) |
| simp only [_root_.add_apply, _root_.sum_apply, chartTransport_apply, |
| transportCoefficient_apply, hder, image_eq_zero_of_notMem_tsupport hk, hk', zero_mul, |
| Finset.sum_const_zero, add_zero] |
|
|
| variable [MeasurableSpace E] [BorelSpace E] |
|
|
| lemma word_deriv_comm (ws : List E) (v : E) (f : 𝓢(E, ℂ)) : |
| schwartzWord ws (∂_{v} f) = ∂_{v} (schwartzWord ws f) := by |
| induction ws with |
| | nil => rfl |
| | cons w ws ih => |
| change ∂_{w} (schwartzWord ws (∂_{v} f)) = ∂_{v} (∂_{w} (schwartzWord ws f)) |
| rw [ih, schwartz_deriv_comm] |
|
|
| include he in |
| lemma chartTransport_word_bound (hs : ContDiffOn ℝ ∞ e.symm e.target) (ws : List E) |
| (κ : ChartCutoff e.source) {s : ℝ} (hws : (ws.length : ℝ) ≤ s) : |
| CoreBound s 0 (fun f => schwartzWord ws (chartTransport e he κ f)) := by |
| induction ws generalizing κ s with |
| | nil => |
| have hzero : CoreBound 0 0 (chartTransport e he κ) := chartTransport_l2_bound e he hs κ |
| exact hzero.comp (CoreBound.id (by simpa using hws)) |
| | cons w ws ih => |
| have hw : (ws.length : ℝ) + 1 ≤ s := by simpa only [List.length_cons, Nat.cast_add, Nat.cast_one] using hws |
| have hfirst := ih (κ.deriv w) (s := s) (by linarith) |
| have hrest (j : Fin (Module.finrank ℝ E)) := |
| (ih (transportCoefficient e he κ w j) (s := s - 1) (by linarith)).comp |
| (coreBound_deriv (stdOrthonormalBasis ℝ E j) (s := s) (t := s - 1) le_rfl) |
| have hh := hfirst.add (CoreBound.sum Finset.univ _ (fun j _ => hrest j)) |
| convert hh using 1 |
| funext f |
| change ∂_{w} (schwartzWord ws (chartTransport e he κ f)) = _ |
| rw [← word_deriv_comm, chartTransport_deriv, map_add, map_sum] |
| rfl |
|
|
| |
|
|
| theorem chartTransport_even_bound (hs : ContDiffOn ℝ ∞ e.symm e.target) |
| (κ : ChartCutoff e.source) (k : ℕ) : |
| CoreBound (2 * (k : ℝ)) (2 * (k : ℝ)) (chartTransport e he κ) := by |
| apply coreBound_of_words k |
| intro ws hw |
| apply chartTransport_word_bound e he hs ws κ |
| exact_mod_cast hw |
|
|
| end GlobalElliptic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
