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159357f | 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 75 76 | import Mathlib
import OAI.Analysis.Conductivity.Branching.SmoothFamilyCorrectionTransport
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology MeasureTheory
variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P]
theorem exists_parametric_source_pullback
(Y : OpenPartialHomeomorph (P×Coord3) (P×Coord3))
(hY : ∀ q x,(Y (q,x)).1=q)
(hsm : ContDiff ℝ (↑(⊤:ℕ∞)) Y)
(hinv : ContDiffOn ℝ (↑(⊤:ℕ∞)) Y.symm Y.target)
(p : P) {V : Set P} {O W K : Set Coord3}
(hV : IsOpen V) (hp : p∈V) (hVb : Bornology.IsBounded V)
(hO : IsOpen O) (hOb : Bornology.IsBounded O)
(hK : IsCompact K) (hKO : K⊆O) (hVO : V×ˢO⊆Y.target)
(hW : ∀ q∈V,W⊆(fiberSlice Y hY q).symm '' K)
{r : Fin 2 → P×Coord3 → ℝ} (hr : SmoothVanishingPairFamily p W r) :
∃ R : Fin 2 → P×Coord3 → ℝ,SmoothVanishingPairFamily p O R ∧
∀ᶠ q in 𝓝 p,(fun j x => R j (q,x))=(fun j => localPiolaSource (fiberSlice Y hY q) (fun x => r j (q,x))) := by
obtain ⟨χ,hχ,hχc,hχs,_,hχone⟩ := exists_smooth_core_cutoff
(isCompact_singleton.prod hK) (hV.prod hO) (hVb.prod hOb)
(prod_mono (singleton_subset_iff.mpr hp) hKO)
let w : P×Coord3 → ℝ := fun q => inverseSourceWeight (fiberSlice Y hY q.1).symm q.2
have hw := parametric_inverseSourceWeight_smooth Y hY hsm hinv
let g (j : Fin 2) : P×Coord3 → ℝ := fun q => w q*r j (q.1,(Y.symm q).2)
have hg (j) : ContDiffOn ℝ (↑(⊤:ℕ∞)) (g j) Y.target :=
hw.mul ((hr.1 j).contDiffOn.comp (contDiff_fst.contDiffOn.prodMk
(contDiff_snd.contDiffOn.comp hinv (fun _ _ => mem_univ _))) (fun _ _ => mem_univ _))
let R : Fin 2 → P×Coord3 → ℝ := fun j q => χ q*g j q
have hR (j) : ContDiff ℝ (↑(⊤:ℕ∞)) (R j) :=
smooth_mul_of_support_in_open Y.open_target (hg j) hχ (hχs.trans hVO)
have hRq (q : P) (j) : tsupport (fun x => R j (q,x))⊆Prod.snd '' tsupport χ := by
intro x hx
have hc : (q,x)∈tsupport χ := (tsupport_comp_subset_preimage χ (continuous_const.prodMk continuous_id))
(tsupport_mul_subset_left hx)
exact ⟨(q,x),hc,rfl⟩
refine ⟨R,⟨hR,fun j x => by simp only [R,g,hr.2.1 j _,mul_zero],
Prod.snd '' tsupport χ,hχc.image continuous_snd,
fun x hx => by rcases hx with ⟨z,hz,rfl⟩; exact (hχs hz).2,
hRq⟩,?_⟩
have hχnear : ∀ᶠ q in 𝓝 p,∀ x∈K,χ (q,x)=1 := by
apply hK.eventually_forall_of_forall_eventually
intro x hx
exact hχone (p,x) ⟨mem_singleton p,hx⟩
filter_upwards [hV.mem_nhds hp,hχnear] with q hq hone
funext j x
let X := (fiberSlice Y hY q).symm
have hXs : K⊆X.source := fun y hy => hVO ⟨hq,hKO hy⟩
have hs : tsupport (fun x => r j (q,x))⊆X '' K := (hr.supported q j).trans (hW q hq)
have hpr : tsupport (localPiolaSource X.symm (fun x => r j (q,x)))⊆K :=
localPiolaSource_inverse_support X _ (hr.compact q j).2 hXs hs
by_cases hx : (q,x)∈Y.target
· have he : localPiolaSource (fiberSlice Y hY q) (fun y => r j (q,y)) x=g j (q,x) := by
simp only [localPiolaSource,ite_eq_left (show x∈(fiberSlice Y hY q).target from hx)]
rfl
rw [he]
by_cases hxK : x∈K
· simp only [R,hone x hxK,one_mul]
· have hz : g j (q,x)=0 := he.symm.trans (image_eq_zero_of_notMem_tsupport (fun h => hxK (hpr h)))
simp only [R,hz,mul_zero]
· have hc : χ (q,x)=0 := image_eq_zero_of_notMem_tsupport (fun hh => hx (hVO (hχs hh)))
simp only [R,hc,zero_mul,localPiolaSource,ite_eq_right (show x∉(fiberSlice Y hY q).target from hx)]
end ScalarConductivity
end
end
end OAI
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