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a458723 | 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 77 78 | import Mathlib
import OAI.Analysis.Conductivity.Sources.ParametricPullback
import OAI.Analysis.Conductivity.Variational.LocalBoundedPiolaCorrection
namespace OAI
section
noncomputable section
namespace ScalarConductivity
open Set Filter Topology Matrix MeasureTheory
open scoped Matrix.Norms.Elementwise
variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P]
def VanishingCorrectionRegion.piola
{u v : P×Coord3 → Fin 2 → ℝ}
(hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) (hv : ContDiff ℝ (↑(⊤:ℕ∞)) v) (p : P)
(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)
{V : Set P} (hV : IsOpen V) (hp : p∈V) (hVb : Bornology.IsBounded V)
{A U W K : Set Coord3} (hA : IsOpen A) (B : VanishingCorrectionRegion v p A)
(hK : IsCompact K) (hKB : K⊆B.region) (hVA : V×ˢA⊆Y.target)
(hW : ∀ q∈V,W⊆(fiberSlice Y hY q).symm '' K)
(hWo : IsOpen W) (hWn : W.Nonempty) (hWb : Bornology.IsBounded W) (hWU : W⊆U)
(himage : ∀ q∈V,(fiberSlice Y hY q).symm '' A⊆U)
(hpot : ∀ q∈V,∀ y∈A,u (q,(fiberSlice Y hY q).symm y)=v (q,y))
{C : ℝ} (hC : 0<C)
(hbound : ∀ q∈V,∀ y∈A,
let X := (fiberSlice Y hY q).symm
9*|(fderiv ℝ X y).det|⁻¹*‖operatorMatrix (fderiv ℝ X y)‖*
‖operatorMatrix (fderiv ℝ X y)‖≤C) :
VanishingCorrectionRegion u p U := by
refine ⟨W,hWo,hWn,hWU,hWb,?_⟩
intro r hr ε hε
obtain ⟨R,hR,hRe⟩ := exists_parametric_source_pullback Y hY hsm hinv p hV hp hVb
B.region_open B.region_bounded hK hKB (fun z hz => hVA ⟨hz.1,B.region_subset hz.2⟩) hW hr
have hsolve := B.solve R hR (ε/C) (div_pos hε hC)
filter_upwards [hRe,hsolve,hV.mem_nhds hp] with q he solve hq hm
let X := (fiberSlice Y hY q).symm
have hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source := (fiberSlice_smooth Y hY hsm hinv q).2
have hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target :=
(fiberSlice_smooth Y hY hsm hinv q).1.contDiffOn
have hAs : A⊆X.source := fun y hy => hVA ⟨hq,hy⟩
have huq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => u (q,y)) := hu.comp (contDiff_const.prodMk contDiff_id)
have hvq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => v (q,y)) := hv.comp (contDiff_const.prodMk contDiff_id)
have hps : EqOn (fun y => u (q,y)) ((fun y => v (q,y))∘X.symm) (X '' A) := by
rintro _ ⟨y,hy,rfl⟩
change u (q,X y)=v (q,X.symm (X y))
rw [X.left_inv (hAs hy)]
exact hpot q hq y hy
have hrX (j) : tsupport (fun y => r j (q,y))⊆X.target :=
(hr.supported q j).trans ((hW q hq).trans (by
rintro _ ⟨y,hy,rfl⟩
exact X.map_source (hAs (B.region_subset (hKB hy)))))
have hre : (fun j => localPiolaSource X (fun y => R j (q,y)))=(fun j y => r j (q,y)) := by
funext j
rw [congrFun he j]
exact localPiolaSource_inverse X (hX.differentiableOn (by simp))
(hXi.differentiableOn (by simp)) _ ((subset_tsupport _).trans (hrX j))
have hRs : PairSupported (fun j y => R j (q,y)) A := (hR.supported q).mono B.region_subset
have hRm : physicalSourceMoment (fun y => v (q,y)) (fun j y => R j (q,y))=0 := by
have hh := physicalSourceMoment_piola_local X hX hXi huq hvq hAs hps (hR.compact q) hRs
rw [hre,hm] at hh
exact hh.symm
have hh := (solve hRm).piola_local X hX hXi huq hvq hA hAs hps hC.le
(hbound q hq) (hR.compact q) hRs
rw [hre,mul_div_cancel₀ ε hC.ne'] at hh
exact hh.region_mono (himage q hq)
end ScalarConductivity
end
end
end OAI
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