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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.Sources.LocalPiola
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma localPiolaSource_support (X : OpenPartialHomeomorph E E) (r : E → ℝ) :
Function.support (localPiolaSource X r)⊆X '' (Function.support r∩X.source) := by
intro y hy
by_cases hyt : y∈X.target
· refine ⟨X.symm y,⟨?_,X.map_target hyt⟩,X.right_inv hyt⟩
intro hz
exact hy (by simp [localPiolaSource,hyt,hz])
· exact False.elim (hy (by simp [localPiolaSource,hyt]))
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
lemma localPiolaSource_compact (X : OpenPartialHomeomorph E E)
(r : E → ℝ) (hc : HasCompactSupport r) (hs : tsupport r⊆X.source) :
HasCompactSupport (localPiolaSource X r) ∧ tsupport (localPiolaSource X r)⊆X.target := by
have hci : IsCompact (X '' tsupport r) := hc.image_of_continuousOn (X.continuousOn.mono hs)
have ht : tsupport (localPiolaSource X r)⊆X '' tsupport r :=
closure_minimal ((localPiolaSource_support X r).trans
(image_mono (fun _ hx => subset_closure hx.1))) hci.isClosed
exact ⟨hci.of_isClosed_subset isClosed_closure ht,ht.trans (by
rintro y ⟨x,hx,rfl⟩; exact X.map_source (hs hx))⟩
omit [MeasurableSpace E] [BorelSpace E] in
lemma localPiolaSource_smooth (X : OpenPartialHomeomorph E E)
(hX : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X X.source)
(hXi : ContDiffOn ℝ (↑(⊤ : ℕ∞)) X.symm X.target)
(r : E → ℝ) (hr : ContDiff ℝ (↑(⊤ : ℕ∞)) r)
(hc : HasCompactSupport r) (hs : tsupport r⊆X.source) :
ContDiff ℝ (↑(⊤ : ℕ∞)) (localPiolaSource X r) := by
apply contDiff_of_contDiffOn_tsupport X.open_target _ (localPiolaSource_compact X r hc hs).2
apply X.open_target.contDiffOn_iff.mpr
intro y hy
have hxi := hXi.contDiffAt (X.open_target.mem_nhds hy)
have hx := hX.contDiffAt (X.open_source.mem_nhds (X.map_target hy))
have hd : ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun z => fderiv ℝ X (X.symm z)) y :=
(hx.fderiv_right (by simp)).comp y hxi
have hd0 := local_fderiv_det_ne_zero X (hX.differentiableOn (by simp))
(hXi.differentiableOn (by simp)) (X.map_target hy)
have hdet := ((contDiff_clm_det (E:=E) (↑(⊤ : ℕ∞))).contDiffAt.comp y hd).abs hd0
have hformula := (hdet.inv (abs_ne_zero.mpr hd0)).mul (hr.contDiffAt.comp y hxi)
apply hformula.congr_of_eventuallyEq
filter_upwards [X.open_target.mem_nhds hy] with z hz
simp only [localPiolaSource,ite_eq_left hz]
rfl
lemma localPiolaSource_coordinate_pairing (μ : Measure E) [μ.IsAddHaarMeasure]
(X : OpenPartialHomeomorph E E) (hX : DifferentiableOn ℝ X X.source)
(hXi : DifferentiableOn ℝ X.symm X.target)
(r : E → ℝ) (hs : Function.support r⊆X.source) (φ : E → ℝ) :
(∫ y,φ (X.symm y)*localPiolaSource X r y ∂μ)=(∫ x,φ x*r x ∂μ) := by
rw [localPiolaSource_pairing μ X hX
(fun _ hx => local_fderiv_det_ne_zero X hX hXi hx) r hs (fun y => φ (X.symm y))]
apply integral_congr_ae
filter_upwards [] with x
by_cases hx : x∈Function.support r
· simp only [X.left_inv (hs hx)]
· rw [Function.notMem_support.mp hx,mul_zero,mul_zero]
end ScalarConductivity
end
end OAI
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