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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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import Mathlib
import OAI.Analysis.Conductivity.Variational.CompactFiberPrimitive
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Matrix Filter Topology
variable {ι P : Type*}
[TopologicalSpace P] [FirstCountableTopology P] [LocallyCompactSpace P]
def compactCrossMoment (χ : ℝ → ℝ) (w v : ℝ → ι → ℝ) : Matrix ι ι ℝ :=
fun i j => ∫ s, χ s*w s j*v s i
lemma compactCrossMoment_restrict (χ : ℝ → ℝ) (w v : ℝ → ι → ℝ) (i j : ι) :
compactCrossMoment χ w v i j=∫ s in tsupport χ, χ s*w s j*v s i := by
dsimp only [compactCrossMoment]
nth_rw 1 [←setIntegral_univ]
apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero MeasurableSet.univ (subset_univ _)
intro s hs
simp only [image_eq_zero_of_notMem_tsupport hs.2,zero_mul]
lemma compactCrossMoment_continuous {χ : ℝ → ℝ} {w : ℝ → ι → ℝ}
{v : P → ℝ → ι → ℝ} (hc : Continuous χ) (hs : HasCompactSupport χ)
(hw : ∀ j,Continuous (fun s => w s j))
(hv : ∀ i,Continuous (fun ps : P × ℝ => v ps.1 ps.2 i)) :
Continuous (fun p => compactCrossMoment χ w (v p)) := by
apply continuous_pi; intro i
apply continuous_pi; intro j
simp only [compactCrossMoment_restrict]
apply continuous_parametric_integral_of_continuous (μ := volume) _ hs
exact ((hc.comp continuous_snd).mul ((hw j).comp continuous_snd)).mul (hv i)
lemma compactCrossMoment_self (χ : ℝ → ℝ) (w : ℝ → ι → ℝ) :
compactCrossMoment χ w w=compactMomentGram χ w := by
ext i j
apply integral_congr_ae
exact Filter.Eventually.of_forall (fun s => by dsimp; ring)
variable [Fintype ι] [DecidableEq ι]
lemma compactCrossMoment_solve {χ : ℝ → ℝ} {w v : ℝ → ι → ℝ}
(hc : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (hs : HasCompactSupport χ)
(hw : ∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => w s j))
(hv : ∀ i,Continuous (fun s => v s i))
(hM : IsUnit (compactCrossMoment χ w v)) (b : ι → ℝ) :
∃ a : ι → ℝ, compactCrossMoment χ w v*ᵥa=b ∧
ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => χ s*(a ⬝ᵥ w s)) ∧
HasCompactSupport (fun s => χ s*(a ⬝ᵥ w s)) ∧
tsupport (fun s => χ s*(a ⬝ᵥ w s))⊆tsupport χ ∧
(∀ i,(∫ s,χ s*(a ⬝ᵥ w s)*v s i)=b i) := by
obtain ⟨a,ha⟩ := Matrix.mulVec_surjective_iff_isUnit.mpr hM b
refine ⟨a,ha,hc.mul (ContDiff.sum fun j _ => contDiff_const.mul (hw j)),
hs.mul_right,tsupport_mul_subset_left,?_⟩
intro i
rw [←congrFun ha i]
change (∫ s,χ s*(a ⬝ᵥ w s)*v s i)=∑ j,compactCrossMoment χ w v i j*a j
calc
_ = ∫ s,∑ j,(χ s*w s j*v s i)*a j := by
apply integral_congr_ae
exact Filter.Eventually.of_forall fun s => by
simp only [dotProduct,Finset.mul_sum,Finset.sum_mul]
apply Finset.sum_congr rfl; intro j _; ring
_ = _ := by
have hi (j : ι) : Integrable (fun s => (χ s*w s j*v s i)*a j) := by
exact (((hc.continuous.mul (hw j).continuous).mul (hv i)).integrable_of_hasCompactSupport
(hs.mul_right.mul_right)).mul_const _
rw [integral_finsetSum _ (fun j _ => hi j)]
simp only [integral_mul_const,compactCrossMoment]
theorem compactMoment_perturbed_surjective {χ : ℝ → ℝ} {w : ℝ → ι → ℝ}
{v : P → ℝ → ι → ℝ} {p₀ : P}
(hc : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (hs : HasCompactSupport χ)
(hw : ∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => w s j))
(hv : ∀ i,Continuous (fun ps : P × ℝ => v ps.1 ps.2 i))
(hlim : v p₀=w) (hG : (compactMomentGram χ w).PosDef) :
∀ᶠ p in 𝓝 p₀,∀ b : ι → ℝ,
∃ f : ℝ → ℝ,ContDiff ℝ (↑(⊤ : ℕ∞)) f ∧ HasCompactSupport f ∧
tsupport f⊆tsupport χ ∧ (∀ i,(∫ s,f s*v p s i)=b i) := by
have hM := compactCrossMoment_continuous hc.continuous hs (fun j => (hw j).continuous) hv
have hM₀ : IsUnit (compactCrossMoment χ w (v p₀)) := by
rw [hlim,compactCrossMoment_self]
exact hG.isUnit
have hd₀ : (compactCrossMoment χ w (v p₀)).det≠0 :=
isUnit_iff_ne_zero.mp ((Matrix.isUnit_iff_isUnit_det _).mp hM₀)
have hed : ∀ᶠ p in 𝓝 p₀,(compactCrossMoment χ w (v p)).det≠0 :=
(isOpen_ne_fun hM.matrix_det continuous_const).mem_nhds hd₀
have he : ∀ᶠ p in 𝓝 p₀,IsUnit (compactCrossMoment χ w (v p)) :=
hed.mono (fun p hp => (Matrix.isUnit_iff_isUnit_det _).mpr (isUnit_iff_ne_zero.mpr hp))
filter_upwards [he] with p hp
intro b
obtain ⟨a,ha,hf,hfs,hsub,hb⟩ := compactCrossMoment_solve hc hs hw
(fun i => (hv i).comp (continuous_const.prodMk continuous_id)) hp b
exact ⟨_,hf,hfs,hsub,hb⟩
end ScalarConductivity
end
end OAI
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