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import OAI.Analysis.Conductivity.Variational.ChangeOld
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
open Matrix Filter Topology
open Set MeasureTheory Filter Topology
open scoped BigOperators
open Set MeasureTheory Filter Topology
open scoped Manifold
open Set Filter
open scoped Topology
open Set Filter MeasureTheory
open scoped Topology Manifold ENNReal
open Set
namespace ScalarConductivity
open Matrix Set MeasureTheory Filter Topology
open scoped Matrix.Norms.Elementwise
structure CompactSelectedReplacement (J : Set (Fin 2)) (μ : Measure Coord3) (O : Set Coord3)
(u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) where
du : Coord3 → Fin 2 → ℝ
dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ
tensor : Coord3 → Symmetric3
smooth_du : ContDiff ℝ (↑(⊤ : ℕ∞)) du
compact_du : HasCompactSupport du
support_du : tsupport du ⊆ O
smooth_dF : ContDiff ℝ (↑(⊤ : ℕ∞)) dF
compact_dF : HasCompactSupport dF
support_dF : tsupport dF ⊆ O
cauchy_dF : ∀ j ∈ J, ∀ (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x ((dF x).col j) ∂μ) = 0
smooth_tensor : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (tensor x).val) O
rank : ∀ x ∈ O, LinearIndependent ℝ
(gradientColumns (fderiv ℝ (fun y => u y+du y) x)).col
constitutive : ∀ x ∈ O, (tensor x).val *
gradientColumns (fderiv ℝ (fun y => u y+du y) x) = conductivityFlux u A x + dF x
theorem synchronized_selected_replacement
(J : Set (Fin 2))
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
(u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u)
(A : Coord3 → Symmetric3) {U O K : Set Coord3}
(hU : IsOpen U) (hUb : Bornology.IsBounded U) (hOU : O ⊆ U)
(hK : IsCompact K) (hKO : K ⊆ O)
(hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U)
(hdiv : ∀ j ∈ J, ∀ (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(Y : Coord3 ≃ Coord3) (hY : ContDiff ℝ (↑(⊤ : ℕ∞)) Y)
(hYi : ContDiff ℝ (↑(⊤ : ℕ∞)) Y.symm) (hoff : ∀ x ∉ K, Y x = x)
(z : Coord3 → ℝ) (hz : ContDiff ℝ (↑(⊤ : ℕ∞)) z)
(C : ℝ → Symmetric3)
(hC : ∀ x ∈ O, ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun t => (C t).val) (z x))
(δ : Fin 2 → Coord3 → Coord3)
(hδ : ∀ j, ContDiff ℝ (↑(⊤ : ℕ∞)) (δ j))
(hsδ : ∀ j, tsupport (δ j) ⊆ K)
(hδdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ →
(∫ x, fderiv ℝ ψ x (δ j x) ∂μ) = 0)
(hstate : ∀ x ∈ O,
let E := gradientColumns (fderiv ℝ u x)
let Ft := conductivityFlux u A x + Matrix.of (fun i j => δ j x i)
let J := operatorMatrix (fderiv ℝ Y x)
let At := repairPushed (C (z x)) J E Ft
0 < J.det ∧ LinearIndependent ℝ E.col ∧
At.val * gradientColumns (fderiv ℝ (u ∘ Y.symm) (Y x)) = pushFlux J Ft) :
∃ R : CompactSelectedReplacement J μ O u A, ∀ x ∈ O,
R.tensor (Y x) = repairPushed (C (z x)) (operatorMatrix (fderiv ℝ Y x))
(gradientColumns (fderiv ℝ u x))
(conductivityFlux u A x + Matrix.of (fun i j => δ j x i)) := by
classical
let Ft : Coord3 → Matrix (Fin 3) (Fin 2) ℝ :=
fun x => conductivityFlux u A x + Matrix.of (fun i j => δ j x i)
let G : Coord3 → Matrix (Fin 3) (Fin 2) ℝ :=
fun x => Matrix.of (fun i j => piolaFlux Y (fun t => (Ft t).col j) x i)
let dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ := fun x => G x - conductivityFlux u A x
let B : Coord3 → Symmetric3 := fun x => repairPushed (C (z x))
(operatorMatrix (fderiv ℝ Y x)) (gradientColumns (fderiv ℝ u x)) (Ft x)
let du : Coord3 → Fin 2 → ℝ := fun x => u (Y.symm x)-u x
have heq : (fun x => u x+du x) = u ∘ Y.symm := by
funext x; simp only [du, Function.comp_apply]; abel
have him : MapsTo Y.symm O O := (equiv_mapsTo_of_fixed_compl Y O
(fun x hx => hoff x (fun hk => hx (hKO hk)))).2
have hF := conductivityFlux_contDiffOn u hu A hU hA
have hcol (j : Fin 2) : ContDiffOn ℝ (↑(⊤ : ℕ∞))
(fun x => (conductivityFlux u A x).col j) U := by
apply contDiffOn_pi.mpr; intro i
exact contDiffOn_pi.mp (contDiffOn_pi.mp hF i) j
have hdcol (j : Fin 2) := local_compact_updated_flux_regular Y hY hYi hU hUb hK
(hKO.trans hOU) hoff (fun x => (conductivityFlux u A x).col j) (δ j)
(hcol j) (hδ j) (hsδ j)
have hdFe (j : Fin 2) : (fun x => (dF x).col j) =
fun x => piolaFlux Y (fun t => (conductivityFlux u A t).col j + δ j t) x -
(conductivityFlux u A x).col j := by rfl
have hdFs : ContDiff ℝ (↑(⊤ : ℕ∞)) dF := by
apply contDiff_pi.mpr; intro i
apply contDiff_pi.mpr; intro j
exact contDiff_pi.mp (hdFe j ▸ (hdcol j).1) i
have hdsupp : tsupport dF ⊆ K := tsupport_matrix_subset dF hK.isClosed
(fun j => hdFe j ▸ (hdcol j).2.2)
have hdu := local_compact_recomposition_difference Y hYi hU hUb hK
(hKO.trans hOU) hoff u hu.contDiffOn
have hBs : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (B x).val) O := by
intro x hx
have hd := (hstate x hx).1.ne'
have hi := (hstate x hx).2.1
have hFt : ContDiffAt ℝ (↑(⊤ : ℕ∞)) Ft x := by
apply ((hF x (hOU hx)).contDiffAt (hU.mem_nhds (hOU hx))).add
apply contDiffAt_pi.mpr; intro i
apply contDiffAt_pi.mpr; intro j
exact contDiffAt_pi.mp (hδ j).contDiffAt i
exact (contDiffAt_repairPushed (fun x => C (z x)) _ _ _
((hC x hx).comp x hz.contDiffAt)
((operatorMatrix_contDiff (m := Fin 3) (n := Fin 3)).contDiffAt.comp x (hY.contDiffAt.fderiv_right (by simp)))
(contDiff_gradientColumns u hu).contDiffAt hFt hd hi).contDiffWithinAt
refine ⟨{
du := du, dF := dF, tensor := B ∘ Y.symm
smooth_du := hdu.1, compact_du := hdu.2.1, support_du := hdu.2.2.trans hKO
smooth_dF := hdFs, compact_dF := hK.of_isClosed_subset isClosed_closure hdsupp
support_dF := hdsupp.trans hKO
cauchy_dF := ?_
smooth_tensor := hBs.comp hYi.contDiffOn him
rank := ?_
constitutive := ?_ }, ?_⟩
· intro j hj ψ hψ
have hh := local_compact_updated_flux μ Y hY hYi hU hUb hK (hKO.trans hOU) hoff
(fun x => (conductivityFlux u A x).col j) (δ j) (hcol j) (hdiv j hj)
(hδ j) (hsδ j) (hδdiv j)
simpa only [congrFun (hdFe j)] using hh.2.2.2 ψ hψ
· intro x hx
rw [heq, ← Y.apply_symm_apply x, gradientColumns_recomposition Y
(hY.differentiable (by simp)) (hYi.differentiable (by simp)) u (hu.differentiable (by simp))]
exact pushGradient_rank _ (hstate (Y.symm x) (him hx)).1.ne' _
(hstate (Y.symm x) (him hx)).2.1
· intro x hx
rw [heq]
have hh := (hstate (Y.symm x) (him hx)).2.2
rw [Y.apply_symm_apply] at hh
change (B (Y.symm x)).val * gradientColumns (fderiv ℝ (u ∘ Y.symm) x) =
conductivityFlux u A x + dF x
rw [hh]
have he : conductivityFlux u A x+dF x=G x := by simp only [dF]; abel
rw [he]
ext i j
have hd : 0 < (fderiv ℝ Y (Y.symm x)).det := by
rw [← operatorMatrix_det]; exact (hstate (Y.symm x) (him hx)).1
have hp := piolaFlux_columns Y Ft hd j
rw [Y.apply_symm_apply] at hp
exact congrFun hp.symm i
· intro x _
simp only [Function.comp_apply, Y.symm_apply_apply, B, Ft]
theorem exists_selected_split_on_subpatch
(J : Set (Fin 2))
(μ : Measure Coord3) [μ.IsAddHaarMeasure]
(u : Coord3 → Fin 2 → ℝ) (hu : ContDiff ℝ (↑(⊤ : ℕ∞)) u)
(A : Coord3 → Symmetric3) {p : Coord3} (hAp : ContinuousAt A p)
{U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U) (hpU : p ∈ U)
(hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U)
(hdiv : ∀ j ∈ J, ∀ (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ →
tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0)
(hD : Function.Surjective (fderiv ℝ u p))
(d : Coord3) (L : Coord3 →L[ℝ] ℝ) (hLd : L d = 1)
(B : Mat3) (hB : B.IsSymm) (hBn : ∀ v, L (B *ᵥ v) = 0)
{θ c m : ℝ} (hθ : 0 < θ) (hθ1 : θ < 1)
(hm : 0 < m) (hpath : ∀ z ∈ Icc (-θ) (1-θ), m ≤ 1+c*z)
(C : ℝ → Symmetric3) (hC : ∀ z ∈ Icc (-θ) (1-θ), ContinuousAt C z)
(hCs : ∀ z ∈ Icc (-θ) (1-θ), ContDiffAt ℝ (↑(⊤ : ℕ∞)) (fun t => (C t).val) z)
(hdet : ∀ z ∈ Icc (-θ) (1-θ), 0 < (splitJacobian d L c z).det)
(hconstit : ∀ z ∈ Icc (-θ) (1-θ),
(C z).val = (splitJacobian d L c z).det⁻¹ •
(splitJacobian d L c z * ((A p).val + z • B) * (splitJacobian d L c z)ᵀ))
{G : Set (ℝ × Symmetric3)} (hG : IsOpen G)
(hCG : ∀ z ∈ Icc (-θ) (1-θ), (z, C z) ∈ G) :
∃ W : Set Coord3, IsOpen W ∧ p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧
∀ O : Set Coord3, IsOpen O → O ⊆ W → ∀ ε : ℝ, 0 < ε →
∃ (R : CompactSelectedReplacement J μ O u A) (q : Coord3 → ℝ),
ContDiff ℝ (↑(⊤ : ℕ∞)) q ∧
(∀ x, q x ∈ Icc (-θ) (1-θ)) ∧
(∀ x ∈ O, (q x, R.tensor x) ∈ G) ∧
μ {x | x ∈ O ∧ q x ≠ -θ ∧ q x ≠ 1-θ} ≤ ENNReal.ofReal ε := by
obtain ⟨X, W, hW, hpW, hWU, hWc, hWX, hX, hXi, hop⟩ :=
exists_local_split_on_subpatch_retained μ u hu A hAp hD d L hLd B hB hBn
hθ hθ1 hm hpath C hC hdet hconstit hG hCG hU hpU
refine ⟨W, hW, hpW, hWU, hWc, ?_⟩
intro O hO hOW ε hε
have hOU : O ⊆ U := hOW.trans (subset_closure.trans hWU)
have hOX : O ⊆ X.source := hOW.trans hWX
obtain ⟨χ, H, F, hχc, hχO, hχb, hHr, hF, hFs, hFdiv, hn⟩ := hop O hO hOW ε hε
obtain ⟨n, hn⟩ := hn.exists
obtain ⟨Y, hY, hYi, hoffO, _, hform, hpure, hstate⟩ := hn
have hχt : tsupport χ.val ⊆ X.target := hχO.trans (by
rintro y ⟨x, hx, rfl⟩; exact X.map_source (hOX hx))
let K := X.symm '' tsupport χ.val
have hK : IsCompact K := hχc.image_of_continuousOn (X.symm.continuousOn.mono hχt)
have hKO : K ⊆ O := by
rintro x ⟨y, hy, rfl⟩
obtain ⟨t, ht, rfl⟩ := hχO hy
simpa only [X.left_inv (hOX ht)] using ht
have hχp := localPullback_tsupport X χ.val hχc hχt
have hoff : ∀ x ∉ K, Y x = x := by
intro x hx
have hz := image_eq_zero_of_notMem_tsupport (fun h => hx (hχp.2.1 h))
rw [hform x, hz, mul_zero, zero_mul, zero_smul, add_zero]
let z : Coord3 → ℝ := fun x => localPullback X χ.val x *
(smoothDirection 1 H).val ((n+1 : ℝ) * L (X x))
have hz : ContDiff ℝ (↑(⊤ : ℕ∞)) z :=
(localPullback_contDiff X hX.contDiffOn χ.val (smoothScalar_contDiff χ) hχc hχt).mul
((smoothScalar_contDiff (smoothDirection 1 H)).comp
(contDiff_const.mul (L.contDiff.comp hX)))
have hzr : ∀ x, z x ∈ Icc (-θ) (1-θ) := fun x =>
scalar_cutoff_segment hθ.le (by linarith) (localPullback_bounds X χ.val hχb x) (hHr _)
obtain ⟨R, hRt⟩ := synchronized_selected_replacement J μ u hu A hU hUb hOU hK hKO hA hdiv
Y hY hYi hoff z hz C (fun x _ => hCs _ (hzr x)) (F (n+1 : ℝ))
(hF _) (fun j => (hFs _ j).2.2) (hFdiv _) (fun x hx =>
⟨(hstate x hx).1, (hstate x hx).2.1, (hstate x hx).2.2.2⟩)
let q : Coord3 → ℝ := z ∘ Y.symm
have him : MapsTo Y.symm O O := (equiv_mapsTo_of_fixed_compl Y O hoffO).2
refine ⟨R, q, hz.comp hYi, fun x => hzr _, ?_, hpure⟩
intro x hx
have ht := hRt (Y.symm x) (him hx)
rw [Y.apply_symm_apply] at ht
rw [ht]
exact (hstate (Y.symm x) (him hx)).2.2.1
end ScalarConductivity
end
end OAI
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