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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 99 100 101 102 103 104 | import Mathlib
import OAI.Analysis.Conductivity.Variational.UniformC1Bounds
import OAI.Analysis.Conductivity.Sources.LocalPhysicalCorrection
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set MeasureTheory Filter Topology Matrix
open scoped Matrix.Norms.Elementwise
def inverseSourceWeight (X : OpenPartialHomeomorph Coord3 Coord3) (x : Coord3) : ℝ :=
|(fderiv ℝ X.symm (X x)).det|⁻¹
lemma inverseSourceWeight_smooth
(X : OpenPartialHomeomorph Coord3 Coord3)
(hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source)
(hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target) :
ContDiffOn ℝ (↑(⊤:ℕ∞)) (inverseSourceWeight X) X.source := by
apply X.open_source.contDiffOn_iff.mpr
intro x hx
have hxs := hX.contDiffAt (X.open_source.mem_nhds hx)
have hxis := hXi.contDiffAt (X.open_target.mem_nhds (X.map_source hx))
have hd : ContDiffAt ℝ (↑(⊤:ℕ∞)) (fun y => fderiv ℝ X.symm (X y)) x :=
(hxis.fderiv_right (by trivial)).comp x hxs
have hd0 := local_fderiv_det_ne_zero X.symm (hXi.differentiableOn (by simp))
(hX.differentiableOn (by simp)) (X.map_source hx)
exact ((((contDiff_clm_det (E := Coord3) (↑(⊤:ℕ∞))).contDiffAt.comp x hd).abs hd0).inv
(abs_ne_zero.mpr hd0))
lemma compact_smooth_C1_bound {f : Coord3 → ℝ} {U K : Set Coord3}
(hU : IsOpen U) (hf : ContDiffOn ℝ (↑(⊤:ℕ∞)) f U)
(hK : IsCompact K) (hKU : K⊆U) :
∃ M : ℝ,0<M ∧ ∀ x∈K,|f x|≤M ∧ ‖fderiv ℝ f x‖≤M := by
have hd : ContinuousOn (fderiv ℝ f) U := by
intro x hx
exact ((hf.contDiffAt (hU.mem_nhds hx)).fderiv_right (m := 0) (by simp)).continuousAt.continuousWithinAt
obtain ⟨A,hA⟩ := (hK.image_of_continuousOn (hf.continuousOn.mono hKU)).isBounded.exists_norm_le
obtain ⟨B,hB⟩ := (hK.image_of_continuousOn (hd.mono hKU)).isBounded.exists_norm_le
refine ⟨|A|+|B|+1,by positivity,fun x hx => ⟨?_,?_⟩⟩
· exact (hA _ (mem_image_of_mem _ hx)).trans (by linarith [le_abs_self A,abs_nonneg B])
· exact (hB _ (mem_image_of_mem _ hx)).trans (by linarith [le_abs_self B,abs_nonneg A])
theorem localPiolaSource_inverse_C1
(X : OpenPartialHomeomorph Coord3 Coord3)
(hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source)
(hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target)
{K : Set Coord3} (hK : IsCompact K) (hKs : K⊆X.source) :
∃ L : ℝ,0<L ∧ ∀ (r : Coord3 → ℝ),ContDiff ℝ (↑(⊤:ℕ∞)) r →
HasCompactSupport r → tsupport r⊆X '' K → ∀ M : ℝ,0≤M →
UniformC1Bound r M → UniformC1Bound (localPiolaSource X.symm r) (L*M) := by
let w := inverseSourceWeight X
have hw := inverseSourceWeight_smooth X hX hXi
obtain ⟨B,hB,hwb⟩ := compact_smooth_C1_bound X.open_source hw hK hKs
have hdX : ContinuousOn (fderiv ℝ X) X.source := by
intro x hx
exact ((hX.contDiffAt (X.open_source.mem_nhds hx)).fderiv_right (m := 0) (by simp)).continuousAt.continuousWithinAt
obtain ⟨D,hD⟩ := (hK.image_of_continuousOn (hdX.mono hKs)).isBounded.exists_norm_le
have hDX (x) (hx : x∈K) : ‖fderiv ℝ X x‖≤|D|+1 :=
(hD _ (mem_image_of_mem _ hx)).trans (by linarith [le_abs_self D])
refine ⟨B*(|D|+2),by positivity,fun r hr hc hs M hM hb => ?_⟩
let p := localPiolaSource X.symm r
have hps : tsupport p⊆K := localPiolaSource_inverse_support X r hc hKs hs
intro x
change |p x| ≤ B*(|D|+2)*M ∧ ‖fderiv ℝ p x‖ ≤ B*(|D|+2)*M
by_cases hx : x∈K
· have hxs := hX.contDiffAt (X.open_source.mem_nhds (hKs hx))
have hws := hw.contDiffAt (X.open_source.mem_nhds (hKs hx))
have he : p=ᶠ[𝓝 x] (fun y => w y*r (X y)) := by
filter_upwards [X.open_source.mem_nhds (hKs hx)] with y hy
simp [p,localPiolaSource,hy,w,inverseSourceWeight]
have hrc : DifferentiableAt ℝ (fun y => r (X y)) x :=
(hr.differentiable (by simp) (X x)).comp x (hxs.differentiableAt (by simp))
refine ⟨?_,?_⟩
· rw [he.eq_of_nhds,abs_mul]
exact (mul_le_mul (hwb x hx).1 (hb (X x)).1 (abs_nonneg _) hB.le).trans
(by nlinarith [mul_nonneg hB.le hM,abs_nonneg D])
· rw [he.fderiv_eq,fderiv_fun_mul (hws.differentiableAt (by simp)) hrc]
have hchain := ((hr.differentiable (by simp) (X x)).hasFDerivAt.comp x
(hxs.differentiableAt (by simp)).hasFDerivAt).fderiv
have hcomp : ‖fderiv ℝ (fun y => r (X y)) x‖≤M*(|D|+1) := by
change ‖fderiv ℝ (r ∘ X) x‖ ≤ _
rw [hchain]
exact (ContinuousLinearMap.opNorm_comp_le _ _).trans
(mul_le_mul (hb (X x)).2 (hDX x hx) (norm_nonneg _) hM)
calc
_ ≤ ‖w x • fderiv ℝ (fun y => r (X y)) x‖+‖r (X x) • fderiv ℝ w x‖ := norm_add_le _ _
_ = |w x| *‖fderiv ℝ (fun y => r (X y)) x‖+|r (X x)| *‖fderiv ℝ w x‖ := by
simp only [norm_smul,Real.norm_eq_abs]
_ ≤ B*(M*(|D|+1))+M*B := add_le_add
(mul_le_mul (hwb x hx).1 hcomp (norm_nonneg _) hB.le)
(mul_le_mul (hb (X x)).1 (hwb x hx).2 (norm_nonneg _) hM)
_ = _ := by ring
· have hn : x∉tsupport p := fun h => hx (hps h)
rw [image_eq_zero_of_notMem_tsupport hn,fderiv_of_notMem_tsupport ℝ hn,abs_zero,norm_zero]
exact ⟨by positivity,by positivity⟩
end ScalarConductivity
end
end OAI
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