| 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 | |