| import Mathlib | |
| import OAI.Analysis.Conductivity.Scalarization.FineScalarSequence | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set Filter Topology | |
| lemma dense_property_in_closure | |
| {X : Type*} [TopologicalSpace X] (S : Set X) (P : X → Prop) | |
| (h : ∀ x ∈ S, ∃ z : ℕ → X, Tendsto z atTop (𝓝 x) ∧ | |
| (∀ n, z n ∈ S) ∧ ∀ᶠ n in atTop, P (z n)) : | |
| Dense {x : closure S | P x.val} := by | |
| rw [Subtype.dense_iff] | |
| apply closure_minimal _ isClosed_closure | |
| intro x hx | |
| obtain ⟨z,hz,hzS,hzP⟩ := h x hx | |
| apply mem_closure_of_tendsto hz | |
| filter_upwards [hzP] with n hn | |
| exact ⟨⟨z n,subset_closure (hzS n)⟩,hn,rfl⟩ | |
| end ScalarConductivity | |
| end | |
| end | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open MeasureTheory Set Filter Topology TopologicalSpace | |
| open scoped ENNReal | |
| def finiteUpdatePairs | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| (U : Set Coord3) (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) | |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) (a b : ℝ) : | |
| Set (Lp FieldVector 2 (μ.restrict U) × Lp FieldVector 2 (μ.restrict U)) := | |
| {Z | ∃ R : CompactGlobalUpdate μ U u A, | |
| μ (U \ regularRegion (fun x => u x+R.du x) R.tensor U) = 0 ∧ | |
| (∀ᵐ x ∂μ, x ∈ U → R.tensor x ∈ matrixFiniteLaminate a b) ∧ | |
| Z = R.fieldPair μ hE hF} | |
| lemma finiteUpdatePairs_nonempty | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| {U : Set Coord3} {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} | |
| (hAm : Measurable A) | |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) | |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) {a b : ℝ} | |
| (hreg : μ (U \ regularRegion u A U) = 0) | |
| (hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b) : | |
| (finiteUpdatePairs μ U u A hE hF a b).Nonempty := by | |
| refine ⟨(CompactGlobalUpdate.refl μ U u A hAm).fieldPair μ hE hF, | |
| CompactGlobalUpdate.refl μ U u A hAm,?_,hgraph,rfl⟩ | |
| simpa only [CompactGlobalUpdate.refl,Pi.zero_apply,add_zero] using hreg | |
| lemma dense_ApproxScalar_in_finiteUpdateClosure | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ] | |
| {a b : ℝ} (ha : 0 < a) (hab : a < b) | |
| {U : Set Coord3} (hUb : Bornology.IsBounded U) (hUm : MeasurableSet U) | |
| [IsFiniteMeasure (μ.restrict U)] [(μ.restrict U).WeaklyRegular] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) | |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) | |
| (hint : SmoothFluxIntegrable μ U (conductivityFlux u A)) | |
| (hdiv : ∀ 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) | |
| (hreg : μ (U \ regularRegion u A U) = 0) {ε : ℝ} (hε : 0 < ε) : | |
| Dense {z : closure (toWeakSpace ℝ _ '' finiteUpdatePairs μ U u A hE hF a b) | | |
| ApproxScalar (μ.restrict U) a b ε ((toWeakSpace ℝ _).symm z.val)} := by | |
| apply dense_property_in_closure | |
| rintro _ ⟨Z,⟨R,hregR,hgraphR,rfl⟩,rfl⟩ | |
| obtain ⟨hER,hFR⟩ := R.fields_memLp μ hUm hreg hE hF | |
| obtain ⟨hintR,hdivR⟩ := R.preserves_harmonic μ hint hdiv | |
| obtain ⟨S,hregS,hgraphS,hw,happrox⟩ := exists_fine_scalar_sequence μ ha hab hUb hUm | |
| (fun x => u x+R.du x) R.tensor R.measurable_tensor hER hFR hintR hdivR hregR hgraphR | |
| have he : ∀ n, (R.trans μ (S n)).fieldPair μ hE hF = (S n).fieldPair μ hER hFR := | |
| fun n => R.fieldPair_trans μ hUm (S n) hreg hregR hE hF hER hFR | |
| refine ⟨fun n => toWeakSpace ℝ _ ((R.trans μ (S n)).fieldPair μ hE hF),?_,?_,?_⟩ | |
| · simpa only [he,R.fieldPair_eq μ hUm hreg hE hF hER hFR] using hw | |
| · intro n | |
| refine ⟨_,⟨R.trans μ (S n),?_,hgraphS n,rfl⟩,rfl⟩ | |
| simpa only [CompactGlobalUpdate.trans,Pi.add_apply,add_assoc] using hregS n | |
| · filter_upwards [happrox ε hε] with n hn | |
| change ApproxScalar (μ.restrict U) a b ε ((R.trans μ (S n)).fieldPair μ hE hF) | |
| rwa [he n] | |
| theorem exact_scalar_in_finiteUpdateClosure | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] [Measure.InnerRegularCompactLTTop μ] | |
| {a b : ℝ} (ha : 0 < a) (hab : a < b) | |
| {U : Set Coord3} (hUb : Bornology.IsBounded U) (hUm : MeasurableSet U) | |
| [IsFiniteMeasure (μ.restrict U)] [(μ.restrict U).WeaklyRegular] | |
| [SeparableSpace (Lp FieldVector 2 (μ.restrict U))] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) (hAm : Measurable A) | |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) | |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) | |
| (hint : SmoothFluxIntegrable μ U (conductivityFlux u A)) | |
| (hdiv : ∀ 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) | |
| (hreg : μ (U \ regularRegion u A U) = 0) | |
| (hgraph : ∀ᵐ x ∂μ, x ∈ U → A x ∈ matrixFiniteLaminate a b) : | |
| ∃ (Z : Lp FieldVector 2 (μ.restrict U) × Lp FieldVector 2 (μ.restrict U)) | |
| (s : Coord3 → ℝ), | |
| toWeakSpace ℝ _ Z ∈ closure (toWeakSpace ℝ _ '' finiteUpdatePairs μ U u A hE hF a b) ∧ | |
| Measurable s ∧ (∀ᵐ x ∂μ.restrict U, s x ∈ Icc a b ∧ Z.2 x = s x • Z.1 x) := by | |
| apply exact_scalar_of_weak_density (μ.restrict U) hab.le | |
| · apply isCompact_weak_closure_of_pair_bounds | |
| rintro Z ⟨R,hregR,hgraphR,rfl⟩ | |
| exact R.fieldPair_energy_bound μ hUm hreg hE hF hint hdiv ha hab hgraphR | |
| · exact ((finiteUpdatePairs_nonempty μ hAm hE hF hreg hgraph).image _).closure | |
| · intro ε hε | |
| exact dense_ApproxScalar_in_finiteUpdateClosure μ ha hab hUb hUm u A hE hF hint hdiv hreg hε | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |