| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.CompactUpdateLp |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open MeasureTheory Set Filter Topology |
|
|
| def CompactGlobalUpdate.fieldPair |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| {U : Set Coord3} {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} |
| (R : CompactGlobalUpdate μ U u A) |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) : |
| Lp FieldVector 2 (μ.restrict U) × Lp FieldVector 2 (μ.restrict U) := |
| (hE.toLp _ + (R.memLp_gradient_correction μ).toLp _, |
| hF.toLp _ + (R.memLp_flux_correction μ).toLp _) |
|
|
| lemma CompactGlobalUpdate.fieldPair_eq |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| {U : Set Coord3} (hUm : MeasurableSet U) |
| {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} |
| (R : CompactGlobalUpdate μ U u A) (hreg : μ (U \ regularRegion u A U) = 0) |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) |
| (hE' : MemLp (voltageGradient (fun x => u x+R.du x)) 2 (μ.restrict U)) |
| (hF' : MemLp (voltageFlux (fun x => u x+R.du x) R.tensor) 2 (μ.restrict U)) : |
| R.fieldPair μ hE hF = (hE'.toLp _,hF'.toLp _) := by |
| exact Prod.ext (R.gradient_toLp_eq μ hUm hreg hE hE').symm |
| (R.flux_toLp_eq μ hF hF').symm |
|
|
| lemma CompactGlobalUpdate.fieldPair_ae |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| {U : Set Coord3} (hUm : MeasurableSet U) |
| {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} |
| (R : CompactGlobalUpdate μ U u A) (hreg : μ (U \ regularRegion u A U) = 0) |
| (hE : MemLp (voltageGradient u) 2 (μ.restrict U)) |
| (hF : MemLp (voltageFlux u A) 2 (μ.restrict U)) : |
| ∀ᵐ x ∂μ.restrict U, |
| (R.fieldPair μ hE hF).1 x = voltageGradient (fun y => u y+R.du y) x ∧ |
| (R.fieldPair μ hE hF).2 x = voltageFlux (fun y => u y+R.du y) R.tensor x := by |
| obtain ⟨hE',hF'⟩ := R.fields_memLp μ hUm hreg hE hF |
| rw [R.fieldPair_eq μ hUm hreg hE hF hE' hF'] |
| exact hE'.coeFn_toLp.and hF'.coeFn_toLp |
|
|
| lemma CompactGlobalUpdate.fieldPair_energy_bound |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] |
| {U : Set Coord3} (hUm : MeasurableSet U) |
| {u : Coord3 → Fin 2 → ℝ} {A : Coord3 → Symmetric3} |
| (R : CompactGlobalUpdate μ U u A) (hreg : μ (U \ regularRegion u A U) = 0) |
| (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) |
| {a b : ℝ} (ha : 0 < a) (hab : a < b) |
| (hgraph : ∀ᵐ x ∂μ, x ∈ U → R.tensor x ∈ matrixFiniteLaminate a b) : |
| ‖(R.fieldPair μ hE hF).1‖ ≤ (b/a)*‖hE.toLp _‖ ∧ |
| ‖(R.fieldPair μ hE hF).2‖ ≤ (b^2/a)*‖hE.toLp _‖ := by |
| obtain ⟨hE',hF'⟩ := R.fields_memLp μ hUm hreg hE hF |
| rw [R.fieldPair_eq μ hUm hreg hE hF hE' hF'] |
| exact R.Lp_energy_bound μ hUm hreg hE hE' hF' hint hdiv ha hab hgraph |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Matrix Set MeasureTheory |
| open scoped Matrix.Norms.Elementwise |
|
|
| lemma diagonal_conjugate_smul_sub {Q : Mat3} (hQ : Q ∈ orthogonalFrames) |
| (d : DiagonalTriple) (s : ℝ) : |
| Q * diagonal d * Qᵀ - s • (1 : Mat3) = Q * diagonal (d - fun _ => s) * Qᵀ := by |
| change Q * diagonal d * Qᵀ - s • (1 : Mat3) = Q * diagonal (fun i => d i - s) * Qᵀ |
| rw [← diagonal_sub, Matrix.mul_sub, Matrix.sub_mul] |
| have he : diagonal (fun _ : Fin 3 => s) = s • (1 : Mat3) := by |
| ext i j |
| change (if i = j then s else 0) = s * (if i = j then 1 else 0) |
| split_ifs <;> simp |
| rw [he, Matrix.mul_smul, Matrix.smul_mul, Matrix.mul_one, hQ.2] |
|
|
| lemma diagonal_conjugate_dot_bound {Q : Mat3} (hQ : Q ∈ orthogonalFrames) |
| {d : DiagonalTriple} {c : ℝ} (hc : 0 ≤ c) (hd : ∀ i, |d i| ≤ c) (v : Coord3) : |
| ((Q * diagonal d * Qᵀ) *ᵥ v) ⬝ᵥ ((Q * diagonal d * Qᵀ) *ᵥ v) ≤ |
| c^2 * (v ⬝ᵥ v) := by |
| rw [← mulVec_mulVec,← mulVec_mulVec,orthogonal_dotProduct hQ] |
| have hv := orthogonal_dotProduct (Q := Qᵀ) ⟨by simpa using hQ.2,by simpa using hQ.1⟩ v v |
| rw [← hv] |
| simp only [dotProduct,mulVec_diagonal,Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro i _ |
| have hs : (d i)^2 ≤ c^2 := by |
| simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hc).mpr (hd i) |
| nlinarith [mul_le_mul_of_nonneg_right hs (sq_nonneg ((Qᵀ *ᵥ v) i))] |
|
|
| lemma diagonal_conjugate_field_bound {Q : Mat3} (hQ : Q ∈ orthogonalFrames) |
| {d : DiagonalTriple} {c : ℝ} (hc : 0 ≤ c) (hd : ∀ i, |d i| ≤ c) |
| (E : Matrix (Fin 3) (Fin 2) ℝ) : |
| ‖fieldVector ((Q * diagonal d * Qᵀ)*E)‖ ≤ c * ‖fieldVector E‖ := by |
| have hs : ‖fieldVector ((Q * diagonal d * Qᵀ)*E)‖^2 ≤ c^2 * ‖fieldVector E‖^2 := by |
| rw [← real_inner_self_eq_norm_sq,← real_inner_self_eq_norm_sq,fieldVector_inner,fieldVector_inner] |
| change (∑ j, ((Q * diagonal d * Qᵀ) *ᵥ E.col j) ⬝ᵥ ((Q * diagonal d * Qᵀ) *ᵥ E.col j)) ≤ |
| c^2 * (∑ j, E.col j ⬝ᵥ E.col j) |
| rw [Finset.mul_sum] |
| exact Finset.sum_le_sum fun j _ => diagonal_conjugate_dot_bound hQ hc hd (E.col j) |
| apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hc (norm_nonneg _))).mp |
| simpa only [mul_pow] using hs |
|
|
| lemma matrixFiniteLaminate_diagonal_bounds {a b : ℝ} (ha : 0 < a) |
| {A : Symmetric3} (hA : A ∈ matrixFiniteLaminate a b) (i : Fin 3) : |
| A.val i i ∈ Icc a b := by |
| have h := matrixFiniteLaminate_quadratic_bounds ha hA (Pi.single i 1) |
| simpa [dotProduct, Matrix.mulVec, Pi.single_apply] using h |
|
|
| lemma scalarNeighbourhood_defect_bound {a b η : ℝ} (hη : 0 ≤ η) |
| {A : Symmetric3} (hA : A ∈ spectralExtension (scalarNeighbourhood a b η)) |
| (E : Matrix (Fin 3) (Fin 2) ℝ) : |
| ‖fieldVector (A.val*E) - (A.val 0 0) • fieldVector E‖ ≤ (2*η) * ‖fieldVector E‖ := by |
| obtain ⟨Q,hQ,d,⟨hd,s,hs⟩,he⟩ := hA |
| have hds : ∀ i, |d i - s| ≤ η := by |
| intro i |
| exact (norm_le_pi_norm (d - fun _ => s) i).trans hs.le |
| have hshift : A.val - s • (1 : Mat3) = Q * diagonal (d - fun _ => s) * Qᵀ := by |
| rw [he] |
| exact diagonal_conjugate_smul_sub hQ d s |
| have hnorm : ∀ F : Matrix (Fin 3) (Fin 2) ℝ, |
| ‖fieldVector (A.val*F) - s • fieldVector F‖ ≤ η * ‖fieldVector F‖ := by |
| intro F |
| have hb := diagonal_conjugate_field_bound (d := d - fun _ => s) hQ hη hds F |
| rw [← hshift, Matrix.sub_mul, Matrix.smul_mul, Matrix.one_mul] at hb |
| exact hb |
| have hssq : |A.val 0 0 - s| ≤ η := by |
| have hb := diagonal_conjugate_dot_bound (d := d - fun _ => s) hQ hη hds (Pi.single 0 1) |
| rw [← hshift] at hb |
| have hl : (A.val 0 0 - s)^2 ≤ |
| ((A.val - s • (1 : Mat3)) *ᵥ Pi.single 0 1) ⬝ᵥ |
| ((A.val - s • (1 : Mat3)) *ᵥ Pi.single 0 1) := by |
| have he0 : ((A.val - s • (1 : Mat3)) *ᵥ Pi.single 0 1) 0 = A.val 0 0 - s := by |
| simp [Matrix.mulVec, dotProduct, Pi.single_apply] |
| rw [← he0] |
| simpa only [dotProduct,pow_two] using |
| Finset.single_le_sum (fun i _ => mul_self_nonneg (((A.val - s • (1 : Mat3)) *ᵥ Pi.single 0 1) i)) |
| (Finset.mem_univ (0 : Fin 3)) |
| have hr : (A.val 0 0 - s)^2 ≤ η^2 := hl.trans (by simpa [dotProduct, Pi.single_apply] using hb) |
| exact (sq_le_sq₀ (abs_nonneg _) hη).mp (by simpa only [sq_abs] using hr) |
| have hsplit : fieldVector (A.val*E) - A.val 0 0 • fieldVector E = |
| (fieldVector (A.val*E) - s • fieldVector E) + (s-A.val 0 0) • fieldVector E := by |
| rw [sub_smul] |
| abel |
| rw [hsplit] |
| calc |
| _ ≤ ‖fieldVector (A.val*E) - s • fieldVector E‖ + ‖(s-A.val 0 0) • fieldVector E‖ := norm_add_le _ _ |
| _ ≤ η * ‖fieldVector E‖ + η * ‖fieldVector E‖ := add_le_add (hnorm E) (by |
| rw [norm_smul,Real.norm_eq_abs,abs_sub_comm] |
| exact mul_le_mul_of_nonneg_right hssq (norm_nonneg _)) |
| _ = _ := by ring |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
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| end OAI |
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