| import Mathlib |
| import OAI.Analysis.Conductivity.Sobolev.WallPrimitive |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
|
|
| section |
| open Set MeasureTheory Filter Topology |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] |
|
|
| def fiberPrimitive (a : ℝ) (f : E × ℝ → ℝ) (p : E × ℝ) : ℝ := |
| wallPrimitive f p-wallPrimitive f (p.1,a) |
|
|
| lemma fiberPrimitive_smooth (a : ℝ) {f : E × ℝ → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) : |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fiberPrimitive a f) := |
| (wallPrimitive_smooth hf).sub ((wallPrimitive_smooth hf).comp (contDiff_fst.prodMk contDiff_const)) |
|
|
| omit [NormedSpace ℝ E] [ProperSpace E] in |
| lemma fiberPrimitive_eq_integral (a : ℝ) {f : E × ℝ → ℝ} (hf : Continuous f) (p : E × ℝ) : |
| fiberPrimitive a f p=∫ t in a..p.2, f (p.1,t) := by |
| have hc : Continuous (fun t : ℝ => f (p.1,t)) := hf.comp (continuous_const.prodMk continuous_id) |
| change wallPrimitive f (p.1,p.2)-wallPrimitive f (p.1,a)=_ |
| simp only [wallPrimitive_eq_intervalIntegral] |
| exact intervalIntegral.integral_interval_sub_left (μ := volume) (hc.intervalIntegrable _ _) (hc.intervalIntegrable _ _) |
|
|
| lemma fiberPrimitive_derivative (a : ℝ) {f : E × ℝ → ℝ} |
| (hf : ContDiff ℝ (↑(⊤ : ℕ∞)) f) (p : E × ℝ) : |
| wallDerivative (fiberPrimitive a f) p=f p := by |
| have h := ((fiberPrimitive_smooth a hf).differentiable (by simp) p).hasFDerivAt.comp_hasDerivAt p.2 |
| ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2)) |
| have hd : HasDerivAt (fun t : ℝ => fiberPrimitive a f (p.1,t)) (f p) p.2 := by |
| exact (wallPrimitive_hasDerivAt hf.continuous p.1 p.2).sub_const (wallPrimitive f (p.1,a)) |
| exact h.unique hd |
|
|
| omit [NormedSpace ℝ E] [ProperSpace E] |
|
|
| lemma fiberPrimitive_of_zero_fiber (a : ℝ) {f : E × ℝ → ℝ} (hf : Continuous f) |
| (x : E) (hx : ∀ t, f (x,t)=0) (z : ℝ) : fiberPrimitive a f (x,z)=0 := by |
| rw [fiberPrimitive_eq_integral a hf] |
| simp only [hx,intervalIntegral.integral_zero] |
|
|
| lemma fiberPrimitive_below {a b : ℝ} {f : E × ℝ → ℝ} (hf : Continuous f) |
| (hv : ∀ p, f p≠0 → p.2∈Ioo a b) (p : E × ℝ) (hp : p.2≤a) : |
| fiberPrimitive a f p=0 := by |
| rw [fiberPrimitive_eq_integral a hf] |
| calc |
| _ = ∫ t in a..p.2, (0:ℝ) := by |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_ge hp] at ht |
| by_contra hn |
| exact (not_lt_of_ge ht.2) (hv (p.1,t) hn).1 |
| _ = 0 := by simp |
|
|
| lemma fiberPrimitive_above {a b : ℝ} {f : E × ℝ → ℝ} (hf : Continuous f) |
| (hv : ∀ p, f p≠0 → p.2∈Ioo a b) |
| (hz : ∀ x, (∫ t in a..b, f (x,t))=0) (p : E × ℝ) (hp : b≤p.2) : |
| fiberPrimitive a f p=0 := by |
| have hc : Continuous (fun t : ℝ => f (p.1,t)) := hf.comp (continuous_const.prodMk continuous_id) |
| rw [fiberPrimitive_eq_integral a hf,←intervalIntegral.integral_add_adjacent_intervals |
| (hc.intervalIntegrable a b) (hc.intervalIntegrable b p.2),hz,zero_add] |
| calc |
| _ = ∫ t in b..p.2, (0:ℝ) := by |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_le hp] at ht |
| by_contra hn |
| exact (not_lt_of_ge ht.1) (hv (p.1,t) hn).2 |
| _ = 0 := by simp |
|
|
| theorem fiberPrimitive_compact {a b : ℝ} {f : E × ℝ → ℝ} (hf : Continuous f) |
| (hs : HasCompactSupport f) (hv : ∀ p, f p≠0 → p.2∈Ioo a b) |
| (hz : ∀ x, (∫ t in a..b, f (x,t))=0) : |
| HasCompactSupport (fiberPrimitive a f) ∧ |
| tsupport (fiberPrimitive a f) ⊆ (Prod.fst '' tsupport f) ×ˢ Icc a b := by |
| have hK : IsCompact ((Prod.fst '' tsupport f) ×ˢ Icc a b) := |
| (hs.image continuous_fst).prod isCompact_Icc |
| have hsub : Function.support (fiberPrimitive a f) ⊆ (Prod.fst '' tsupport f) ×ˢ Icc a b := by |
| intro p hp |
| refine ⟨?_,?_,?_⟩ |
| · by_contra hx |
| have hz' : ∀ t, f (p.1,t)=0 := by |
| intro t |
| by_contra hn |
| exact hx ⟨(p.1,t),subset_tsupport f hn,rfl⟩ |
| exact hp (fiberPrimitive_of_zero_fiber a hf p.1 hz' p.2) |
| · by_contra h |
| exact hp (fiberPrimitive_below hf hv p (le_of_lt (lt_of_not_ge h))) |
| · by_contra h |
| exact hp (fiberPrimitive_above hf hv hz p (le_of_lt (lt_of_not_ge h))) |
| refine ⟨HasCompactSupport.of_support_subset_isCompact hK hsub,?_⟩ |
| exact closure_minimal hsub hK.isClosed |
|
|
| end |
|
|
| open Set MeasureTheory Matrix |
|
|
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def compactMomentGram (χ : ℝ → ℝ) (w : ℝ → ι → ℝ) : Matrix ι ι ℝ := |
| fun i j => ∫ s, χ s*w s i*w s j |
|
|
| omit [Fintype ι] [DecidableEq ι] in |
| lemma compactMoment_integrable {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} |
| (hc : Continuous χ) (hs : HasCompactSupport χ) (hw : ∀ i, Continuous (fun s => w s i)) |
| (i j : ι) : Integrable (fun s => χ s*w s i*w s j) := |
| ((hc.mul (hw i)).mul (hw j)).integrable_of_hasCompactSupport ((hs.mul_right).mul_right) |
|
|
| omit [DecidableEq ι] in |
| lemma compactMoment_dot_integrable {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} |
| (hc : Continuous χ) (hs : HasCompactSupport χ) (hw : ∀ i, Continuous (fun s => w s i)) |
| (a : ι → ℝ) : Integrable (fun s => χ s*(a ⬝ᵥ w s)^2) := by |
| apply Continuous.integrable_of_hasCompactSupport |
| · exact hc.mul ((continuous_finsetSum _ (fun i _ => continuous_const.mul (hw i))).pow 2) |
| · exact hs.mul_right |
|
|
| omit [DecidableEq ι] in |
| lemma compactMoment_quadratic {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} |
| (hc : Continuous χ) (hs : HasCompactSupport χ) (hw : ∀ i, Continuous (fun s => w s i)) |
| (a : ι → ℝ) : a ⬝ᵥ ((compactMomentGram χ w)*ᵥa)=∫ s, χ s*(a ⬝ᵥ w s)^2 := by |
| have hi (i j : ι) : Integrable (fun s => a i*(χ s*w s i*w s j)*a j) := |
| ((compactMoment_integrable hc hs hw i j).const_mul _).mul_const _ |
| calc |
| _ = ∑ i, ∑ j, ∫ s, a i*(χ s*w s i*w s j)*a j := by |
| simp only [compactMomentGram,mulVec,dotProduct,Finset.mul_sum, |
| integral_mul_const,integral_const_mul] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| _ = ∫ s, ∑ i, ∑ j, a i*(χ s*w s i*w s j)*a j := by |
| rw [integral_finsetSum _ (fun i _ => integrable_finsetSum _ (fun j _ => hi i j))] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (integral_finsetSum _ (fun j _ => hi i j)).symm |
| _ = _ := by |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall fun s => by |
| simp only [dotProduct,pow_two,Finset.sum_mul,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
|
| omit [DecidableEq ι] in |
| theorem compactMomentGram_posDef {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} |
| (hc : Continuous χ) (hs : HasCompactSupport χ) (hw : ∀ i, Continuous (fun s => w s i)) |
| (hn : ∀ s, 0≤χ s) {l r : ℝ} (hp : ∀ s∈Ioo l r, 0<χ s) |
| (hind : ∀ a : ι → ℝ, (∀ s∈Ioo l r, a ⬝ᵥ w s=0) → a=0) : |
| (compactMomentGram χ w).PosDef := by |
| apply Matrix.posDef_iff_dotProduct_mulVec.mpr |
| refine ⟨?_,?_⟩ |
| · ext i j |
| simp only [conjTranspose_apply,star_trivial,compactMomentGram] |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall (fun s => by ring) |
| · intro a ha |
| simp only [star_trivial] |
| rw [compactMoment_quadratic hc hs hw] |
| have hnon : ∀ s, 0≤χ s*(a ⬝ᵥ w s)^2 := fun s => mul_nonneg (hn s) (sq_nonneg _) |
| apply (integral_nonneg hnon).lt_of_ne' |
| intro hz |
| have hae := (integral_eq_zero_iff_of_nonneg hnon |
| (compactMoment_dot_integrable hc hs hw a)).mp hz |
| have he := MeasureTheory.Measure.eq_of_ae_eq hae |
| (hc.mul ((continuous_finsetSum _ (fun i _ => continuous_const.mul (hw i))).pow 2)) |
| continuous_zero |
| apply ha |
| apply hind |
| intro s hs |
| have h := congrFun he s |
| have hsq : (a ⬝ᵥ w s)^2=0 := (mul_eq_zero.mp h).resolve_left (hp s hs).ne' |
| nlinarith [sq_nonneg (a ⬝ᵥ w s)] |
|
|
| theorem compactMoment_surjective {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} |
| (hc : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (hs : HasCompactSupport χ) |
| (hw : ∀ i, ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => w s i)) |
| (hG : (compactMomentGram χ w).PosDef) (b : ι → ℝ) : |
| ∃ f : ℝ → ℝ, ContDiff ℝ (↑(⊤ : ℕ∞)) f ∧ HasCompactSupport f ∧ |
| tsupport f ⊆ tsupport χ ∧ (∀ i, (∫ s, f s*w s i)=b i) := by |
| obtain ⟨a,ha⟩ := Matrix.mulVec_surjective_iff_isUnit.mpr hG.isUnit b |
| refine ⟨fun s => χ s*(a ⬝ᵥ w s),hc.mul ?_,hs.mul_right,tsupport_mul_subset_left,?_⟩ |
| · exact ContDiff.sum (fun i _ => contDiff_const.mul (hw i)) |
| · intro i |
| rw [←congrFun ha i] |
| change (∫ s, χ s*(a ⬝ᵥ w s)*w s i)=∑ j, compactMomentGram χ w i j*a j |
| calc |
| _ = ∫ s, ∑ j, (χ s*w s i*w s j)*a j := by |
| apply integral_congr_ae |
| exact Filter.Eventually.of_forall (fun s => by |
| simp only [dotProduct,Finset.mul_sum,Finset.sum_mul] |
| apply Finset.sum_congr rfl; intro j _; ring) |
| _ = _ := by |
| rw [integral_finsetSum _ (fun j _ => |
| (compactMoment_integrable hc.continuous hs (fun j => (hw j).continuous) i j).mul_const _)] |
| simp only [integral_mul_const,compactMomentGram] |
|
|
| end ScalarConductivity |
|
|
| end |
|
|
| end OAI |
|
|
