File size: 9,282 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 | 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
|