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namespace OAI
open scoped BigOperators ENNReal NNReal Topology ContDiff
open MeasureTheory
namespace AlternatingNS
namespace Spatial
noncomputable def d {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(i : Fin 3) (F : Space → E) (x : Space) : E := fderiv ℝ F x (e i)
lemma coord_smooth (i : Fin 3) : ContDiff ℝ ∞ (fun x : Space => x i) :=
(PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i).contDiff
lemma coord_hasFDeriv (i : Fin 3) (x : Space) :
HasFDerivAt (fun x : Space => x i) (PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i) x :=
(PiLp.proj 2 (𝕜 := ℝ) (fun _ : Fin 3 => ℝ) i).hasFDerivAt
lemma e_apply (i j : Fin 3) : e i j = if j = i then 1 else 0 := by
simp [e, EuclideanSpace.single, PiLp.single_apply]
lemma d_smooth {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(i : Fin 3) (F : Space → E) (hF : ContDiff ℝ ∞ F) : ContDiff ℝ ∞ (d i F) := by
exact (hF.fderiv_right (by simp : (∞ : ℕ∞ω) + 1 ≤ (∞ : ℕ∞ω))).clm_apply contDiff_const
lemma d_tsupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(i : Fin 3) (F : Space → E) : tsupport (d i F) ⊆ tsupport F :=
tsupport_fderiv_apply_subset ℝ (e i)
lemma d_comm (i j : Fin 3) (F : Space → ℝ) (hF : ContDiff ℝ ∞ F) :
d i (d j F) = d j (d i F) := by
ext x
have hd := (hF.fderiv_right (by simp : (∞ : ℕ∞ω) + 1 ≤ (∞ : ℕ∞ω))).differentiable (by simp)
have heq (a b : Fin 3) : d a (d b F) x = fderiv ℝ (fderiv ℝ F) x (e a) (e b) := by
unfold d
rw [fderiv_clm_apply (hd x) (differentiableAt_const _)]
simp
rw [heq, heq]
exact hF.contDiffAt.isSymmSndFDerivAt (by norm_num) _ _
noncomputable def curl (i j : Fin 3) (φ : Space → ℝ) (x : Space) : Space :=
d j φ x • e i - d i φ x • e j
lemma curl_smooth (i j : Fin 3) (φ : Space → ℝ) (hφ : ContDiff ℝ ∞ φ) :
ContDiff ℝ ∞ (curl i j φ) :=
((d_smooth j φ hφ).smul contDiff_const).sub ((d_smooth i φ hφ).smul contDiff_const)
lemma curl_tsupport (i j : Fin 3) (φ : Space → ℝ) : tsupport (curl i j φ) ⊆ tsupport φ := by
apply (tsupport_sub _ _).trans
apply Set.union_subset
· exact (tsupport_smul_subset_left _ _).trans (d_tsupport j φ)
· exact (tsupport_smul_subset_left _ _).trans (d_tsupport i φ)
lemma curl_div (i j : Fin 3) (φ : Space → ℝ) (hφ : ContDiff ℝ ∞ φ) (x : Space) :
(∑ k : Fin 3, d k (curl i j φ) x k) = 0 := by
have hd (k : Fin 3) := (d_smooth k φ hφ).differentiable (by simp)
have hc (k : Fin 3) : d k (curl i j φ) x k =
d k (d j φ) x * (if k = i then 1 else 0) -
d k (d i φ) x * (if k = j then 1 else 0) := by
change (fderiv ℝ ((fun y => d j φ y • e i) - (fun y => d i φ y • e j)) x (e k)) k = _
rw [fderiv_sub ((hd j x).smul_const _) ((hd i x).smul_const _),
fderiv_smul_const (hd j x), fderiv_smul_const (hd i x)]
simp [e_apply, d]
simp_rw [hc]
rw [Finset.sum_sub_distrib]
simp only [mul_ite, mul_one, mul_zero, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
rw [d_comm i j φ hφ, sub_self]
def box : Set Space := {x | ∀ i : Fin 3, |x i| ≤ 3}
def plateau : Set Space := {x | ∀ i : Fin 3, |x i| < 2}
noncomputable def cutoff (x : Space) : ℝ :=
letI := threeAtLeastTwo
∏ i : Fin 3, Real.smoothTransition (3 - x i) * Real.smoothTransition (3 + x i)
lemma cutoff_smooth : ContDiff ℝ ∞ cutoff := by
unfold cutoff
apply contDiff_prod
intro i _
have hi := coord_smooth i
fun_prop
lemma isClosed_box : IsClosed box := by
have heq : box = ⋂ i : Fin 3, {x : Space | |x i| ≤ 3} := by ext; simp [box]
rw [heq]
exact isClosed_iInter (fun i => isClosed_le (coord_smooth i).continuous.abs continuous_const)
lemma box_compact : IsCompact box := by
apply (isCompact_closedBall (0 : Space) 9).of_isClosed_subset isClosed_box
intro x hx
rw [Metric.mem_closedBall, dist_zero_right]
rw [EuclideanSpace.norm_eq, Real.sqrt_le_iff]
refine ⟨by norm_num, ?_⟩
calc
∑ i : Fin 3, ‖x i‖ ^ 2 ≤ ∑ _i : Fin 3, (3 : ℝ) ^ 2 := by
apply Finset.sum_le_sum
intro i _
exact pow_le_pow_left₀ (norm_nonneg _) (hx i) 2
_ ≤ 9 ^ 2 := by norm_num
lemma plateau_open : IsOpen plateau := by
have heq : plateau = ⋂ i : Fin 3, {x : Space | |x i| < 2} := by ext; simp [plateau]
rw [heq]
exact isOpen_iInter_of_finite (fun i => isOpen_lt (coord_smooth i).continuous.abs continuous_const)
lemma cutoff_one (x : Space) (hx : x ∈ plateau) : cutoff x = 1 := by
apply Finset.prod_eq_one
intro i _
have hi := abs_lt.1 (hx i)
rw [Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 3 - x i),
Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 3 + x i), one_mul]
lemma cutoff_one_near (x : Space) (hx : x ∈ plateau) :
cutoff =ᶠ[𝓝 x] (fun _ => 1) :=
Filter.eventually_of_mem (plateau_open.mem_nhds hx) (fun y hy => cutoff_one y hy)
lemma cutoff_support : tsupport cutoff ⊆ box := by
apply closure_minimal _ isClosed_box
intro x hx i
by_contra hi
have hn : 3 < |x i| := lt_of_not_ge hi
have hz : Real.smoothTransition (3 - x i) * Real.smoothTransition (3 + x i) = 0 := by
rcases (lt_abs.1 hn) with hn | hn
· rw [Real.smoothTransition.zero_of_nonpos (by linarith : 3 - x i ≤ 0), zero_mul]
· rw [Real.smoothTransition.zero_of_nonpos (by linarith : 3 + x i ≤ 0), mul_zero]
exact hx (Finset.prod_eq_zero (Finset.mem_univ i) hz)
noncomputable def shearPotential (d i j : Fin 3) (L E : ℝ → ℝ) (x : Space) : ℝ :=
x j * L (x d) - x i * E (x d)
noncomputable def localizedShear (d i j : Fin 3) (L E : ℝ → ℝ) : Space → Space :=
curl i j (fun x => cutoff x * shearPotential d i j L E x)
lemma shearPotential_smooth (d i j : Fin 3) (L E : ℝ → ℝ)
(hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) :
ContDiff ℝ ∞ (shearPotential d i j L E) :=
((coord_smooth j).mul (hL.comp (coord_smooth d))).sub
((coord_smooth i).mul (hE.comp (coord_smooth d)))
lemma localizedShear_smooth (d i j : Fin 3) (L E : ℝ → ℝ)
(hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) :
ContDiff ℝ ∞ (localizedShear d i j L E) :=
curl_smooth i j _ (cutoff_smooth.mul (shearPotential_smooth d i j L E hL hE))
lemma localizedShear_support (d i j : Fin 3) (L E : ℝ → ℝ) :
tsupport (localizedShear d i j L E) ⊆ box :=
(curl_tsupport _ _ _).trans (tsupport_mul_subset_left.trans cutoff_support)
lemma localizedShear_div (d i j : Fin 3) (L E : ℝ → ℝ)
(hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E) (x : Space) :
(∑ k : Fin 3, Spatial.d k (localizedShear d i j L E) x k) = 0 :=
curl_div i j _ (cutoff_smooth.mul (shearPotential_smooth d i j L E hL hE)) x
lemma localizedShear_plateau (d i j : Fin 3) (hij : i ≠ j) (hdi : d ≠ i) (hdj : d ≠ j)
(L E : ℝ → ℝ) (hL : ContDiff ℝ ∞ L) (hE : ContDiff ℝ ∞ E)
(x : Space) (hx : x ∈ plateau) :
localizedShear d i j L E x = L (x d) • e i + E (x d) • e j := by
have heq : (fun y => cutoff y * shearPotential d i j L E y) =ᶠ[𝓝 x]
shearPotential d i j L E := by
filter_upwards [cutoff_one_near x hx] with y hy
rw [hy, one_mul]
have hL' := ((hL.differentiable (by simp)) (x d)).hasFDerivAt.comp x (coord_hasFDeriv d x)
have hE' := ((hE.differentiable (by simp)) (x d)).hasFDerivAt.comp x (coord_hasFDeriv d x)
have hφ := ((coord_hasFDeriv j x).mul hL').sub ((coord_hasFDeriv i x).mul hE')
have hφeq := hφ.fderiv
change fderiv ℝ (shearPotential d i j L E) x = _ at hφeq
simp only [localizedShear, curl, Spatial.d, heq.fderiv_eq, hφeq]
simp [ContinuousLinearMap.comp_apply, PiLp.proj_apply, e_apply, hij, hij.symm, hdi,
hdj, sub_eq_add_neg]
end Spatial
end AlternatingNS
end OAI
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