ederivWithin

DeRhamCohomology/DifferentialForm.lean

@@ -4,7 +4,7 @@ import DeRhamCohomology.ContinuousAlternatingMap.FDeriv
 
 noncomputable section
 
-open Filter ContinuousAlternatingMap
+open Filter ContinuousAlternatingMap Set
 open scoped Topology
 
 variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
@@ -43,3 +43,73 @@ theorem ederiv_ederiv_apply (ω : Ω^n⟮E, F⟯) {x} (h : ContDiffAt ℝ 2 ω x
 
 theorem ederiv_ederiv (ω : Ω^n⟮E, F⟯) (h : ContDiff ℝ 2 ω) : ederiv (ederiv ω) = 0 :=
   funext fun _ ↦ ederiv_ederiv_apply ω h.contDiffAt
+
+/- Exterior derivative of a differential form within a set -/
+def ederivWithin (ω : Ω^n⟮E, F⟯) (s : Set E) : Ω^n + 1⟮E, F⟯ :=
+  fun (x : E) ↦ .uncurryFin (fderivWithin ℝ ω s x)
+
+@[simp]
+theorem ederivWithin_univ (ω : Ω^n⟮E, F⟯) :
+    ederivWithin ω univ = ederiv ω := by
+  ext1 x
+  rw[ederivWithin, ederiv, fderivWithin_univ]
+
+theorem Filter.EventuallyEq.ederivWithin_eq {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : ω₁ =ᶠ[𝓝[s] x] ω₂) (hx : ω₁ x = ω₂ x) : ederivWithin ω₁ s x = ederivWithin ω₂ s x := by
+  simp only[ederivWithin, uncurryFin, hs.fderivWithin_eq hx]
+
+theorem Filter.EventuallyEq.ederivWithin_eq_of_mem {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : ω₁ =ᶠ[𝓝[s] x] ω₂) (hx : x ∈ s) : ederivWithin ω₁ s x = ederivWithin ω₂ s x :=
+  hs.ederivWithin_eq (mem_of_mem_nhdsWithin hx hs :)
+
+theorem Filter.EventuallyEq.ederivWithin_eq_of_insert {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : ω₁ =ᶠ[𝓝[insert x s] x] ω₂) : ederivWithin ω₁ s x = ederivWithin ω₂ s x := by
+  apply Filter.EventuallyEq.ederivWithin_eq (nhdsWithin_mono _ (subset_insert x s) hs)
+  exact (mem_of_mem_nhdsWithin (mem_insert x s) hs :)
+
+theorem Filter.EventuallyEq.ederivWithin' {ω₁ ω₂ : Ω^n⟮E, F⟯} {s t : Set E} {x : E}
+    (hs : ω₁ =ᶠ[𝓝[s] x] ω₂) (ht : t ⊆ s) : ederivWithin ω₁ t =ᶠ[𝓝[s] x] ederivWithin ω₂ t :=
+  (eventually_eventually_nhdsWithin.2 hs).mp <|
+    eventually_mem_nhdsWithin.mono fun _y hys hs =>
+      EventuallyEq.ederivWithin_eq (hs.filter_mono <| nhdsWithin_mono _ ht)
+        (hs.self_of_nhdsWithin hys)
+
+protected theorem Filter.EverntuallyEq.ederivWithin {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : ω₁ =ᶠ[𝓝[s] x] ω₂) : ederivWithin ω₁ s =ᶠ[𝓝[s] x] ederivWithin ω₂ s :=
+  hs.ederivWithin' Subset.rfl
+
+theorem Filter.EventuallyEq.ederivWithin_eq_nhds {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (h : ω₁ =ᶠ[𝓝 x] ω₂) : ederivWithin ω₁ s x = ederivWithin ω₂ s x :=
+  (h.filter_mono nhdsWithin_le_nhds).ederivWithin_eq h.self_of_nhds
+
+theorem ederivWithin_congr {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : EqOn ω₁ ω₂ s) (hx : ω₁ x = ω₂ x) : ederivWithin ω₁ s x = ederivWithin ω₂ s x :=
+  (hs.eventuallyEq.filter_mono inf_le_right).ederivWithin_eq hx
+
+theorem ederivWithin_congr' {ω₁ ω₂ : Ω^n⟮E, F⟯} {s : Set E} {x : E}
+    (hs : EqOn ω₁ ω₂ s) (hx : x ∈ s) : ederivWithin ω₁ s x = ederivWithin ω₂ s x :=
+  ederivWithin_congr hs (hs hx)
+
+theorem ederivWithin_apply (ω : Ω^n⟮E, F⟯) {s : Set E} {x : E}
+    (h : DifferentiableWithinAt ℝ ω s x) (hs : UniqueDiffWithinAt ℝ s x) (v : Fin (n + 1) → E) :
+    ederivWithin ω s x v = ∑ i, (-1) ^ i.val • fderivWithin ℝ (ω · (i.removeNth v)) s x (v i) := by
+  simp only [ederivWithin, ContinuousAlternatingMap.uncurryFin_apply,
+    ContinuousAlternatingMap.fderivWithin_apply h hs]
+
+theorem ederivWithin_ederivWithin_apply (ω : Ω^n⟮E, F⟯) {s : Set E} {t : Set (E →L[ℝ] E [⋀^Fin n]→L[ℝ] F)} {x}
+    (hxx : x ∈ closure (interior s)) (hx : x ∈ s) (hst : MapsTo (fderivWithin ℝ ω s) s t)
+    (h : ContDiffWithinAt ℝ 2 ω s x) (hs : UniqueDiffOn ℝ s) :
+    ederivWithin (ederivWithin ω s) s x = 0 := calc
+  ederivWithin (ederivWithin ω s) s x =
+    uncurryFin (fderivWithin ℝ (fun y ↦ uncurryFin (fderivWithin ℝ ω s y)) s x) := rfl
+  _ = uncurryFin (uncurryFinCLM.comp <| fderivWithin ℝ (fderivWithin ℝ ω s) s x) := by
+    congr 1
+    have : DifferentiableWithinAt ℝ (fderivWithin ℝ ω s) s x := (h.fderivWithin_right hs le_rfl hx).differentiableWithinAt le_rfl
+    exact (uncurryFinCLM.hasFDerivWithinAt.comp x this.hasFDerivWithinAt hst).fderivWithin (hs.uniqueDiffWithinAt hx)
+  _ = 0 :=
+    uncurryFin_uncurryFinCLM_comp_of_symmetric <| h.isSymmSndFDerivWithinAt le_rfl hs hxx hx
+
+theorem ederivWithin_ederivWithin (ω : Ω^n⟮E, F⟯) {s : Set E} {t : Set (E →L[ℝ] E [⋀^Fin n]→L[ℝ] F)}
+    (hst : MapsTo (fderivWithin ℝ ω s) s t) (h : ContDiffOn ℝ 2 ω s) (hs : UniqueDiffOn ℝ s) :
+    EqOn (ederivWithin (ederivWithin ω s) s) 0 (s ∩ (closure (interior s))) :=
+  fun _ ⟨ hx, hxx ⟩ => ederivWithin_ederivWithin_apply ω hxx hx hst (h.contDiffWithinAt hx) hs