nevanlinna/Nevanlinna/diffOp.lean

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import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
/-
Let E, F, G be vector spaces over nontrivally normed field 𝕜, a homogeneus
linear differential operator of order n is a map that attaches to every point e
of E a linear evaluation
{Continuous 𝕜-multilinear maps E → F in n variables} → G
In other words, homogeneus linear differential operator of order n is an
instance of the type:
D : E → (ContinuousMultilinearMap 𝕜 (fun _ : Fin n ↦ E) F) →ₗ[𝕜] G
Given any map f : E → F, one obtains a map D f : E → G by sending a point e to
the evaluation (D e), applied to the n.th derivative of f at e
fun e ↦ D e (iteratedFDeriv 𝕜 n f e)
-/
@[ext]
class HomLinDiffOp
(𝕜 : Type*) [NontriviallyNormedField 𝕜]
(n : )
(E : Type*) [NormedAddCommGroup E] [NormedSpace 𝕜 E]
(F : Type*) [NormedAddCommGroup F] [NormedSpace 𝕜 F]
(G : Type*) [NormedAddCommGroup G] [NormedSpace 𝕜 G]
where
tensorfield : E → ( E [×n]→L[𝕜] F) →L[𝕜] G
-- tensorfield : E → (ContinuousMultilinearMap 𝕜 (fun _ : Fin n ↦ E) F) →ₗ[𝕜] G
namespace HomLinDiffOp
noncomputable def toFun
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{n : }
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
(o : HomLinDiffOp 𝕜 n E F G)
: (E → F) → (E → G) :=
fun f z ↦ o.tensorfield z (iteratedFDeriv 𝕜 n f z)
noncomputable def Laplace
{𝕜 : Type*} [RCLike 𝕜]
{n : }
: HomLinDiffOp 𝕜 2 (EuclideanSpace 𝕜 (Fin n)) 𝕜 𝕜
where
tensorfield := by
intro _
let v := stdOrthonormalBasis 𝕜 (EuclideanSpace 𝕜 (Fin n))
rw [finrank_euclideanSpace_fin] at v
exact {
toFun := fun f' ↦ ∑ i, f' ![v i, v i]
map_add' := by
intro f₁ f₂
exact Finset.sum_add_distrib
map_smul' := by
intro m f
exact Eq.symm (Finset.mul_sum Finset.univ (fun i ↦ f ![v i, v i]) m)
cont := by
simp
apply continuous_finset_sum
intro i _
exact ContinuousMultilinearMap.continuous_eval_const ![v i, v i]
}
noncomputable def Gradient
{𝕜 : Type*} [RCLike 𝕜]
{n : }
: HomLinDiffOp 𝕜 1 (EuclideanSpace 𝕜 (Fin n)) 𝕜 (EuclideanSpace 𝕜 (Fin n))
where
tensorfield := by
intro _
let v := stdOrthonormalBasis 𝕜 (EuclideanSpace 𝕜 (Fin n))
rw [finrank_euclideanSpace_fin] at v
exact {
toFun := fun f' ↦ ∑ i, (f' ![v i]) • (v i)
map_add' := by
intro f₁ f₂
simp; simp_rw [add_smul, Finset.sum_add_distrib]
map_smul' := by
intro m f
simp; simp_rw [Finset.smul_sum, ←smul_assoc,smul_eq_mul]
cont := by
simp
apply continuous_finset_sum
intro i _
apply Continuous.smul
exact ContinuousMultilinearMap.continuous_eval_const ![v i]
exact continuous_const
}
end HomLinDiffOp