105 lines
3.0 KiB
Plaintext
105 lines
3.0 KiB
Plaintext
|
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 : 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 : 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 : 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 : 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
|