Prof. Hans Wilhelm Alt: Script Analysis IV -- Notations

Linear differential operators

Proof

Index

Notations

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General notations

Ω⊂Rⁿ

open set,

C^k(Ω;R^N)

{u : Ω → R^N; u is k-times continuously differentiable in Ω},

C^k(Ω)

C^k(Ω;R),

C^k₀(Ω;R^N)

{u∈C^k(Ω;R^N); supp u compact subset of Ω},

C^k₀(Ω)

C₀^k(Ω;R).

We shall use the following definitions for partial derivatives:

∂_i u

partial derivative in i-th direction,

∂_i^l u

∂_i...∂_i

l times

∂^αu

∂₁^α₁...∂_n^α_nu.

Here α is a multiindex, that is

(α₁,...,α_n), α_i∈Z, α_i≧0,

|α|

∑

i=1

α_i order of α.

If Ω⊂Rⁿ is an open bounded set, then for k∈N

C^k(clos(Ω);R^N) := {

f : clos(Ω) → R^N ; f is k-times contiuously differentiable in Ω and

all partial derivatives of order ≦k

have a continuous continuation on clos(Ω) }.

For norms on this function space see the definitions following sect:1-14.

Version 1.5
H.W. Alt - 02.01.2007

Linear differential operators

Proof

Index