Prof. Hans Wilhelm Alt: Script Analysis IV -- Transposed differential operator

Motivation

Linear differential operators

Examples

Index

Transposed differential operator

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Definition (Transposed operator) [sect:1-6]

Let L : C^m(Ω; R^N) → C⁰(Ω; R^M) be a classical linear differential operator of order m with C^m -coefficients: (as in sect:1-1):

L(u) =

∑

|α|≦m

a_α∂^αu , a_α∈C^m(Ω; R^M×N).

Then there exists a unique classical linear differential operator L^T : C^m(Ω; R^M) → C⁰(Ω; R^N) of order m with

∫

L^T(v)•u dLⁿ =

∫

v•L(u) dLⁿ

for all u∈C^m(Ω; R^N) and all v∈C^m₀(Ω; R^M) . This differential operator is given by

L^T(v) =

∑

|α|≦m

(-1)^|α| ∂^α(a_α^T v) .

We call L^T the transposed operator of L or the formally adjoint operator of L .

Proof. For u , v as in the assertion

∫

v•L(u) dLⁿ

∑

|α|≦m

∫

v•(a_α∂^αu) dLⁿ

∑

|α|≦m

∫

(a_α^T v) •∂^αu dLⁿ

∑

|α|≦m

(-1)^|α|

∫

∂^α(a_α^T v)•u dLⁿ (partial integration)

∫

L^T(v)•u dLⁿ

with L^T as in the assertion. We have to show, that the differential operator L^T has a representation as in sect:1-1. This follows using the LEIBNIZ rule (Proof see below)

∂^α(vw) =

∑

β : 0≦β≦α

(

)

∂^α-βv ∂^βw

for functions v, w∈C^m(Ω; R) , where

(

)

∏

i=1

(

α_i

β_i

)

Applying this LEIBNIZ rule to the transposed operator we obtain

L^T(v)

∑

|α|≦m

∑

β : 0≦β≦α

(-1)^|α|

(

)

∂^α-β a^T_α ∂^βv

∑

|β|≦m

(

∑

α : |α|≦m, α≧β

(-1)^|α|

(

)

∂^α-βa^T_α ) ∂^βv ,

which is a representation of L^T as in sect:1-1.

Remark: This also shows, that it is sufficient to assume a_α∈C^|α|(Ω' ; R^M×N) (see sect:1-7-(3) below).

It remains to prove the uniqueness of L^T . For this let M be another operator with the same properties as L^T in sect:1-6. Then for all u , v

∫

( L^T(v)-M(v) ) •u = 0 .

Using the Fundamental Lemma of the Caculus of Variations (see [Analysis III:Fundamental lemma] ) we obtain, that L^T(v)-M(v)=0 for all v∈C₀^m(Ω; R^M) . We claim, that this then also holds for all v∈C^m(Ω;R^M) . This would imply L^T=M .

For the proof of this claim let v∈C^m(Ω; R^M) and x∈Ω . Choose a function η∈C₀^m(Ω) so, that η=1 in a small neighbourhood of x . Then L^T(ηv)=M(ηv) . Moreover, ∂^α(ηv)(x)=∂^αv(x) for all |α|≦m . From the representation of the operators L^T and M as linear partial differential operators in sect:1-1 it therefore follows, that L^T(ηv)(x) = L^T(v)(x) and M(ηv)(x) = M(v)(x) . Since this holds for all x∈Ω , we have proved that L^T(v) = M(v) .

Proof of the LEIBNIZ rule. We have to show that

[eq:1-leibniz]

∂^α(vw)=

∑

0≦β≦α

(

)

∂^α-βv ∂^βw .

This can be proved by induction with respect to α : With γ=α+e_i

∂_i∂^α-β v

∂^γ-βv ,

∂_i∂^βw

∂^β+e_i w .

This gives

∂^γ(vw)

∂_i∂^α(vw)

∑

β: 0≦β≦α=γ-e_i

(

)

∂^γ-βv ∂^βw +

∑

e_i≦β≦γ

(

β-e_i

)

∂^γ-βv ∂^βw

∑

β: e_i≦β≦γ-e_i

(

γ-e_i

)

(

γ-e_i

β-e_i

)

) ∂^γ-βv ∂^βw

∑

β: β_i=0

(

)

∂^γ-βv ∂^βw +

∑

β: β_i=γ_i

(

γ-e_i

β-e_i

)

∂^γ-βv ∂^βw

∑

β: e_i≦β≦γ-e_i

(

)

∂^γ-βv ∂^βw

∑

β: β_i=0

(

)

∂^γ-βv ∂^βw +

∑

β: β_i=γ_i

(

)

∂^γ-βv ∂^βw

∑

β: 0≦γ≦β

(

)

∂^γ-βv ∂^βw .

Remark [sect:1-7]

[sect:1-7-(1)] Let L(u)= ( ∑ _j=1^N L_ij(u_j) ) _i=1,... M . Then

(L_ij)^T=(L^T)_ji.

[sect:1-7-(2)] (L^T)^T=L .

[sect:1-7-(3)] For the existence of the transposed operator L^T it suffices to assume that the coefficients satisfy a_α∈C^|α|(Ω; R^M×N) .

Proof sect:1-7-(1). By the representation of L in sect:1-6

L_ij(w) =

∑

α: |α|≦m

(a_α)_ij∂^αw ,

hence

(L_ij)^T(w) =

∑

α: |α|≦m

(-1)^|α| ∂^α ( (a_α)_ji w ) ,

and by the representation of L^T in sect:1-6

(L^T)_ij(w) =

∑

α: |α|≦m

(-1)^|α| ∂^α ( (a^T_α)_ij w ) .

And for the coefficients (a^T_α)_ij=(a_α)_ji .

Proof sect:1-7-(2). For u∈C₀^m(Ω; R^N) and v∈C₀^m(Ω; R^M) by sect:1-6

∫

v •L(u) dLⁿ =

∫

L^T(v)•u dLⁿ =

∫

u•L^T(v) dLⁿ =

∫

( L^T ) ^T(u)•v dLⁿ

Hence by the Fundamental Lemma in Calculus of Variations (see [Analysis III:Fundamental lemma] )

L(u) = ( L^T ) ^T(u) for all u∈C₀^m(Ω; R^N) .

As in the proof of uniqueness in sect:1-6 this then follows also for all u∈C^m(Ω; R^N) , that is, L= ( L^T ) ^T .

Proof sect:1-7-(3). See the proof of sect:1-6.

Version 1.5
H.W. Alt - 02.01.2007

Motivation

Linear differential operators

Examples

Index