Linear differential operators

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Holomorphic functions

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Fundamental solutions
[chap:2]

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Fundamental solutions play, as we shall see, a central role in the treatment of differential operators with constant coefficients. For a given differential operator these fundamental solutions are functions with a characteristic singularity. Moreover, fundamental solutions are essential for the derivation of integral representations of solutions of the corresponding differential equation.

Let us start with the essential definitions:

DIRAC distribution [sect:2-1]

Let x₀∈Rⁿ . Then

δ_x₀(ζ) := ζ(x₀) für ζ∈C₀^∞(Rⁿ)

defines a distribution δ_x₀∈D'(Rⁿ) , called the DIRAC-Distribution at the point x₀ . This is a distribution as defined in sect:1-11, since

|δ_x₀(ζ)| ≦ || ζ || _C⁰(Ω).

Connection to DIRAC sequences

In Analysis III (see [Analysis III:LAPLACE operator] and [Analysis III:CAUCHY formula] ) we already have seen so called singularity functions for special differential operators. This has been for

∂_conj(z)

the function

z |→

z-z₀

-Δ

for n=3 the function

x |→

|x-x₀|

∂_t-Δ

the function

(t,x) |→

t^n/2

exp ( -

|x|²

) .

In course of this lecture we shall see, that these functions, up to normalization, are exactly the fundamental solutions of the corresponding differential operators. Here is the general definition of a fundamental solution:

Definition (Fundamental solution) [sect:2-2]

Let

L : C^m(Rⁿ;R^N) → C⁰(Rⁿ;R^M)

be a linear differential operator as in sect:1-1 with constant coefficients. Then

F=(F_jk)_{j=1,...,N ; k=1,...,M} with F_jk∈D'(Rⁿ)

is called fundamental solution of L , if

∑

j=1

L_ij(F_jk) =

δ₀

for i=k ,

for i≠k .

Hence fundamental solutions are distributional solutions (see sect:1-12). (This system of distributional equations can also be written as L(F)=δ₀ Id , where Id denotes the unit in the set of M×M -matrices.)

Spezial case of a single equation (N=M=1): Let

L : C^m(R;R) → C⁰(R;R)

be a linear differential operator with constant coefficients. Then a distribution F∈D'(Rⁿ) is called fundamental solution of L , if

L(F) = δ₀ .

Notice: The definition of fundamental solutions for systems usually is not covered in literature.

Spezial case of a function as fundamental solution: Let F=(F_jk)_jk∈L¹_loc(Rⁿ;R^N×M) . If [F]:=([F_jk])_jk is a fundamental solution, then also F is called fundamental solution. In this lecture all fundamental solutions are L¹_loc -functions (except the elementary example in sect:2-8).

As most simple example we begin with the fundamental solutions of two ordinary differential operators.

Ordinary differential operators

Fundamental solutions can be used to solve the inhomogeneous differential equation on the entire space. For example, consider the ODE-case in sect:2-4-(1) and set f(y):=0 for x∉I . Moreover, let F be the fundamental solution from sect:2-3-(1). Then for x∈R

(F∗ f)(x) =

∫

F(x-y)f(y) dy =

∫

-∞

f(y) dy =

∫

f(y) dy.

It follows from theorem sect:2-4 that u:=F∗ f satisfies the differential equation [u]′=[f] .

Now we try to prove an analogue for general differential operators.

Motivation

Theorem [sect:2-5]

Let F=(F_jk)_jk∈L¹_loc(Rⁿ;R^N×M) be a fundamental solution of L as in sect:2-2, and f∈L¹(Rⁿ;R^M) with compact support in Rⁿ . Then

u := F∗ f = (

∑

F_jk ∗ f_k ) _{j=1,..., N} ∈L¹_loc(Rⁿ;R^N)

and solves

∑

L_ij[u_j] = [f_i] for i=1,..., M .

Reminder: Here [u_j] and [f_i] denote the distributions corresponding to the functions u_j and f_i in sect:1-9.

Proof

In this lecture we shall get to know the most important fundamental solutions. Here a list:

L(u)=u′ (see sect:2-3-(1))
L(u)=u′′ (see sect:2-3-(2))
L(u)=u′-Au (see sect:2-6)
L(u)= div u (see sect:2-7)
L(u)=∂_iu (see sect:2-8)
L(u)=∂_conj(z)u (see sect:3-1)
L(u)=-Δu (see sect:5-1)
L(u)=∂_tu-Δu (see sect:6-1)
L(u)=∂_t²u-Δu (see sect:7-1)

The next example is a general system of linear first order ordinary differential equations with constant coefficients ( n=1 , m=1 , M=N in sect:2-2).

Theorem (ODE system) [sect:2-6]

Let L(u):=u′-Au for u∈C¹(R;R^N) (that is n=1 , m=1 , M=N in sect:2-2), where A∈R^N×N is a N×N -matrix. Then

F(t) :=

e^tA

for t>0 ,

for t<0 ,

defines a fundamental solution F∈L¹_loc(R;R^N×N) of L . Every other L¹_loc -fundamental solution has the form

t |→ F(t)+e^tAC

with a matrix C∈R^N×N .

Proof

The next example deals with the divergence operator.

Theorem (Divergence operator) [sect:2-7]

Let L(u):= div (u) for u∈C¹(Rⁿ; Rⁿ) (hence M=1 , N=n , M=1 in sect:2-2). Then

F(x):=

ϭ_n

|x|ⁿ

, wobei ϭ_n:=H^n-1(∂B₁(0)) ,

defines a fundamental solution F∈L¹_loc(Rⁿ; Rⁿ) of L .

Important notice: This is in fact only one fundamental solution of the divergence operator. There exist many other fundamental solutions with a different behaviour at the origin. This means, that a uniqueness statement as for example in theorem sect:2-6 is inpossible.

Proof

So far, all fundamental solutions have been smooth functions away from the origin. However, for a general differential operator with constant coefficients fundamental solutions can be of different type. We close this section with two elementary examples: One example, for which fundamental solutions are measures, and another example, for which no fundamental solution exists.

Further examples

Another important property of differential operators with constant coefficients is, that distributional solutions locally can be approximated by C^∞ -solutions. The proof of this fact uses approximation by convolution. In following chapters we shall use this result for the derivation of integral representations of weak solutions.

Approximation theorem

Version 1.5
H.W. Alt - 02.01.2007

Linear differential operators

Titlepage

Holomorphic functions

Index

Fundamental solutions [chap:2]

Fundamental solutions
[chap:2]