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© 2002-2007 Prof. Dr. Hans Wilhelm Alt, University of Bonn, Germany

Linear differential operators
[chap:1]


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In this lecture we shall use the following notations:

Let us start with the definition of linear differential operators.

Definition (Classical linear differential operators)    [sect:1-1]
A classical linear differential operator of order  m≧0  on  Ω  is a mapping
L : Cm(Ω;RN) → C0(Ω;RM)
with the property, that for  u∈Cm(Ω;RN)  and  x∈Ω  the value  L(u)(x)∈RM  is a linear combination of the partial deritives  ∂αu(x)  for  |α|≦m . Here  N  and  M  are integers. Thus  L  has the following representation for all  u∈Cm(Ω;RN)  and all  x∈Ω :
L(u)(x) =
 
∑
|α|≦m
 aα(x)∂αu(x) .
Here  aα(x)∈RM×N  are  M×N -matrices. The international common short notation for this is
L(u) =
 
∑
|α|≦m
 aα∂αu     in Ω.
As a consequence of this definition one obtains:
  • [sect:1-1-(1)] The functions  aα  are continuous.
  • [sect:1-1-(2)] The functions  aα  are uniquely determined.
  • [sect:1-1-(3)] The operator  L  can be written as
    L(u) = (
    N
    ∑
    j=1
    		 Lij(uj) ) i=1,...,M ,
    where  Lij : Cm(Ω) → C0(Ω)  are scalar differential operators with representation
    Lij(v)(x) =
     
    ∑
    |α|≦m
    		 (aα)ij(x)∂αv(x) ,
    aα(x) = ( (aα)ij(x) ) i=1,...,M ; j=1,...,N .
The uniquely determined terms  aα  are called coefficients of  L . The operator  L  is called linear differential operator with constant coefficients (resp.  C∞ -coefficients or analytic coefficienten, etc.), if the coefficients  x |→ aα(x)  are independent of  x  (resp. infinitely often differentiable or real analytic, etc.).
Every differential operator as above gives rise to a differential equation, more precisely, to a system of  M  differential equations for  N  independent functions. The goal is, to solve such differential equations and to study properties of the solutions.

Corresponding differential equation:
  • Given:   f∈C0(Ω;RM) .
  • Find:   u∈Cm(Ω;RN)  with  L(u)=f  in  Ω .
Then  u  is called strong solution of the differential equation
L(u) = f     in Ω.
This differential equation in fact is a system consisting of  M  scalar differential equations for  N  unknown functions.

Questions:
  • For which  f  there exists a solution to the differential equation?
    (For example, if  f=0 , then  u=0  is a solution since  L(0)=0 . Moreover, the set of right-hand sides  f , for which a solution exists, is a subspace of  C0(Ω;RM) . This is a consequence of the linearity of  L .)
  • How many solutions exist for a given right-hand side  f ?
    (The set of solutions either is empty or an affine subspace of  Cm(Ω;RN) . This again is a consequence of the linearity of  L .)

Here a list of the most commonly used linear differential operators:

Examples    [sect:1-2]
  • [sect:1-2-(6)] Heat operator ( m=2 , N=M=1 ). 
  • [sect:1-2-(7)] Wave operator ( m=2 ,  N=M=1 ). 

Next we present some special classes of solutions of the corresponding homogeneous differential equation . For a given linear differential operator  L  this is the differential equation  L(u)=0  with an unknown function  u . We restrict considerations to operators  L  with constant coefficients.

Special solutions    [sect:1-3]
Let  L  be a linear differential operator with constant coefficients. In the following we present some methods to derive special classes of solutions for the homogeneous differential equation  L(u)=0 .

The following theorem sect:1-4 is related to example sect:1-2-(1). We prove, that the necessary condition derived in sect:1-2-(1) for solving the differential equation  ∇u=f  indeed is a sufficient condition. This is a special case of the general POINCARÉ Lemma.

Existence theorem for Gradient (POINCARÉ Lemma)    [sect:1-4]
Let  Ω⊂Rn  be open and  f∈C1(Ω;Rn)  with  ∂jfi - ∂ifj = 0  for  i, j=1,...,n . If every closed curve in  Ω  is contractable to a point, then there exists a function  u∈C2(Ω)  with  ∇u = f .
Addendum: If  Ω  in addition is (path-)connected (then  Ω  is called simply connected), then for given  x0∈Ω  all solutions  u  of the differential equation have the representation
u(x) = u(x0) +
1
∫
0
 f(γ(s)) •γ′(s) ds ,
where  γ : [0,1] → Ω  is any continuous, piecewise differentiable mapping satisfying  γ(0)=x0  and  γ(1)=x .
The differential equation  ∇u=f  also is well defined in the classical sense for  u∈C1(Ω;Rn)  and  f∈C0(Ω;R) . However, then condition sect:1-4 on  f  cannot be formulated in the classical sense. In order to deal with this more general situation let us perform some considerations.

Thus we have motivated a weak version of the necessary condition on  f  for the differential equation  ∇u=f . As we shall see it is appropriate to formulate this in a more general phramework. This is done by introducing the notion of a transposed differential operator.

Here we mention, that integrals involving test functions always are linear mappings on the function space  C0∞(Ω;Rl)  (see also Analysis III in chapter [Analysis III:7] , as well as [Analysis III:EULER-LAGRANGE equation] , [Analysis III:Divergence equation] , [Analysis III:Conservation law] ). In the following we deal with this situation by a systematic approach using the notion of distributions.

Definition (Distribution of a function)    [sect:1-9]
Let  Ω⊂Rn  be an open set.
  • [sect:1-9-(1)] For every  u∈L1loc(Ω)  a linear mapping  [u]  from  C0∞(Ω)  to  R  is defined by
    [u](ζ):=
     
    ∫
    Ω
    		 ζu dLn .
    It is easy to see, that the mapping  u |→ [u]  is linear and injektive. The mapping  [u]  is called the distribution corresponding to  u .
    Notice: In this lecture we shall use systematically the notion  [u]  for the distribution induces by  u . Later, and also common in mathematical literature, we shall drop the brackets defining the distribution. We mention, that in literature about the theory of distributions and also in books about partial differential equations there is no notational distinction between a function and its corresponding distribution.
  • [sect:1-9-(2)] Let  S : C0∞(Ω) → R  be linear,  1≦i≦n , then also
    ( ∂iS ) (ζ):=-S ( ∂iζ )
    is a mapping of this type.
  • [sect:1-9-(3)] Let  S : C0∞(Ω) → R  be linear,  a∈C∞(Ω) , then
    ( a S ) (ζ):=S ( a ζ )
    again is a mapping of this type.
  • [sect:1-9-(4)] The set of all linear mappings from  C0∞(Ω)  to  R  is a vector space.
With this definitions the following holds:
  • [sect:1-9-(5)]  ∂j∂iS=∂i∂jS  for  i, j=1...n .
  • [sect:1-9-(6)] All iterated "derivatives" are described by
    ∂αS:=∂1α1...∂nαnS.
    Here  α  is a multiindex.

The main advantage of the notion of a distribution is the possibility to define so called "weak solutions" of differential equations.

Definition (Distributional solution)    [sect:1-12]
Let  L  be a linear differential operator with  C∞ -coefficients as in sect:1-1. Assume  F=(F1, ..., FM)  with  Fi ∈D'(Ω)  are given and consider the equation
N
∑
j=1
 Lij (Uj) = Fi in D' (Ω)      for i = 1, ..., M .
If  U = (U1, ..., Un)  with  Uj ∈D'(Ω)  satisfies this equation, we call  U  a distributional solution.
Special case: Given  f ∈L1loc(Ω;RM)  a function  u ∈L1loc(Ω;RN)  is called weak solution of  L(u) = f , if
N
∑
j=1
 Lij ( [uj] ) = [fi] in D' (Ω)      for i = 1, ..., M .

We have seen in sect:1-9-(2), that for functions  u ∈L1loc(Ω)  arbitrary derivatives  ∂α[u]  are defined in the space  D'(Ω)  of distributions. In general these distributional derivatives really are distributions. The special case, that such a distributional derivative can be represented by a function, is of particular importance. We define:

Definition (Weak derivative)    [sect:1-13]
Let  α  be a multiindex and  u ∈L1loc(Ω;RN) . Then  f ∈L1loc(Ω;RN)  is called weak derivative with respect to  α , if
∂α[uj] = [fj]      in D' (Ω) for j = 1, ..., N .
By definition sect:1-10 this means that
(-1)|α|
 
∫
Ω
 ∂α ζ·uj dLn =
 
∫
Ω
 ζfj dLn
for all  ζ∈C0∞(Ω)  and  j=1,...,N .

An immediate question arises: What is the connection between this weak derivative and classical strong derivatives?

From Analysis II the following is known: Let  Ω⊂Rn  is an open set and  u : Ω → R . If  ∂i u = 0  in  Ω  for  i=1, ...,n  in classical sense, then  u  is constant in each connected component of  Ω .

More general the following holds: If  ∂α u = 0  in  Ω  for all  α  with  |α| = k , then  u , in each connected component of  Ω  ia a polynomial of degree at most  k-1 .

Question: In which sense are these statements true for weak derivatives?

Conjecture: The same statements hold in the weak sense.

This is the content of the following theorem, which is a generalized version of the "Fundamental Lemma of Calculus of Variations".

Theorem    [sect:1-16]
Let  u ∈L1loc(Ω) ,  Ω⊂Rn  an open connected set, and  k ∈N∪{0} . Assume that  ∂α [u] = 0  for all multiindices  α  with  |α| = k . Then  u  coincides almost everywhere with a polynomial of order less or equal  k-1  (if  k=0 , then  u=0 ).
Special cases:
  • The case  k=0  is the Fundamental Lemma of Calculus of Variations for  L1 -functions (see the proof of sect:1-9).
  • For  k=1  the theorem states: If  ∂i [u]= 0  for  i=1,...,n , then it follows that  u  almost everywhere is a constant function.

Above we have motivated the notion of distributions and weak solutions by the goal, to solve the equation

∇u = f
also for continuous right-hand side  f . Now we are able to perform such an existence proof.

Theorem    [sect:1-18]
Let  Ω⊂Rn  be an open set with the property in sect:1-4. Moreover, let  f∈C0(Ω; Rn)  with
∂j[fi] - ∂i[fj] = 0      in D'(Ω)
for  i,j=1,...,n . Then equation eq:14 defines a solution  u∈C1(Ω; R)  of
∇u = f .

Version 1.5
H.W. Alt - 02.01.2007

Contents (LaTeX) Contents (LaTeX)
Titlepage Titlepage
Fundamental solutions Fundamental solutions
Fundamental solutions Index
© 2002-2007 Prof. Dr. Hans Wilhelm Alt, University of Bonn, Germany