Partial Differential Equations/Fundamental solutions, Green's functions and Green's kernels

Partial Differential Equations
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In the last two chapters, we have studied test function spaces and distributions. In this chapter we will demonstrate a method to obtain solutions to linear partial differential equations which uses test function spaces and distributions.

Distributional and fundamental solutionsEdit

In the last chapter, we had defined multiplication of a distribution with a smooth function and derivatives of distributions. Therefore, for a distribution  , we are able to calculate such expressions as

 

for a smooth function   and a  -dimensional multiindex  . We therefore observe that in a linear partial differential equation of the form

 

we could insert any distribution   instead of   in the left hand side. However, equality would not hold in this case, because on the right hand side we have a function, but the left hand side would give us a distribution (as finite sums of distributions are distributions again due to exercise 4.1; remember that only finitely many   are allowed to be nonzero, see definition 1.2). If we however replace the right hand side by   (the regular distribution corresponding to  ), then there might be distributions   which satisfy the equation. In this case, we speak of a distributional solution. Let's summarise this definition in a box.

Definition 5.1:

Let   be open, let

 

be a linear partial differential equation, and let  .   is called a distributional solution to the above linear partial differential equation if and only if

 .

Definition 5.2:

Let   be open and let

 

be a linear partial differential equation. If   has the two properties

  1.   is continuous and
  2.  ,

we call   a fundamental solution for that partial differential equation.

For the definition of   see exercise 4.5.

Lemma 5.3:

Let   be open and let   be a set of distributions, where  . Let's further assume that for all  , the function   is continuous and bounded, and let   be compactly supported. Then

 

is a distribution.

Proof:

Let   be the support of  . For  , let us denote the supremum norm of the function   by

 .

For   or  ,   is identically zero and hence a distribution. Hence, we only need to treat the case where both   and  .

For each  ,   is a compact set since it is bounded and closed. Therefore, we may cover   by finitely many pairwise disjoint sets   with diameter at most   (for convenience, we choose these sets to be subsets of  ). Furthermore, we choose  .

For each  , we define

 

, which is a finite linear combination of distributions and therefore a distribution (see exercise 4.1).

Let now   and   be arbitrary. We choose   such that for all  

 .

This we may do because continuous functions are uniformly continuous on compact sets. Further, we choose   such that

 .

This we may do due to dominated convergence. Since for  

 ,

 . Thus, the claim follows from theorem AI.33. 

Theorem 5.4:

Let   be open, let

 

be a linear partial differential equation such that   is integrable and has compact support. Let   be a fundamental solution of the PDE. Then

 

is a distribution which is a distributional solution for the partial differential equation.

Proof: Since by the definition of fundamental solutions the function   is continuous for all  , lemma 5.3 implies that   is a distribution.

Further, by definitions 4.16,

 . 

Lemma 5.5:

Let  ,  ,   and  . Then

 .

Proof:

By theorem 4.21 2., for all  

 . 

Theorem 5.6:

Let   be a solution of the equation

 ,

where only finitely many   are nonzero, and let  . Then   solves

 .

Proof:

By lemma 5.5, we have

 . 

Partitions of unityEdit

In this section you will get to know a very important tool in mathematics, namely partitions of unity. We will use it in this chapter and also later in the book. In order to prove the existence of partitions of unity (we will soon define what this is), we need a few definitions first.

Definitions 5.7:

Let   be a set. We define:

  •  
  •  

  is called the boundary of   and   is called the interior of  . Further, if  , we define

 .

We also need definition 3.13 in the proof, which is why we restate it now:

Definition 3.13:

For  , we define

 .

Theorem and definitions 5.8: Let   be an open set, and let   be open subsets of   such that   (i. e. the sets   form an open cover of  ). Then there exists a sequence of functions   in   such that the following conditions are satisfied:

  1.  
  2.  
  3.  
  4.  

The sequence   is called a partition of unity for   with respect to  .

Proof: We will prove this by explicitly constructing such a sequence of functions.

1. First, we construct a sequence of open balls   with the properties

  •  
  •  
  •  .

In order to do this, we first start with the definition of a sequence compact sets; for each  , we define

 .

This sequence has the properties

  •  
  •  .

We now construct   such that

  •   and
  •  

for some  . We do this in the following way: To meet the first condition, we first cover   with balls by choosing for every   a ball   such that   for an  . Since these balls cover  , and   is compact, we may choose a finite subcover  .

To meet the second condition, we proceed analogously, noting that for all     is compact and   is open.

This sequence of open balls has the properties which we wished for.

2. We choose the respective functions. Since each  ,   is an open ball, it has the form

 

where   and  .

It is easy to prove that the function defined by

 

satisfies   if and only if  . Hence, also  . We define

 

and, for each  ,

 .

Then, since   is never zero, the sequence   is a sequence of   functions and further, it has the properties 1. - 4., as can be easily checked. 

Green's functions and Green's kernelsEdit

Definition 5.9:

Let

 

be a linear partial differential equation. A function   such that for all     is well-defined and

 

is a fundamental solution of that partial differential equation is called a Green's function of that partial differential equation.

Definition 5.10:

Let

 

be a linear partial differential equation. A function   such that the function

 

is a Greens function for that partial differential equation is called a Green's kernel of that partial differential equation.

Theorem 5.11:

Let

 

be a linear partial differential equation (in the following, we will sometimes abbreviate PDE for partial differential equation) such that  , and let   be a Green's kernel for that PDE. If

 

exists and   exists and is continuous, then   solves the partial differential equation.

Proof:

We choose   to be a partition of unity of  , where the open cover of   shall consist only of the set  . Then by definition of partitions of unity

 .

For each  , we define

 

and

 .

By Fubini's theorem, for all   and  

 .

Hence,   as given in theorem 4.11 is a well-defined distribution.

Theorem 5.4 implies that   is a distributional solution to the PDE

 .

Thus, for all   we have, using theorem 4.19,

 .

Since   and   are both continuous, they must be equal due to theorem 3.17. Summing both sides of the equation over   yields the theorem. 

Theorem 5.12:

Let   and let   be open. Then for all  , the function   is continuous.

Proof:

If  , then

 

for sufficiently large  , where the maximum in the last expression converges to   as  , since the support of   is compact and therefore   is uniformly continuous by the Heine–Cantor theorem. 

The last theorem shows that if we have found a locally integrable function   such that

 ,

we have found a Green's kernel   for the respective PDEs. We will rely on this theorem in our procedure to get solutions to the heat equation and Poisson's equation.

ExercisesEdit

SourcesEdit

Partial Differential Equations
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