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 solutions
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.
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
Let be open and let
be a linear partial differential equation. If has the two properties
- is continuous and
we call a fundamental solution for that partial differential equation.
For the definition of see exercise 4.5.
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.
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.
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,
Let , , and . Then
By theorem 4.21 2., for all
Let be a solution of the equation
where only finitely many are nonzero, and let . Then solves
By lemma 5.5, we have
Partitions of unity
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.
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:
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:
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
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 kernels
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.
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.
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.
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
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.
Let and let be open. Then for all , the function is continuous.
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.