Partial Differential Equations/Fundamental solutions, Green's functions and Green's kernels
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 iff
Let be open and let
be a linear partial differential equation. If has the two properties
, we call a fundamental solution for that partial differential equation.
For the definition of see exercise 4.5.
Let be a family of distributions, where . Let's further assume that for all , the function is continuous and bounded, and let . Then
is a distribution.
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. Let be a fundamental solution of the equation such that , the function is bounded. 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
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
is locally integrable and sufficiently often differentiable such that is continuous, then solves the partial differential equation.
Since is locally integrable, as given in theorem 4.11 is a well-defined distribution. Furthermore, by Fubini's theorem, for all
Hence, theorem 5.4 implies that is a distributional solution to the PDE.
Thus, for all we have, using theorem 4.19,
From this follows that almost everywhere. But since and are both continuous, they must be equal everywhere.
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.