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 formula_1, we are able to calculate such expressions as
for a smooth function formula_3 and a formula_4-dimensional multiindex formula_5. We therefore observe that in a linear partial differential equation of the form
we could insert any distribution formula_1 instead of formula_8 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 formula_9 are allowed to be nonzero, see definition 1.2). If we however replace the right hand side by formula_10 (the regular distribution corresponding to formula_11), then there might be distributions formula_1 which satisfy the equation. In this case, we speak of a "distributional solution". Let's summarise this definition in a box.
For the definition of formula_13 see exercise 4.5.
Proof:
Let formula_14 be the support of formula_11. For formula_16, let us denote the supremum norm of the function formula_17 by
For formula_19 or formula_20, formula_1 is identically zero and hence a distribution. Hence, we only need to treat the case where both formula_22 and formula_23.
For each formula_24, formula_25 is a compact set since it is bounded and closed. Therefore, we may cover formula_26 by finitely many pairwise disjoint sets formula_27 with diameter at most formula_28 (for convenience, we choose these sets to be subsets of formula_26). Furthermore, we choose formula_30.
For each formula_24, we define
, which is a finite linear combination of distributions and therefore a distribution (see exercise 4.1).
Let now formula_33 and formula_34 be arbitrary. We choose formula_35 such that for all formula_36
This we may do because continuous functions are uniformly continuous on compact sets. Further, we choose formula_38 such that
This we may do due to dominated convergence. Since for formula_40
formula_42. Thus, the claim follows from theorem AI.33.
Proof: Since by the definition of fundamental solutions the function formula_43 is continuous for all formula_16, lemma 5.3 implies that formula_1 is a distribution.
Further, by definitions 4.16,
Lemma 5.5:
Let formula_47, formula_48, formula_5 and formula_50. Then
Proof:
By theorem 4.21 2., for all formula_52
Proof:
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.
We also need definition 3.13 in the proof, which is why we restate it now:
"Proof": We will prove this by explicitly constructing such a sequence of functions.
1. First, we construct a sequence of open balls formula_55 with the properties
In order to do this, we first start with the definition of a sequence compact sets; for each formula_24, we define
This sequence has the properties
We now construct formula_55 such that
for some formula_66. We do this in the following way: To meet the first condition, we first cover formula_67 with balls by choosing for every formula_68 a ball formula_69 such that formula_70 for an formula_71. Since these balls cover formula_67, and formula_67 is compact, we may choose a finite subcover formula_74.
To meet the second condition, we proceed analogously, noting that for all formula_75 formula_76 is compact and formula_77 is open.
This sequence of open balls has the properties which we wished for.
2. We choose the respective functions. Since each formula_78, formula_24 is an open ball, it has the form
where formula_81 and formula_82.
It is easy to prove that the function defined by
satisfies formula_84 if and only if formula_85. Hence, also formula_86. We define
and, for each formula_24,
Then, since formula_90 is never zero, the sequence formula_91 is a sequence of formula_92 functions and further, it has the properties 1. - 4., as can be easily checked.
Green's functions and Green's kernels.
Proof:
We choose formula_91 to be a partition of unity of formula_94, where the open cover of formula_94 shall consist only of the set formula_94. Then by definition of partitions of unity
For each formula_24, we define
and
By Fubini's theorem, for all formula_101 and formula_24
Hence, formula_104 as given in theorem 4.11 is a well-defined distribution.
Theorem 5.4 implies that formula_104 is a distributional solution to the PDE
Thus, for all formula_101 we have, using theorem 4.19,
Since formula_109 and formula_110 are both continuous, they must be equal due to theorem 3.17. Summing both sides of the equation over formula_111 yields the theorem.
Proof:
If formula_115, then
for sufficiently large formula_117, where the maximum in the last expression converges to formula_118 as formula_119, since the support of formula_120 is compact and therefore formula_120 is uniformly continuous by the Heine–Cantor theorem.
The last theorem shows that if we have found a locally integrable function formula_122 such that
we have found a Green's kernel formula_122 for the respective PDEs. We will rely on this theorem in our procedure to get solutions to the heat equation and Poisson's equation.