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 $$\mathcal T$$, we are able to calculate such expressions as
 * $$a \cdot \partial_\alpha \mathcal T$$

for a smooth function $$a: \mathbb R^d \to \mathbb R$$ and a $$d$$-dimensional multiindex $$\alpha \in \mathbb N_0^d$$. We therefore observe that in a linear partial differential equation of the form
 * $$\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)$$

we could insert any distribution $$\mathcal T$$ instead of $$u$$ 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 $$a_\alpha$$ are allowed to be nonzero, see definition 1.2). If we however replace the right hand side by $$\mathcal T_f$$ (the regular distribution corresponding to $$f$$), then there might be distributions $$\mathcal T$$ 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 $$\delta_x$$ see exercise 4.5.

Proof:

Let $$C \subset \mathbb R^d$$ be the support of $$f$$. For $$\varphi \in \mathcal D(O)$$, let us denote the supremum norm of the function $$C \to \mathbb R, x \mapsto \mathcal T_x(\varphi)$$ by
 * $$\|\mathcal T_\cdot(\varphi)\|_\infty$$.

For $$\|f\|_{L_1} = 0$$ or $$\|\mathcal T_\cdot(\varphi)\|_\infty = 0$$, $$\mathcal T$$ is identically zero and hence a distribution. Hence, we only need to treat the case where both $$\|f\|_{L_1} \neq 0$$ and $$\|\mathcal T_\cdot(\varphi)\|_\infty \neq 0$$.

For each $$n \in \mathbb N$$, $$\overline{B_n(0)}$$ is a compact set since it is bounded and closed. Therefore, we may cover $$\overline{B_n(0)} \cap S$$ by finitely many pairwise disjoint sets $$Q_{n, 1}, \ldots, Q_{n, k_n}$$ with diameter at most $$1/n$$ (for convenience, we choose these sets to be subsets of $$\overline{B_n(0)} \cap S$$). Furthermore, we choose $$x_{n, 1} \in Q_{n, 1}, \ldots, x_{n, k_n} \in Q_{n, k_n}$$.

For each $$n \in \mathbb N$$, we define
 * $$\mathcal T_n(\varphi) := \sum_{j=1}^{k_n} \int_{Q_{n, j}} f(x) \mathcal T_{x_{n, j}}(\varphi) dx$$

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

Let now $$\vartheta \in \mathcal D(O)$$ and $$\epsilon > 0$$ be arbitrary. We choose $$N_1 \in \mathbb N$$ such that for all $$n \ge N_1$$
 * $$\forall x \in B_{R_n}(0) \cap S : y \in B_{1/n} (x) \Rightarrow |\mathcal T_x(\varphi) - \mathcal T_y(\varphi)| < \frac{\epsilon}{2 \|f\|_{L^1}}$$.

This we may do because continuous functions are uniformly continuous on compact sets. Further, we choose $$N_2 \in \mathbb N$$ such that
 * $$\int_{S \setminus B_n(0)} | f(x) | dx < \frac{\epsilon}{2 \|\mathcal T_\cdot(\varphi)\|_\infty}$$.

This we may do due to dominated convergence. Since for $$n \ge N := \max \{N_1, N_2\}$$
 * $$|\mathcal T_n(\varphi) - \mathcal T(\varphi)| < \sum_{j=1}^{k_n} \int_{Q{n, j}} |f(x)| |\mathcal T_{\lambda_{x_{n, j}}}(\varphi) - \mathcal T_x(\varphi)| d x + \frac{\epsilon \|\mathcal T_\cdot (\varphi)\|_\infty}{2 \|T_\cdot(\varphi)\|_\infty} < \epsilon$$,

$$\forall \varphi \in \mathcal D(O) : \mathcal T_l(\varphi) \to \mathcal T(\varphi)$$. Thus, the claim follows from theorem AI.33.

Proof: Since by the definition of fundamental solutions the function $$x \mapsto F(x)(\varphi)$$ is continuous for all $$\varphi \in \mathcal D(O)$$, lemma 5.3 implies that $$\mathcal T$$ is a distribution.

Further, by definitions 4.16,
 * $$\begin{align}

\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T(\varphi) & = \mathcal T\left( \sum_{\alpha \in \mathbb N_0^d} \partial_\alpha (a_\alpha \varphi) \right) \\ & = \int_{\mathbb R^d} f(x) F(x)\left( \sum_{\alpha \in \mathbb N_0^d} \partial_\alpha (a_\alpha \varphi) \right) dx \\ & = \int_{\mathbb R^d} f(x) \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha F(x)(\varphi) dx \\ & = \int_{\mathbb R^d} f(x) \delta_x(\varphi) dx \\ & = \int_{\mathbb R^d} f(x) \varphi(x) dx \\ & = \mathcal T_f(\varphi) \end{align}$$.

Lemma 5.5:

Let $$\varphi \in \mathcal D(\mathbb R^d)$$, $$f \in \mathcal C^\infty(\mathbb R^d)$$, $$\alpha \in \mathbb N_0^d$$ and $$\mathcal T \in \mathcal D(\mathbb R^d)^*$$. Then
 * $$f \partial_\alpha (\mathcal T * \varphi) = (f \partial_\alpha \mathcal T) * \varphi$$.

Proof:

By theorem 4.21 2., for all $$x \in \mathbb R^d$$
 * $$\begin{align}

f \partial_\alpha (\mathcal T * \varphi)(x) & = f \mathcal T * (\partial_\alpha \varphi)(x) \\ & = f \mathcal T((\partial_\alpha \varphi)(x - \cdot)) \\ & = f \mathcal T \left( (-1)^{|\alpha|} \partial_\alpha (\varphi(x - \cdot)) \right) \\ & = f (\partial_\alpha \mathcal T) (\varphi(x - \cdot)) \\ & = (\partial_\alpha \mathcal T) (f \varphi(x - \cdot)) \\ & = (f \partial_\alpha \mathcal T) (\varphi(x - \cdot)) = (f \partial_\alpha \mathcal T) * \varphi (x) \\ \end{align}$$.

Proof:

By lemma 5.5, we have
 * $$\begin{align}

\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u(x) & = \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha (\mathcal T * \vartheta)(x) \\ & = \sum_{\alpha \in \mathbb N_0^d} a_\alpha (\partial_\alpha \mathcal T) * \vartheta(x) \\ & = \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T(\vartheta(x - \cdot)) \\ & = \delta_0(\vartheta(x - \cdot)) = \vartheta(x) \end{align}$$.

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 $$(B_l)_{l \in \mathbb N}$$ with the properties
 * $$\forall n \in \N : \exists \upsilon \in \Upsilon : \overline{B_n} \subseteq U_\upsilon$$
 * $$\forall x \in O : |\{n \in \mathbb N | x \in \overline{B_n}\}| < \infty$$
 * $$\bigcup_{j \in \N} B_j = O$$.

In order to do this, we first start with the definition of a sequence compact sets; for each $$n \in \mathbb N$$, we define
 * $$K_n := \left\{ x \in O \big| \text{dist}(\partial O, x) \ge \frac{1}{n}, \|x\| \le n \right\}$$.

This sequence has the properties
 * $$\bigcup_{j \in \mathbb N} K_j = O$$
 * $$\forall n \in \mathbb N : K_n \subset \overset{\circ}{K_{n+1}}$$.

We now construct $$(B_l)_{l \in \mathbb N}$$ such that for some $$k_1, k_2, \ldots \in \mathbb N$$. We do this in the following way: To meet the first condition, we first cover $$K_1$$ with balls by choosing for every $$x \in K_1$$ a ball $$B_x$$ such that $$B_x \subseteq U_\upsilon \cap \overset{\circ}{K_2}$$ for an $$\upsilon \in \Upsilon$$. Since these balls cover $$K_1$$, and $$K_1$$ is compact, we may choose a finite subcover $$B_1, \ldots B_{k_1}$$.
 * $$K_1 \subset \bigcup_{1 \le j \le k_1} B_j \subseteq \overset{\circ}{K_2}$$ and
 * $$\forall n \in \mathbb N : K_{n+1} \setminus \overset{\circ}{K_n} \subset \bigcup_{k_n < j \le k_{n+1}} B_j \subseteq \overset{\circ}{K_{n+2}} \setminus K_{n-1}$$

To meet the second condition, we proceed analogously, noting that for all $$n \in \mathbb N_{\ge 2}$$ $$K_{n+1} \setminus \overset{\circ}{K_n}$$ is compact and $$\overset{\circ}{K_{n+2}} \setminus K_{n-1}$$ is open.

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

2. We choose the respective functions. Since each $$B_n$$, $$n \in \mathbb N$$ is an open ball, it has the form
 * $$B_n = B_{R_n}(x_n)$$

where $$R_n \in \mathbb R$$ and $$x_n \in \mathbb R^d$$.

It is easy to prove that the function defined by
 * $$\tilde \eta_n (x) := \eta_{R_n}(x - x_n)$$

satisfies $$\tilde \eta_n(x) = 0$$ if and only if $$x \in B_n$$. Hence, also $$\text{supp } \tilde \eta_n = \overline{B_n}$$. We define
 * $$\eta(x) := \sum_{j=1}^\infty \tilde \eta_j(x)$$

and, for each $$n \in \mathbb N$$,
 * $$\eta_n := \frac{\tilde \eta_n}{\eta}$$.

Then, since $$\eta$$ is never zero, the sequence $$(\eta_l)_{l \in \mathbb N}$$ is a sequence of $$\mathcal D(\mathbb R^d)$$ functions and further, it has the properties 1. - 4., as can be easily checked.

Green's functions and Green's kernels
Proof:

We choose $$(\eta_l)_{l \in \mathbb N}$$ to be a partition of unity of $$O$$, where the open cover of $$O$$ shall consist only of the set $$O$$. Then by definition of partitions of unity
 * $$f = \sum_{j \in \mathbb N} \eta_j f$$.

For each $$n \in \mathbb N$$, we define
 * $$f_n := \eta_n f$$

and
 * $$u_n := f_n * K$$.

By Fubini's theorem, for all $$\varphi \in \mathcal D(\R^d)$$ and $$n \in \mathbb N$$
 * $$\begin{align}

\int_{\R^d} T_{K(\cdot - y)}(\varphi) f_n(y) dy & = \int_{\R^d} \int_{\mathbb R^d} K(x - y) \varphi(x) dx f_n(y) dy \\ & = \int_{\R^d} \int_{\mathbb R^d} f_n(y) K(x - y) \varphi(x) dy dx \\ & = \int_{\mathbb R^d} (f_n * K)(x) \varphi(x) dx \\ & = \mathcal T_{u_n}(\varphi) \end{align}$$.

Hence, $$\mathcal T_{u_n}$$ as given in theorem 4.11 is a well-defined distribution.

Theorem 5.4 implies that $$\mathcal T_{u_n}$$ is a distributional solution to the PDE
 * $$\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u_n(x) = f_n(x)$$.

Thus, for all $$\varphi \in \mathcal D(\R^d)$$ we have, using theorem 4.19,
 * $$\begin{align}

\int_{\R^d} \left( \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u_n \right)(x) \varphi(x) dx & = \mathcal T_{\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u_n} (\varphi) \\ & = \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T_{u_n} (\varphi) \\ & = T_{f_n}(\varphi) = \int_{\R^d} f_n(x) \varphi(x) dx \end{align}$$. Since $$\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u_n$$ and $$f_n$$ are both continuous, they must be equal due to theorem 3.17. Summing both sides of the equation over $$n$$ yields the theorem.

Proof:

If $$x_l \to x, l \to \infty$$, then
 * $$\begin{align}

\mathcal T_{K(\cdot - x_l)}(\varphi) - \mathcal T_{K(\cdot - x)}(\varphi) & = \int_{\mathbb R^d} K(y - x_l) \varphi(y) dy - \int_{\R^d} K(y - x) \phi(y) dy \\ & = \int_{\mathbb R^d} K(y) (\varphi(y + x_l) - \varphi(y + x)) dy \\ & \le \max_{y \in \mathbb R^d} |\varphi(y + x_l) - \varphi(y + x)| \underbrace{\int_{\text{supp } \varphi + B_1(x)} K(y) dy}_\text{constant} \end{align}$$ for sufficiently large $$l$$, where the maximum in the last expression converges to $$0$$ as $$l \to \infty$$, since the support of $$\varphi$$ is compact and therefore $$\varphi$$ is uniformly continuous by the Heine–Cantor theorem.

The last theorem shows that if we have found a locally integrable function $$K$$ such that
 * $$\forall x \in \mathbb R^d : \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T_{K(\cdot - x)} = \delta_x$$,

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