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

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Partial Differential Equations
 ← Distributions Fundamental solutions, Green's functions and Green's kernels The heat equation → 

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[edit]

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.

Definition 5.1:

Let O \subseteq \mathbb R^d be open, let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

be a linear partial differential equation, and let \mathcal T \in \mathcal D(O)^*. \mathcal T is called a distributional solution to the above linear partial differential equation if and only if

\forall \varphi \in \mathcal D(O) : \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T(\varphi) = \mathcal T_f (\varphi).

Definition 5.2:

Let O \subseteq \mathbb R^d be open and let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

be a linear partial differential equation. If F : O \to \mathcal D(O)^* has the two properties

  1. \forall \varphi \in \mathcal D(O) : x \mapsto F(x)(\varphi) is continuous and
  2. \forall x \in O : \forall \varphi \in \mathcal D(O) : \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha F(x)(\varphi) = \delta_x(\varphi),

we call F a fundamental solution for that partial differential equation.

For the definition of \delta_x see exercise 4.5.

Lemma 5.3:

Let O \subseteq \mathbb R^d be open and let \{\mathcal T_x | x \in S\} \subseteq \mathcal D(O)^* be a set of distributions, where S \subseteq \mathbb R^d. Let's further assume that for all \varphi \in \mathcal D(O), the function S \to \mathbb R, x \mapsto \mathcal T_x(\varphi) is continuous and bounded, and let f \in L^1(S) be compactly supported. Then

\mathcal T(\varphi) := \int_S f(x) \mathcal T_x(\varphi) d x

is a distribution.

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.

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Theorem 5.4:

Let O \subseteq \mathbb R^d be open, let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

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

\mathcal T: \mathcal D(O) \to \mathbb R, \mathcal T(\varphi) := \int_{\R^d} f(x) F(x)(\varphi) dx

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

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}.
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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}.
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Theorem 5.6:

Let \mathcal T be a solution of the equation

\forall \varphi \in \mathcal D(\mathbb R^d) : \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T(\varphi) = \delta_0,

where only finitely many a_\alpha are nonzero, and let \vartheta \in \mathcal D(\mathbb R^d). Then u := \mathcal T * \vartheta solves

\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u = \vartheta.

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}.
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Partitions of unity[edit]

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 S \subseteq \mathbb R^d be a set. We define:

  • \partial S := \left\{x \in \mathbb R \big| \forall \epsilon > 0 : B_\epsilon(x) \cap S \neq \emptyset \wedge B_\epsilon(x) \cap (\mathbb R^d \setminus S) \neq \emptyset\right\}
  • \overset{\circ}{S} := S \setminus \partial S

\partial S is called the boundary of S and \overset{\circ}{S} is called the interior of S. Further, if x \in \mathbb R^d, we define

\text{dist}(S, x) := \inf_{y \in S} \|x - y\|.

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

Definition 3.13:

For R \in \mathbb R_{>0}, we define

\eta_R : \mathbb R^d \to \mathbb R, \eta_R(x) = \eta\left( \frac{x}{R} \right) \big/ R^d.

Theorem and definitions 5.8: Let O \subseteq \R^d be an open set, and let U_\upsilon, \upsilon \in \Upsilon be open subsets of \mathbb R^d such that \bigcup_{\upsilon \in \Upsilon} U_\upsilon = O (i. e. the sets U_\upsilon, \upsilon \in \Upsilon form an open cover of O). Then there exists a sequence of functions (\eta_l)_{l \in \mathbb N} in \mathcal D (\mathbb R^d) such that the following conditions are satisfied:

  1. \forall n \in \N : \forall x \in O : 0 \le \eta_n(x) \le 1
  2. \forall n \in \N : \exists \upsilon \in \Upsilon : \text{supp } \eta_n \subseteq U_\upsilon
  3. \forall x \in O : |\{n \in \mathbb N | \eta_n(x) \neq 0\}| < \infty
  4. \forall x \in O : \sum_{i=0}^\infty \eta_i(x) = 1

The sequence (\eta_l)_{l \in \mathbb N} is called a partition of unity for O with respect to U_\upsilon, \upsilon \in \Upsilon.

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

  • 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}

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}.

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.

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Green's functions and Green's kernels[edit]

Definition 5.9:

Let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

be a linear partial differential equation. A function G: \mathbb R^d \times \mathbb R^d \to \mathbb R such that for all x \in \mathbb R^d \mathcal T_{G(\cdot, x)} is well-defined and

F(x) := \mathcal T_{G(\cdot, x)}

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

Definition 5.10:

Let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

be a linear partial differential equation. A function K: \mathbb R^d \to \mathbb R such that the function

G(y, x) := K(y - x)

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

Theorem 5.11:

Let

\forall x \in O : \sum_{\alpha \in \mathbb N_0^d} a_\alpha(x) \partial_\alpha u(x) = f(x)

be a linear partial differential equation (in the following, we will sometimes abbreviate PDE for partial differential equation) such that f \in \mathcal C (\mathbb R^d), and let K be a Green's kernel for that PDE. If

u := f * K

exists and \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u exists and is continuous, then u solves the partial differential equation.

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.

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Theorem 5.12:

Let K \in L^1_{\text{loc}} and let O \subseteq \R^d be open. Then for all \varphi \in \mathcal D(O), the function x \mapsto \mathcal T_{K(\cdot - x)}(\varphi) is continuous.

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.

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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.

Exercises[edit]

Sources[edit]

Partial Differential Equations
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