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 Poisson's 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 \Omega : \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 iff

\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) \text{ is continuous}
  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 \{\mathcal T_x | x \in S\} \subseteq \mathcal D(O)^* be a family 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). Then

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

is a distribution.

Proof:

For \varphi \in \mathcal D(O), let us denote the supremum norm of the function 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. Let F be a fundamental solution of the equation such that \forall \varphi \in \mathcal D(O), the function x \mapsto F(x)(\varphi) is bounded. 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}.
////

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}.
////

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

Definition 5.7:

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 5.8:

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 5.9:

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 (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(x) := f * K(x)

is locally integrable and sufficiently often differentiable such that \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u is continuous, then u solves the partial differential equation.

Proof:

Since u is locally integrable, \mathcal T_u as given in theorem 4.11 is a well-defined distribution. Furthermore, by Fubini's theorem, for all \varphi \in \mathcal D(\R^d)

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

Hence, theorem 5.4 implies that \mathcal T_u is a distributional solution to the PDE.

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 \right)(x) \varphi(x) dx & = \mathcal T_{\sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u} (\varphi) \\
& = \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha \mathcal T_u (\varphi) \\
& = T_f(\varphi) = \int_{\R^d} f(x) \varphi(x) dx
\end{align}

and therefore

\int_{\R^d} \left( \left( \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u \right)(x) - f(x) \right) \varphi(x) dx = 0.

From this follows that \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u = f almost everywhere. But since \sum_{\alpha \in \mathbb N_0^d} a_\alpha \partial_\alpha u and f are both continuous, they must be equal everywhere.

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Theorem 5.10:

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)} 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.

////

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 K 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.

Exercises[edit]

Sources[edit]

Partial Differential Equations
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