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

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

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}.
$////$

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

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.

$////$

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.

## Sources

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