# Partial Differential Equations/Fundamental Solutions, Green's functions, Green's kernels and Dirichlet Problems

This chapter will show how you can apply distributions to find solutions for certain PDEs (partial differential equations).

## Definition: Solutions in the sense of distributions

Let $L = \sum_{|\alpha| \le k} a_\alpha (x) \frac{\partial^\alpha}{\partial x^\alpha}$. Consider the equation:

$Lu = f ~ (*)$

We call a distribution $T \in \mathcal D'(\Omega)$ a solution in the sense of distributions for $(*)$, if and only if:

$LT = T_f$

## Fundamental Solutions

### Definition: Fundamendal Solutions

We call a function $K_{(\cdot)} : \Omega \to \mathcal D'(\Omega)$ a fundamental solution for $L$, if and only if

• it is continuous[1], and
• $LK_\xi = \delta_\xi$, where $\delta_\xi(x)$ is the Dirac delta distribution.

### Lemma 2.1

Let $T_n$ be a sequence of distributions in $\mathcal A'$, and let $\mathcal A$ be a function space, which is a locally convex space. Let us furthermore have that $\lim_{n \to \infty} T_n(\phi) = T(\phi)$ for all $\phi$. Then

$T: \mathcal A \to \R, T(\varphi) = \lim_{n \to \infty} T_n(\varphi)$

defines a distribution.

### Lemma 2.2

Let $\mathcal A$ be a function space, which is a locally convex normed space. For $\Lambda \subseteq \R^d$, let $\{T_\lambda : \lambda \in \Lambda\} \subseteq \mathcal A'$ be a family of distributions on that function space. Let's further assume that for all $\varphi \in \mathcal A$, the function $\lambda \mapsto T_\lambda(\varphi)$ is continuous on $\Lambda$ and bounded, and let $f \in L^1(\Lambda)$. Then

$T(\varphi) := \int_\Lambda f(\lambda) T_\lambda(\varphi) d \lambda$

defines a distribution.

Proof: Due to the truncation of $L^p$-functions, we have that there are radii $R_i \in \R_+$ such that

$\int_{\Lambda \setminus B_{R_i}(0)} | f(\lambda) | d\lambda < \frac{1}{2 i \|T_\lambda(\varphi)\|_\infty}$

, where $\|T_\lambda(\varphi)\|_\infty$ is the supremum of the function $\lambda \mapsto T_\lambda(\varphi)$.

$B_{R_i}(0)$ is a compact set, since it is bounded as well as closed. Therefore, we may divide $B_{R_i}(0)$ into finitely many (let's say $n_i$) squares $d_{m_i}$ with diameter at most $\delta_i$, such that

$\forall \nu \in B_{R_i}(0) : \lambda \in B_{\delta_i} (\nu) \Rightarrow |T_\lambda(\varphi) - T_\nu(\varphi)| < \frac{1}{2i \|f\|_{L^1}}$

. This we may do because continuous functions are uniformly continuous on compact sets. At the border, we just round the squares so that they fit in with the sphere. Furthermore, we choose for each square a $\lambda_{m_i}$ inside this square.

We choose now

$T_i(\varphi) := \sum_{m=0}^{n_i} \int_{d_{m_i}} f(\lambda) T_{\lambda_{m_i}}(\varphi) d \lambda$

, which is a finite linear combination of distributions and therefore a distribution. Due to the normal triangle inequality for the absolute value, the triangle inequality for the Lebegue integral, our first calculation and the fundamental integral estimation, we obtain:

$|T_i(\varphi) - T(\varphi)| < \sum_{m=0}^{n_i} \int_{d_{m_i}} | f(\lambda) (T_{\lambda_{m_i}}(\varphi) - T_\lambda(\varphi))| d \lambda + \frac{\|T_\lambda(\varphi)\|_\infty}{2 i \|T_\lambda(\varphi)\|_\infty} \le \frac{1}{i}$

This obviously goes to zero, and this lemma follows with Lemma 2.1.

### Theorem 2.3

Let's assume that in equation $(*)$, $f$ is integrable. Let $K_{(\cdot)}$ be a fundamental solution for $(*)$ with respect to the locally convex normed function space $\mathcal A$, such that $\forall \phi \in \mathcal A$, the function $\xi \mapsto K_\xi(\phi)$ is bounded. Then we can know, that:

$T(\varphi) = \int_{\R^d} f(\xi) K_\xi(\varphi) d\xi$

is well-defined and solves $(*)$ in the sense of distributions.

Proof: Since by the definition of fundamental solutions, the function $\xi \mapsto K_\xi(\phi)$ is continuous, we may apply lemma 2.2, which gives us that $T$ is indeed well-defined.

To show that it really solves $(*)$ in the sense of distributions, we need the following calculation:

$LT(\varphi) = T(L^*\varphi) = \int_{\R^d} f(\xi) K_\xi(L^* \varphi) d\xi = \int_{\R^d} f(\xi) LK_\xi(\varphi) d\xi$
$= \int_{\R^d} f(\xi) \delta_\xi(\varphi) d\xi = \int_{\R^d} f(\xi) \varphi(\xi) d\xi = T_f(\varphi)$

, which is what we wanted to show.

## Green's functions and Green's kernels

### Definition: Green's function

Assume that for each $\xi$, the fundamental solution $K_\xi$ is a regular distribution, i. e. for each $\xi \in \Omega$, there is an integrable function $G( \cdot| \xi)$ such that $K_\xi = T_{G(\cdot | \xi)}$. Then we call this function $G: \R^d \times \Omega \to \R$ a Green's function for $L$.

### Definition: Green's kernel

Let's assume that $L$ has the Green's function $G(\cdot|\xi)$. If there exists a function $\tilde G: \R^d \to \R$ such that

$G(\cdot|\xi) = \tilde G(\cdot - \xi)$

, then we call $\tilde G$ a Green's kernel for $L$.

### Lemma 2.4

Let $\tilde G$ be a locally integrable function, and $\Omega \subseteq \R^d$ be a domain. Then the family of distributions $K_\xi := T_{\tilde G(\cdot - \xi)} \in \mathcal D'(\Omega)$ is well-defined and depends continuously on $\xi$. Furthermore, for each $\phi \in \mathcal D(\Omega)$, the function $\xi \mapsto K_\xi(\phi)$ is bounded.

Proof: Well-definedness follows from Lemma 1.3.

Let $\phi \in \mathcal D(\Omega)$, and let $\xi_n \to \xi$. Then we can calculate the following:

$T_{\tilde G(\cdot - \xi_n)}(\phi) - T_{\tilde G(\cdot - \xi)}(\phi) = \int_{\R^d} \tilde G(x - \xi_n) \phi(x) dx - \int_{\R^d} \tilde G(x - \xi) \phi(x) dx = \int_{\R^d} \tilde G(x) (\phi(x + \xi_n) - \phi(x + \xi)) dx$
$\le \max_{x \in \R^d} |\phi(x + \xi_n) - \phi(x + \xi)| \underbrace{\int_{\text{supp } \phi + B_{2\xi}(0)} \tilde G(x) dx}_\text{constant}$

for sufficiently large $n$, where the last expression goes to $0$ as $n \to \infty$, since the support of $\phi(x)$ is compact and therefore the function is (even uniformly) continuous.

Furthermore, we have

$T_{\tilde G(\cdot - \xi)}(\phi) = \int_{\R^d} \tilde G(x - \xi) \phi(x) dx = \int_{\text{supp } \phi} \tilde G(x) \phi(x + \xi) dx$

, which is zero for $\|\xi\|$ sufficiently large, which is why the function $\xi \mapsto K_\xi(\phi)$ has compact support. But since the function is also continuous, we know that it obtains a maximum and a minimum and is therefore bounded.

#### Remark

This lemma shows that if we have found a locally integrable function $\tilde G$ such that $LT_{\tilde G(\cdot - \xi)} = \delta_\xi$, we already know that it is a Green's kernel, and don't need to check the continuity property.

### Theorem 2.5

Now this theorem finally shows us why distributions are useful:

Let $\tilde G$ be a Green's kernel for $L$, and let $f \in L^\infty(\R^d)$. If

$u(x) = (f * \tilde G)(x)$

is sufficiently often differentiable such that $L u$ is continuous, then it is a solution for $(*)$ in the classical sense.

Proof: From a case of Hölder's inequality (namely $p = 1, q = \infty$, i. e. $\|f \cdot \tilde G\|_{L^1} \le \|\tilde G\|_{L^1} \cdot \|f\|_{L^\infty}$), we obtain that $u$ is locally integrable, which is why $T_u$ is a distribution in $\mathcal D'(\R^d)$.

Furthermore, due to the theorem of Fubini, we have for $\varphi \in \mathcal D(\R^d)$, that

$T_u(\varphi) = \int_{\R^d} (f * \tilde G)(x) \varphi(x) dx = \int_{\R^d} \int_{\R^d} f(y) \tilde G(x - y) \varphi(x) dy dx$
$= \int_{\R^d} \int_{\R^d} \tilde G(x - y) \varphi(x) dx ~ f(y) dy = \int_{\R^d} T_{\tilde G(\cdot - \xi)}(\varphi) f(y) dy$

, which is why $T_u$ solves $(*)$ in the sense of distributions (this is due to theorem 2.3).

Thus, for all $\varphi \in \mathcal D(\R^d)$, we can calculate the following:

$\int_{\R^d} (Lu)(x) \varphi(x) dx = T_{Lu} (\varphi) = LT_u(\varphi) = T_f(\varphi) = \int_{\R^d} f(x) \varphi(x) dx$

and therefore

$\int_{\R^d} ((Lu)(x) - f(x)) \varphi(x) dx = 0$.

From this follows that $Lu = f$ almost everywhere. But since $Lu$ and $f$ are both continuous, they must be equal everywhere. This is what we wanted to prove.

## Definition: Dirichlet problems

If one looks at a domain $\Omega \subset \R^d$ with boundary $\partial \Omega$, then finding a function $u$ which satisfies

$\begin{cases} Lu(x) = f(x) & x \in \Omega \\ u(x) = g(x) & x \in \partial \Omega \end{cases}$

is called a Dirichlet Problem.

## Notes

1. This means here that $\xi \mapsto K_\xi(\phi)$ as a function from $\R^d$ to $\R$ is continuous for every $\phi \in \mathcal D(\Omega)$.