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Dirichlet Green function for minus Laplacian

Statement

Assume the Axiom of Countable Choice, written ${A C}_{ω}$ , and let $n \geq 2$ . Let $Ω \subset R^{n}$ be a bounded domain, and use the kernel $Φ$ fixed by Fundamental solution for the positive operator minus Laplacian. A Dirichlet Green function for $- Δ$ on $Ω$ is a function $G_{Ω} : {(x, y) \in Ω \times Ω : x \neq y} \to R$ such that for each pole $y \in Ω$ there is a harmonic function $H_{y} \in C^{2} (Ω) \cap C (\overline{Ω})$ with $H_{y} (z) = Φ (z - y) (z \in \partial Ω), G_{Ω} (x, y) = Φ (x - y) - H_{y} (x) (x \in Ω ∖ {y}) .$ For fixed $y$ , $G_{Ω} (\cdot, y)$ is harmonic away from $y$ , extends continuously to $\overline{Ω} ∖ {y}$ , and has zero boundary trace. Its locally integrable representative defines $T_{G_{Ω} (\cdot, y)} \in D^{'} (Ω)$ and satisfies $- Δ_{x} T_{G_{Ω} (\cdot, y)} = δ_{y} in D^{'} (Ω) .$ The definition is conditional: it applies only when such a corrector $H_{y}$ exists for every pole $y$ ; it asserts no existence for every bounded domain. No boundary smoothness is required for this definition.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , a bounded domain $Ω$ , and a family of correctors $H_{y}$ with the stated harmonicity, continuity, and boundary values.

[A1]

${A C}_{ω}$ says every countable family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The normalized kernel $Φ$ is locally integrable and its value at its pole may be assigned arbitrarily (Fundamental solution for the positive operator minus Laplacian).

[F2]

The kernel is smooth and harmonic away from its pole (The Laplace fundamental solution is harmonic off its pole).

[F3]

For every $y \in R^{n}$ , $- Δ_{x} T_{Φ (\cdot - y)} = δ_{y}$ in $D^{'} (R^{n})$ under ${A C}_{ω}$ (The negative Laplacian of the fundamental solution is the unit Dirac distribution).

[F4]

If $f \in C^{k} (Ω)$ , then its regular distribution satisfies $\partial^{α} T_{f} = T_{\partial^{α} f}$ for $∣ α ∣ \leq k$ , under ${A C}_{ω}$ (Distributional differentiation is continuous and commutes).

[F5]

A locally integrable function defines its regular functional by integration (Regular distribution from a locally integrable function).

[F6]

Under Countable Choice, locally integrable functions embed as regular distributions in $D^{'} (Ω)$ (Locally integrable functions embed in distributions).

[F7]

The distributional Poisson equation $- Δ T = F$ is an equality of distributions on the open set (Distributional harmonicity and Poisson's equation on an open subset of Rn).

Proof

technique · direct

1.1givenF1F2F4F5F6algebra

Fix $y \in Ω$ . On $Ω ∖ {y}$ , both $x \mapsto Φ (x - y)$ and $H_{y}$ are harmonic by [F2] and the given corrector property, so their difference $G_{Ω} (\cdot, y)$ is harmonic there. Since $Φ (\cdot - y)$ is locally integrable by [F1] and $H_{y} \in C^{2} (Ω)$ has its regular distribution by [F4], their difference is locally integrable on $Ω$ ; choose any value at $y$ , which does not affect its almost-everywhere class or regular distribution.

1.2givenF1F2algebra

Because $y$ is an interior point, some ball $B_{r} (y)$ lies in $Ω$ , and hence every boundary point is distinct from $y$ . The function $x \mapsto Φ (x - y)$ is continuous on $\overline{Ω} ∖ {y}$ by its smoothness away from the pole, and $H_{y}$ is continuous there by hypothesis. Thus their difference gives a continuous extension of $G_{Ω} (\cdot, y)$ to $\overline{Ω} ∖ {y}$ . On $\partial Ω$ the two terms agree, so this extension has boundary value zero. No boundary chart or normal is involved.

2.1A1F3F4F5F6F7step 1.1algebra

Regard the locally integrable functions in step 1.1 as regular distributions using [F5]–[F6]. By [F4] and $Δ H_{y} = 0$ , $- Δ T_{H_{y}} = T_{- Δ H_{y}} = 0$ on $Ω$ . The restriction of [F3] from $R^{n}$ to test functions in $C_{c}^{\infty} (Ω)$ gives $- Δ T_{Φ (\cdot - y)} = δ_{y}$ on $Ω$ . Linearity of the regular functional and of distributional differentiation, together with $G_{Ω} (\cdot, y) = Φ (\cdot - y) - H_{y}$ almost everywhere, therefore gives $- Δ T_{G_{Ω} (\cdot, y)} = δ_{y}$ in the sense of [F7].

3.1

The argument includes both kernel cases already fixed by [F1], namely the logarithmic kernel when $n = 2$ and the power kernel when $n \geq 3$ ; $n = 1$ and dimension zero are excluded by the stated hypothesis. It proves properties of a Green function only after the correctors are given and does not prove that correctors exist. Countable Choice is used through [F3], [F4], and [F6], exactly the named distributional embedding and classical-derivative interfaces; no full Axiom of Choice is used. There is no iff assertion. [A1, F1, F3, F4, F6, given, cases] $□$

Source notes

Teschl §5.4, equations (5.33)–(5.34), defines the harmonic correction with boundary values equal to the fundamental solution and forms the Green function by subtraction; the surrounding text explicitly defines existence only when the harmonic Dirichlet problem is solvable for every pole. Schmidt §2.8, printed pp.44–45, defines the Green function through harmonic cancellation and zero boundary limits, then notes that the singularity has the same type as the fundamental solution. Schmidt uses $Δ F = δ_{0}$ and a nonpositive Green function; the convention here is obtained by $Φ = - F$ and $G_{Ω} = - G_{S c h m i d t}$ . The distributional point-source assertion here is proved from the already established kernel identity rather than inferred from a citation or from the word “Green function.”

Dirichlet Green function for minus Laplacian

Statement

Facts & Assumptions

Proof

Source notes

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