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Zero-Dirichlet Green representation for Poisson data

Statement

Assume Countable Choice and the hypotheses and sign convention of Green representation for classical Poisson data: $n \geq 2$ , a bounded $C^{1}$ domain $Ω$ carrying a Dirichlet Green function for $- Δ$ whose correctors satisfy $H_{y} \in C^{2} (\overline{Ω})$ , and $P_{Ω} = - \partial_{ν_{y}} G_{Ω}$ . If $u \in C^{2} (\overline{Ω})$ is real with $u = 0$ on $\partial Ω$ and $f : = - Δ u,$ then $u (x) = \int_{Ω} G_{Ω} (x, y) f (y) d y (x \in Ω),$ the integral being absolutely finite. No regularity beyond $u \in C^{2} (\overline{Ω})$ is assumed, and no existence of Green functions is claimed.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 2$ , the bounded $C^{1}$ domain $Ω$ , its Dirichlet Green function $G_{Ω}$ with correctors $H_{y} \in C^{2} (\overline{Ω})$ , and the real function $u \in C^{2} (\overline{Ω})$ with $u ∣_{\partial Ω} = 0$ and $f = - Δ u$ .

[A1]

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

[F1]

Under the stated hypotheses the Green representation formula holds for every $x \in Ω$ , $u (x) = \int_{Ω} G_{Ω} (x, y) (- Δ u (y)) d y + \int_{\partial Ω} P_{Ω} (x, y) u (y) d S (y),$ and both integrals are absolutely finite (Green representation for classical Poisson data).

[F2]

The Green function is $G_{Ω} (x, y) = Φ (x - y) - H_{y} (x)$ for $x \neq y$ , with the kernel normalized by $- Δ Φ = δ_{0}$ , and the Poisson kernel is $P_{Ω} = - \partial_{ν_{y}} G_{Ω}$ (Dirichlet Green function for minus Laplacian).

[F3]

Surface integration on a compact embedded $C^{1}$ hypersurface is chart integration; signed integrands with finite absolute integral are integrated through their positive and negative parts, so an integrand that vanishes identically integrates to zero (Surface integration on compact C1 hypersurfaces).

[F4]

The Laplacian is $Δ u = \sum_{i} \partial_{i} \partial_{i} u$ , and $f = - Δ u$ means $Δ u = - f$ pointwise (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field).

Proof

technique · direct

1.1givenA1F1F2F4

Fix $x \in Ω$ . The datum $u$ is real and $C^{2}$ up to the boundary, the correctors satisfy the regularity hypothesis, and $f = - Δ u$ ; so the representation formula [F1] applies at $x$ , and its second integral is the surface integral over the compact hypersurface $\partial Ω$ of the product $y \mapsto P_{Ω} (x, y) u (y)$ . The hypothesis $u ∣_{\partial Ω} = 0$ means that the continuous trace of $u$ vanishes at every boundary point.

1.2givenA1F3

For every $y \in \partial Ω$ we have $u (y) = 0$ by hypothesis, hence $P_{Ω} (x, y) u (y) = 0$ . The boundary integrand is therefore the identically zero function on $\partial Ω$ , and its surface integral vanishes; this uses only the chart definition and the signed-integral convention of [F3], with no appeal to the size of $P_{Ω}$ .

2.1step 1.1step 1.2F1F4algebra

Substituting $f = - Δ u$ and the vanishing boundary integral of step 1.2 into the formula of step 1.1 gives $u (x) = \int_{Ω} G_{Ω} (x, y) f (y) d y$ for the fixed $x$ , and the absolute finiteness asserted there is exactly the absolute finiteness of this integral.

3.1givenA1F1F3step 2.1cases∎

Since $x \in Ω$ was arbitrary, the identity holds for every $x \in Ω$ . If $f = 0$ , then $Δ u = 0$ and $u = 0$ on $\partial Ω$ , and the formula returns $u (x) = \int_{Ω} G_{Ω} (x, y) \cdot 0 d y = 0$ for every $x$ , consistent with the statement; the zero and empty cases are covered by this same substitution. Countable Choice is inherited from the representation theorem and its Green, kernel and surface conventions; no new choice is used in steps 1.1–2.1. For complex-valued $u$ the real result applies to the real and imaginary parts, whose boundary traces also vanish; the statement is formulated for real $u$ .

Source notes

Hunter §§2.5–2.7, printed pp.32–42, constructs the Green function for the Laplacian and states the representation $u = \int G f$ for zero boundary data as the classical motivation for the Green function, after the Green identities of §2.5. Teschl §§5.3–5.4, printed pp.117–129, defines the Green function by the harmonic correction and derives the representation formula for classical data. Neither reference is used here as a proof of the specialization: the corollary is the substitution $f = - Δ u$ into the already proved representation theorem, with the boundary term disposed of by the zero trace. The regularity hypotheses are those of Green representation for classical Poisson data and are not weakened; in particular no weak-boundary or $L^{2}$ -trace statement is made.

Depends on

Used by

Uniqueness of classical Dirichlet and compatible Neumann solutions Corollary

Dependency tree · two levels

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Sources

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript) (standard reference, not scraped)