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Half-space Poisson extension of a plane wave

Example

Assume Countable Choice and $n \geq 3$ . Write $e_{n}$ for the last canonical basis vector $e_{n - 1}$ . Fix $ξ \in R^{n - 1}$ and let $g (x^{'}) = \exp (2 π i ξ \cdot x^{'})$ on $\partial H = R^{n - 1}$ . Then the half-space Poisson integral of $g$ is $U_{g} (x^{'}, t) = \exp (- 2 π ∣ ξ ∣ t) \exp (2 π i ξ \cdot x^{'}), (x^{'}, t) \in H,$ including the case $ξ = 0$ , where the extension is the constant $1$ . Consequently $\partial_{t} U_{g} (x^{'}, 0) = - 2 π ∣ ξ ∣ g (x^{'})$ and the outward normal derivative at the boundary is $+ 2 π ∣ ξ ∣ g$ , so for a single spatial frequency the Dirichlet-to-Neumann map is multiplication by $2 π ∣ ξ ∣$ .

Facts & Assumptions

Given: Countable Choice, an integer $n \geq 3$ , a frequency $ξ \in R^{n - 1}$ and the datum $g (x^{'}) = \exp (2 π i ξ \cdot x^{'})$ on $\partial H$ .

[F1]

For bounded continuous $g$ the half-space Poisson integral $U_{g}$ is the unique bounded harmonic function on $H$ , continuous on $\overline{H}$ , with trace $g$ ; the Poisson kernel is $P_{H} ((x^{'}, t), z) = 2 t / (ω_{n - 1} (∣ x^{'} - z ∣^{2} + t^{2})^{n / 2})$ (Poisson kernel and bounded Dirichlet problem on a half-space).

[F2]

Laplacian and partial derivatives: $Δ f = \sum_{i} \partial_{i} \partial_{i} f$ , and for the exponential $\exp (2 π i ξ \cdot x^{'})$ the tangential derivatives give $Δ_{x^{'}} \exp (2 π i ξ \cdot x^{'}) = - 4 π^{2} ∣ ξ ∣^{2} \exp (2 π i ξ \cdot x^{'})$ by the chain and product rules, while $\partial_{t}^{2} \exp (- 2 π ∣ ξ ∣ t) = 4 π^{2} ∣ ξ ∣^{2} \exp (- 2 π ∣ ξ ∣ t)$ (The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The exponential function is smooth and $(\exp)^{'} = \exp$ , The complex exponential is entire and its complex derivative is itself).

[F3]

In the negative-sign $2 π$ normalisation the Fourier transform of the plane wave $x^{'} \mapsto e^{2 π i ξ \cdot x^{'}}$ is $δ_{ξ}$ (Fourier transform of delta constants plane waves and polynomials).

[F4]

Countable Choice ${A C}_{ω}$ is the standing hypothesis (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

technique · direct

1.1givenF4algebra

Work under [F4] and define $V (x^{'}, t) : = \exp (- 2 π ∣ ξ ∣ t) \exp (2 π i ξ \cdot x^{'})$ on $\overline{H}$ . Since $∣ V (x^{'}, t) ∣ = \exp (- 2 π ∣ ξ ∣ t) \leq 1$ and $V (x^{'}, 0) = g (x^{'})$ , the function $V$ is bounded and continuous on $\overline{H}$ with trace $g$ .

2.1step 1.1F2

By [F2], $Δ V = Δ_{x^{'}} V + \partial_{t}^{2} V = - 4 π^{2} ∣ ξ ∣^{2} V + 4 π^{2} ∣ ξ ∣^{2} V = 0$ on $H$ ; so $V$ is harmonic (all derivatives exist and are continuous, being those of an exponential).

3.1step 1.1step 2.1F1

Applying uniqueness in [F1] to $V$ and to the Poisson integral $U_{g}$ of the bounded continuous datum $g$ gives $U_{g} = V$ , which is the displayed formula; for $ξ = 0$ this reads $U_{g} \equiv 1$ .

4.1step 3.1algebra

Differentiating the formula at $t = 0$ gives $\partial_{t} U_{g} (x^{'}, 0) = - 2 π ∣ ξ ∣ g (x^{'})$ ; the outward unit normal of $H$ at the boundary plane is $- e_{n}$ , so the outward normal derivative is $- \partial_{t} U_{g} (x^{'}, 0) = + 2 π ∣ ξ ∣ g (x^{'})$ . This is the single-mode Dirichlet-to-Neumann computation: the half-space Poisson multiplier $e^{- 2 π ∣ ξ ∣ t}$ differentiates to the boundary multiplier $2 π ∣ ξ ∣$ in the outward normal.

5.1step 3.1F3algebra∎

The same multiplier is visible in the Fourier description: [F3] says the datum $g$ has Fourier transform $δ_{ξ}$ , and the extension multiplies that mode by the factor $e^{- 2 π ∣ ξ ∣ t}$ ; the constant mode $ξ = 0$ is fixed and does not decay.

Depends on

Used by

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Sources

Armin Schikorra, Partial Differential Equations I & II (2025) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript) (standard reference, not scraped)