 Neumann compatibility condition for Laplace's equation

ID: neumann-compatibility-condition-for-laplace-s-equation

For a smooth bounded region

Ω

, prescribing the outward normal derivative

\partial_{n} u = g

\int_{\partial Ω} g d S = 0.

This follows from the divergence theorem applied to the gradient:

\int_{\partial Ω} \partial_{n} u d S = \int_{Ω} \nabla \cdot \nabla u d V = \int_{Ω} Δ u d V = 0.

On the unit disk, the condition becomes

\int_{0}^{2 π} g (θ) d θ = 0

. If

g = \sum_{n \geq 1} (a_{n} cos (n θ) + b_{n} sin (n θ))

is sufficiently regular with zero mean, its regular harmonic expansion on a disk is

u (r, θ) = C + \sum_{n \geq 1} \frac{r ^{n}}{n} (a_{n} cos (n θ) + b_{n} sin (n θ)) .

The factor

1/ n

compensates for the radial derivative. The arbitrary

C

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