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18.8 The Neumann Problem 659

18.8 The Neumann Problem

Recall that, if g is a function of two variables defined on a set of points D in the plane having
boundary C, then the normal derivative ∂g/∂n of g on C is the dot product of the gradient of g
with a unit normal vector n to C:
∂g
=∇g · n.
∂n
We will assume that n is a unit outer normal to D. This means that, if drawn as an arrow from a
point on C, n points away from D, as in Figure 12.19.

A Neumann problem in the plane consists of finding a function that is harmonic on a given
region D, and whose normal derivative assumes given values on the boundary C of D.
This problem has the form
2
∇ u(x, y) = 0on D,
∂u
= g(x, y) for (x, y) in C,
∂n
with g(x, y) a given function defined on the boundary of C.

The following lemma plays an important role in attempting to solve a Neumann problem.

LEMMA 18.1 Green’s First Identity

Let D be a bounded set of points in the plane, having boundary curve C. Assume that C is a
simple, closed, piecewise smooth curve. Let f and g be continuous with continuous first and
second partial derivatives on D and at points of C. Then
∂ f

2
g ds = (g∇ f +∇ f ·∇g)dA.
C ∂n D
The line integral on the left is with respect to arc length along C.
Proof of Lemma 18.1 By Green’s theorem,

∂g
g ds = (g∇ f ) · nds = ∇· (g∇ f )dA.
C ∂n C D
The rest of the proof consists of computing

∂ f ∂ f
∇· (g∇ f ) =∇ · g i + g j
∂x ∂y
∂ ∂ f ∂ ∂ f
= g + g
∂x ∂x ∂y ∂y
2 2
∂ f ∂ f ∂g ∂ f ∂g ∂ f
= g + + +
∂x 2 ∂y 2 ∂x ∂x ∂y ∂
= g∇ f +∇ f ·∇g.
2

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October 14, 2010 15:27 THM/NEIL Page-659 27410_18_ch18_p641-666