20 2. FOUR IMPORTANT LINEAR PDE

2.2. LAPLACE’S EQUATION

Among the most important of all partial diﬀerential equations are undoubt-

edly Laplace’s equation

(1) Δu = 0

and Poisson’s equation

(2) −Δu = f.

∗

In both (1) and (2), x ∈ U and the unknown is u :

¯

U → R, u = u(x),

where U ⊂ Rn is a given open set. In (2) the function f : U → R is also

given. Remember from §A.3 that the Laplacian of u is Δu =

∑n

i=1

uxixi .

DEFINITION. A

C2

function u satisfying (1) is called a harmonic func-

tion.

Physical interpretation. Laplace’s equation comes up in a wide variety

of physical contexts. In a typical interpretation u denotes the density of

some quantity (e.g. a chemical concentration) in equilibrium. Then if V is

any smooth subregion within U, the net flux of u through ∂V is zero:

∂V

F ·

ν

dS = 0,

F denoting the flux density and

ν

the unit outer normal ﬁeld. In view of

the Gauss–Green Theorem (§C.2), we have

V

div F dx =

∂V

F ·

ν

dS = 0,

and so

(3) div F = 0 in U,

since V was arbitrary. In many instances it is physically reasonable to as-

sume the flux F is proportional to the gradient Du but points in the opposite

direction (since the flow is from regions of higher to lower concentration).

Thus

(4) F = −aDu (a 0).

∗I prefer to write (2) with the minus sign, to be consistent with the notation for general

second-order elliptic operators in Chapter 6.