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.
