Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 12

§ 2. The Maximum Principle

An important tool for the analysis of finite difference methods is the discrete
analog of the so-called maximum principle. Before turning to the discrete case,
we examine a simple continuous version.
In the following, denotes a bounded domain in

d . Let

R

d

Lu : =− a ik (x)u x i x k ( 2 . 1 )

i,k = 1

be a linear elliptic differential operator L . This means that the matrix A = (a ik )
is symmetric and positive definite on . For our purposes we need a quantitative
measure of ellipticity.
For convenience, the reader may assume that the coefficients a ik are contin-
uous functions, although the results remain true under less restrictive hypotheses.

2.1 Maximum Principle. Fo r u ∈ C 2 () ∩ C 0 ( ¯ ), let

Lu = f ≤ 0 in .

Then u attains its maximum over ¯ on the boundary of . Moreover, if u attains a
maximum at an interior point of a connected set , then u must be constant on .
Here we prove the first assertion. For a proof of the second one, see Gilbarg
and Trudinger [1983].
(1) We first carry out the proof under the stronger assumption that f< 0.
Suppose that for some x 0 ∈ ,

u(x 0 ) = sup

x ∈

u(x) > sup
x

u(x).

∈ ∂

Applying the linear coordinate transformation x −→ ξ = Ux , the differential
operator becomes

(U t A(x)U) ik u ξ i ξ k

Lu =−

i,k

in the new coordinates. In view of the symmetry, we can find an orthogonal matrix
U so that U T A(x 0 )U is diagonal. By the definiteness of A(x 0 ) , we deduce that
these diagonal elements are positive. Since x 0 is a maximal point,

u ξ i = 0 , ξ i ξ i ≤ 0