27.1.3 The Weak Maximum Principle
There is also a fundamental result having great significance which involves ∇2 called the
maximum principle. This principle says that if ∇2u ≥ 0 on a bounded open set U, then u
achieves its maximum value on the boundary of U.
Theorem 27.1.5 Let U be a bounded open set in ℝn and suppose
such that ∇2u ≥ 0 in U. Then letting ∂U = U ∖ U, it follows that
Proof: If this is not so, there exists x0 ∈ U such that
Since U is bounded, there exists ε > 0 such that
also has its maximum in U
because for ε
for all x ∈ ∂U.
Now let x1 be the point in U at which u
achieves its maximum.
As an exercise you should show that ∇2
. (Why?) Therefore,
a contradiction. ■