The
mean-value property
has a striking consequence. If a harmonic function equals the average of its neighbours at every
point, it can never rise above all of them — there can be no interior peak. This is the
maximum principle.
Let u be harmonic on a bounded region \Omega
and continuous up to its boundary \partial\Omega. Then:
- the maximum and minimum of u are both attained on the boundary;
- (strong form) unless u is constant, it attains them only there — never at an interior point.
Physically: a plate in thermal equilibrium has its hottest and coldest spots on its edges. With
no source inside, heat cannot pile up to a hidden interior peak — it would immediately flow away
to the cooler surroundings.
Why it guarantees uniqueness
The maximum principle is not just a curiosity — it is the cleanest route to uniqueness
for the Dirichlet problem. Suppose two harmonic functions u_1 and
u_2 share the same boundary values. Their difference
w = u_1 - u_2 is harmonic and is 0 on the
entire boundary. By the maximum principle its maximum and minimum are both
0, so w \equiv 0 throughout — meaning
u_1 = u_2. The boundary data determines the solution completely.
The same argument gives stability: if two boundary datasets differ by at most
\varepsilon, their solutions differ by at most
\varepsilon everywhere inside. Existence, uniqueness, and continuous
dependence — the Dirichlet problem for Laplace's equation is
well-posed.