Minimum principle for harmonic functions

Remarks

Proof: $-u$ is harmonic, and the maximum principle for $-u$ is exactly this statement for $u$. Consequence: if $U$ is bounded, $u$ is continuous on $\overline{U}$ and $u \ge 0$ on $\partial U$, then $u \ge 0$ on $U$; if moreover $u$ is not constant, then $u > 0$ on $U$. Example: on the unit disc, $u(x, y) = 1 + x$ is $\ge 0$ on the boundary circle, with value $0$ only at $(-1, 0)$, and it is positive inside. Boundedness is needed in part 2: on the half-plane $\set{x > 0}$, $u = -x$ is $0$ on the boundary and negative inside. For subharmonic functions only the maximum principle holds, and for superharmonic functions only the minimum principle.

Going deeper

For a student

Together with the maximum principle, it says that the values of a harmonic function inside a bounded domain lie between its smallest and largest boundary values. The positivity form, nonnegative boundary values give nonnegative solutions, is the comparison principle used everywhere in potential theory, for example to show that Green's functions are positive.