Positive harmonic function

Remarks

Example: on the upper half-plane $\set{(x, y) : y > 0}$, the function $u(x, y) = y$ is positive harmonic, since $\Delta u = 0$ and $y > 0$ there. Non-example: $u(x, y) = x$ on the same half-plane is harmonic but takes negative values, for instance at $(-1, 1)$. Some authors call a harmonic function with $u \ge 0$ nonnegative harmonic; on a connected open set, the minimum principle shows that such a function is either identically $0$ or positive everywhere. On a ball, positive harmonic functions satisfy Harnack's inequality.