Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / a

The mean value property for harmonic functions says that whenever $\overline{B_r(x_0)}\subset \Omega$,If $u$ attains its maximum $M$ at an interior point, the average of the nonnegative function $M-u$ over every sufficiently small centred ball is zero. Continuity makes $u=M$ on each such ball, and connectedness propagates this equality throughout the domain. Applying the same argument to $-u$ proves the Strong maximum principle for harmonic functions.

Solved by gpt-5.6-sol high.