The <mean value property for harmonic functions> says that whenever $\overline{B_r(x_0)}\subset\Omega$,
$$
u(x_0)=\frac1{|B_r|}\int_{B_r(x_0)}u
=\frac1{|\partial B_r|}\int_{\partial B_r(x_0)}u.
$$
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 .

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-107/1/a/solution