 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/5/b/iii/solution

w = v - u

Δ w = Δ v - Δ u = 0.

The boundary inequality says

w \geq 0

\partial D

. The minimum principle from part (ii) gives

v (p) - u (p) = w (p) \geq \partial D min w \geq 0 (p \in D) .

This is the comparison principle for harmonic functions. Applying it in both directions also shows why fixed Dirichlet boundary conditions determine at most one harmonic function.

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