The mean value property states that for a harmonic function u, u(x₀) equals:
AThe average of u over any sphere centered at x₀
BThe maximum of u on any sphere centered at x₀
CThe average of u over the entire domain
DThe value of u at the nearest boundary point
The mean value property says u(x₀) = (1/|∂B_r|)∫_{∂B_r(x₀)} u dS for any ball B_r(x₀) contained in the domain. This also holds for the volume average: u(x₀) = (1/|B_r|)∫_{B_r(x₀)} u dV. This property characterizes harmonic functions.
Question 2 True / False
A harmonic function on a bounded domain that is zero on the boundary must be identically zero.
TTrue
FFalse
Answer: True
By the maximum principle, a harmonic function achieves its maximum and minimum on the boundary. If u = 0 on the boundary, then max u ≤ 0 and min u ≥ 0 in the domain, so u ≡ 0. This is the uniqueness result for the Dirichlet problem.
Question 3 Short Answer
What is the key regularity property of harmonic functions?
Think about your answer, then reveal below.
Model answer: They are real-analytic (C^ω): infinitely differentiable and locally equal to their convergent power series
Despite Laplace's equation being a second-order equation, its solutions are not merely C² or even C^∞ — they are real-analytic. This follows from the mean value property and can be proved using the Poisson integral formula. This extreme regularity is a hallmark of elliptic equations.
Question 4 True / False
If u is harmonic in ℝⁿ and bounded, then u must be constant.
TTrue
FFalse
Answer: True
This is Liouville's theorem for harmonic functions. It follows from the mean value property: taking the average over larger and larger spheres, the oscillation of u must tend to zero since u is bounded. The analogous result for holomorphic functions in complex analysis is a special case.