Questions: Boundary Value Problems (Dirichlet, Neumann, Robin)

Question 1

For the Neumann problem for Laplace's equation ∂u/∂n = g on ∂Ω, what compatibility condition must g satisfy?

• A. ∫_∂Ω g dS = 0
• B. g must be positive everywhere
• C. g must be continuous
• D. max|g| < 1

Answer: (A)

By the divergence theorem, ∫_∂Ω (∂u/∂n)dS = ∫_Ω Δu dV = 0 for a harmonic function. So the boundary data must have zero total flux. Physically, in steady-state heat conduction with no sources, the net heat flow through the boundary must be zero.

Question 2

A Dirichlet problem for Laplace's equation on a bounded domain has a unique solution.

• T. True
• F. False

Answer: True

Uniqueness follows from the maximum principle: if u₁ and u₂ are both harmonic with the same Dirichlet data, their difference w = u₁ - u₂ is harmonic with w = 0 on ∂Ω, so by the maximum principle w ≡ 0.

Question 3

What physical scenario does the Robin boundary condition αu + β(∂u/∂n) = g model?

Model answer: Newton's law of cooling (convective heat transfer between the body and its surroundings)

The Robin condition models the situation where the heat flux through the boundary is proportional to the difference between the surface temperature and the ambient temperature. The coefficient α/β represents the heat transfer coefficient.

Question 4

The Neumann problem for Laplace's equation, when solvable, has a unique solution.

• T. True
• F. False

Answer: False

Neumann boundary conditions determine the solution only up to an additive constant. If u solves the Neumann problem, so does u + C for any constant C, since constants have zero normal derivative. Uniqueness is restored by adding a normalization condition such as ∫_Ω u dV = 0.