Questions: Maximum Principles (Elliptic and Parabolic)

Question 1

If u is harmonic in a bounded domain Ω and continuous on its closure, where does u attain its maximum?

• A. On the boundary ∂Ω
• B. At a critical point in the interior
• C. At the center of the domain
• D. It could be anywhere

Answer: (A)

The strong maximum principle for harmonic functions states that if u attains its maximum in the interior, then u must be constant throughout Ω. So a non-constant harmonic function achieves its maximum strictly on the boundary.

Question 2

The maximum principle directly implies uniqueness for the Dirichlet problem for Laplace's equation.

• T. True
• F. False

Answer: True

If u₁ and u₂ both solve Δu = 0 with the same boundary data, then w = u₁ - u₂ is harmonic with w = 0 on the boundary. By the maximum principle, w ≤ 0 in Ω, and by the minimum principle, w ≥ 0. So w ≡ 0 and the solution is unique.

Question 3

What is the parabolic boundary of the cylinder Ω × (0,T]?

Model answer: The set (Ω × {0}) ∪ (∂Ω × [0,T]), consisting of the initial time slice and the lateral spatial boundary

For parabolic equations, the maximum principle states extrema occur on the parabolic boundary—the bottom (t=0) and sides (∂Ω) of the space-time cylinder, but not the top (t=T). Information flows forward in time, so the solution at time T is controlled by earlier data.

Question 4

For which type of PDE do maximum principles generally fail?

• A. Elliptic
• B. Parabolic
• C. Hyperbolic
• D. Both parabolic and hyperbolic

Answer: (C)

Hyperbolic equations like the wave equation do not satisfy a maximum principle. The solution u_tt = c²u_xx can achieve interior extrema that exceed the boundary values, as waves can constructively interfere.