Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
Which of the following statements is/are TRUE for solutions of the Laplace equation $\nabla^2 u = 0$ in a bounded domain $D$ with smooth boundary $\partial D$?
Options:
(A) Solutions $u$ are called harmonic functions
(B) By the Maximum Principle, the maximum and minimum values of $u$ must occur on the boundary $\partial D$
(C) At any point $(x_0, y_0)$, the value $u(x_0, y_0)$ equals the average of $u$ over any circle centered at $(x_0, y_0)$ inside $D$ (Mean Value Property)
(D) If Dirichlet boundary conditions $u = g$ on $\partial D$ are prescribed, the solution inside $D$ is unique
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four properties are fundamental theorems of potential theory:
1. Solutions to $\nabla^2 u = 0$ are harmonic functions (A is true).
2. The Strong Maximum Principle states that a non-constant harmonic function achieves its extrema only on the boundary $\partial D$ (B is true).
3. Harmonic functions satisfy Gauss's mean value property (C is true).
4. By energy methods or maximum principle, the Dirichlet problem for Laplace's equation has a unique solution (D is true).