Subtopic: Partial Differential Equations – Laplace, heat and wave equations
Question Statement:
Which of the following statements is/are TRUE for Laplace's equation $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ in a bounded planar domain $D$ with boundary $C$?
Options:
(A) Solutions to Laplace's equation are called harmonic functions
(B) A non-constant harmonic function cannot attain its maximum or minimum at an interior point of $D$
(C) The solution to the Dirichlet problem (prescribed values of $u$ on boundary $C$) is unique
(D) At any point $(x_0, y_0)$, the value $u(x_0, y_0)$ equals the average value of $u$ along any circle centered at $(x_0, y_0)$ lying entirely in $D$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four statements are foundational properties of harmonic functions:
1. **Harmonic Definition (Option A)**: Any $C^2$ function satisfying $\nabla^2 u = 0$ is harmonic (True).
2. **Maximum/Minimum Principle (Option B)**: The extrema of harmonic functions in a bounded domain must lie on the boundary $\partial D$ (True).
3. **Uniqueness (Option C)**: If $u_1, u_2$ satisfy the same boundary values, $w = u_1 - u_2$ satisfies $\nabla^2 w = 0$ with $w = 0$ on $C$. By energy method $\iint_D |\nabla w|^2\,dA = 0$, $w \equiv 0$, establishing uniqueness (True).
4. **Mean Value Property (Option D)**: Gauss's mean value theorem proves that the center point value equals the perimeter circular average (True).