Question

The solution of Laplace’s equation satisfies which condition?

• A. It has a maximum inside the domain
• B. It has a minimum inside the domain
• C. It satisfies the maximum–minimum principle
• D. It diverges at the boundary

Hint:

A physical interpretation: If you have a steady-state temperature distribution (which follows Laplace's eq), the hottest and coldest points must be on the edges/surface, not in the middle of the object.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Laplace's equation is a second-order partial differential equation given by \( \nabla^2 \phi = 0 \).
Solutions to this equation are called "harmonic functions".
Step 2: Detailed Explanation:
Harmonic functions possess unique properties:
1. Mean Value Property: The value of the function at a point is the average of its values on a surrounding circle or sphere.
2. Maximum-Minimum Principle: A non-constant harmonic function cannot have a local maximum or minimum at any point inside the domain where it is defined.
This means the extreme values (maximum and minimum) must occur on the boundaries of the domain.
If a solution had a local maximum inside, the second derivative would have to be negative in all directions, making the Laplacian \( \nabla^2 \phi<0 \), which contradicts \( \nabla^2 \phi = 0 \).
Step 3: Final Answer:
Since the solution cannot have local extrema inside the domain, it must satisfy the maximum–minimum principle.