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: Recall Laplace’s equation and its solutions. 
Laplace’s equation is written as $\nabla^2 \phi = 0$.
Any function that satisfies this equation is known as a harmonic function.

Step 2: Describe the key properties of harmonic functions.
Harmonic functions exhibit several important mathematical properties:

• Mean value property:
  The value of the function at any interior point equals the average of its values over a circle (in two dimensions) or a sphere (in three dimensions) centered at that point.
• Maximum–minimum principle:
  A harmonic function that is not constant cannot attain a maximum or minimum at any interior point of its domain.
  All extreme values must occur on the boundary of the region.

To see why this is true, suppose a harmonic function had a local maximum at an interior point.
At such a point, the curvature would be negative in all directions, implying $\nabla^2 \phi < 0$.
This contradicts Laplace’s equation, which requires $\nabla^2 \phi = 0$.

Step 3: Final conclusion.
Because a solution of Laplace’s equation cannot have interior maxima or minima, it necessarily obeys the maximum–minimum principle.