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GATE 2016 EC – Question 14

Engineering Mathematics · Differential Equations · 1 mark · Multiple choice

Which one of the following is a property of the solutions to the Laplace equation: $\nabla^2 f = 0$?

  1. The solutions have neither maxima nor minima anywhere except at the boundaries.
  2. The solutions are not separable in the coordinates.
  3. The solutions are not continuous.
  4. The solutions are not dependent on the boundary conditions.

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Show answer and explanation

Correct answer: (A) The solutions have neither maxima nor minima anywhere except at the boundaries.

Explanation

By the maximum principle, a solution of Laplace's equation cannot have a local maximum or minimum inside the region, so its extreme values occur on the boundary. Such solutions are smooth, often separable in suitable coordinates, and fully determined by the boundary conditions.