GATE EC 2016 Set 1 — Question 14
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Engineering Mathematics → Differential Equations → Heat, Wave & Laplace Equations
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Question
Which one of the following is a property of the solutions to the Laplace equation: ?
Correct answer
(A) The solutions have neither maxima nor minima anywhere except at the boundaries.
Solution
Solutions to the Laplace equation are known as harmonic functions. A fundamental property of harmonic functions is the Maximum Principle, which states that a non-constant harmonic function cannot attain its maximum or minimum in the interior of its domain; these extrema must occur on the boundary. This is a direct consequence of the mean value property of harmonic functions.
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