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I'd like to draw your attention to the First Uniqueness Theorem in context of Laplace's equation. It states that if we know the value of a function $V$ at surface of a volume, then the solution to Laplace's equation is uniquely defined. I don't really understand the reasoning behind this, given that Laplace's equation itself has infinitely many solutions.

I'm aware that there is another post that answers this doubt analytically, but I am unable to grasp the essence of it. I'd really appreciate an explanation that provides an intuitive understanding as I'm trying to understand this in context of electrostatic potentials. Thanks a lot for your help!

GanTheMan
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    You shouldn't say it has "infinite solutions" if you mean that it has infinitely many solutions. To say it has infinite solutions means it has some solutions, each one of which is an infinite solution. To say it has infinitely many solutions is another matter. – Michael Hardy May 08 '20 at 03:25
  • Noted. I've changed it above :) – GanTheMan May 08 '20 at 08:07

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You shouldn't say it has "infinite solutions" if you mean that it has infinitely many solutions. To say it has infinite solutions means it has some solutions, each one of which is an infinite solution. To say it has infinitely many solutions is another matter.

Laplace's equation has infinitely many solutions. But for every assignment of values on the boundary, it has only one solution. That does not mean that if we know those values, there there is only one solution; rather it means that there is only one solution that has those boundary values.

Michael Hardy
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  • Hi Michael, thanks for your response. Why is it that for every assignment of values on the boundary, it has only one solution? – GanTheMan May 08 '20 at 04:35
  • @MathTakingOvertheWorld : The question in your comment above appears to be different from your original question above, and possibly worth posting as a separate question if it has not already appeared. – Michael Hardy May 08 '20 at 16:14