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Please see this page: The implicit function theorem: A PDE example.

In the implicit function theorem they quote, uniqueness is not mentioned. But the inverse function theorem (which is equivalent to the IMFT), we do get uniqueness. So am I right that the solution got by using the theorem is unique as well?

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  1. Their statement of the Implicit Function theorem does contain a uniqueness part: "$F(x,y)=0$ if and only if".

  2. However, this is a local statement, only for $(x,y)\in U\times V$. It does not exclude the possibility of having other solutions outside of small neighborhood $V$.

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    @hopo2 The neighborhood is taken in the function space, not in the space-time coordinates. So it's not "short-time uniqueness", it's uniqueness among the functions that are close to the particular solution.2012-08-23