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I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this:

enter image description here

I can understand the proof, but I don't know why this solution which is obtained as a limit of approximations must be a minimal solution ( minimal solution means that this solution at each point is less than or equal with the other solutions).

I need only a hint. Thanks.

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  • $\begingroup$ Note the solution one constructed does NOT depends on $ \tilde{u}$ other than the fact that it exists. So now suppose $ w_0$ is another solution, note by the iteration procedure the constructed solution $v$ satisfies $ v \le w$. So it is THE minimal solution... at least i think this is how that goes... $\endgroup$
    – Math604
    Commented Oct 19, 2015 at 23:18
  • $\begingroup$ @Math604, thanks, I was thinking in this way too, but then I suspected that maybe it needs zorn kemma. $\endgroup$
    – Hheepp
    Commented Oct 20, 2015 at 4:38

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