What is a nontrivial minimizer?

Question

I came across a statement that x is a nontrivial minimizer of some function, but couldn't find a definition of "nontrivial minimizer" on the Internet. Can anyone help point out some references for that? Thanks.

Answer 1

"Nontrivial" is used by mathematicians to indicate that there's more to said about something than just the obvious (it's kind of hard to define this without seeming like I'm stating the obvious). Typically, $0$ is a trivial solution to many things, so non-trivial solutions are the non-zero ones in that case.

Consider for example trying to find minimisers $u(x)$ for $$\int_\Omega \sqrt{1+|\nabla u|^2}\ dx = \cal{A}(u)$$ for some $u\in C^1(\Omega)\cap C^0(\partial \Omega)$. The trivial solution is $u\equiv 0$ on $\bar\Omega$, but the nontrivial solutions are the ones we're interested in (as, in this case, they are minimal surfaces).

So your nontrivial minimiser is just a minimiser that (probably) isn't identically zero.