LG has a good 24 hour support line.
1-888-542-2623
I know, because the idiots at Rogers tried to tell me that it was our TV and not their Nextbox that was the problem 2 nights ago. (it wasn't - of course - but they gave me 6 months of HBO/Crave for free for my trouble when I called Rogers back again. PVR needs replacing - again. 3rd time in 2 years - losing all the recordings, of course
)
Bottom line: the monitor isn't receiving a signal.
Have you tried a different cable? Do you have another monitor handy, or can you borrow one, to make sure it's the monitor and not the computer? Because it's also possible that it's the Mac Mini video output not working - and you can't know that unless you test somehow.
Another alternative is to the monitor to a friend or neighbour to try on their computer (PC or Mac - doesn't matter if it has DVI or VGA, at least you can rule out the monitor, though if you can't test the DVI, then you can't rule that out). Unfortunately, you'd need an adapter (DVI or VGA to USB 2) to test the monitor yourself on your other two Macs, as neither have suitable ports, unless the MBP is a lot older than I think it is. And then there is always the potential dodgy adapter issue; it's best to avoid adding another possible cause into the mix!