I think that Mistborn pulled it off rather well: we start out with a look at the Final Empire, and are given a pretty good idea that things for the underclass are really, really bad. It probably helps that hope is introduced fairly soon afterwards, so perhaps that is your avenue: don't let the narrative run for too long before you introduce some element of hope or other positive perspective.
My Games & Writing
So what basically happens in the first part is: a man dies and is mistaken for someone else, (this has to be at least gruesome to the point that it would make sense why the body was mistaken)a family member learns about this, becomes extremely depressed and this contributes to their death. Someone else dies as well (unreleated to previous actions), but at the end of the first part, the previously mentioned character's relatives discover that they are alive and they are all reunited (except for the people who are dead obviously) Does this sound relatively okay?
I know that a Downer Beginning can be pulled off quite well, but usually in cases were the Angst is happening 'offstage.' The beginning I'm working on my story involves some explicitly depressing stuff, but is quickly resolved and the majority of the rest of the book is idealistic. I want to avoid Audience Induced Apathy, could this be a problem?