Belle Reve is the name of the family plantation in their hometown of Laurel. The name is of French origin, and means beautiful dream. This could relate to Blanche and her tendencies to always make up her own fantasies, but in this case it is not made up because even though Blanche had various fantasies, the plantation really existed. It is said that there is a grammatical error in the word because the word Belle is feminine, and the word Reve is masculine, which means that Belle should have been masculine as well. Tennessee Williams perhaps did this on purpose because it reveals the fact that Belle Reve was a dream house that crumbled due to financial issues. It also relates to Blanche's imperfections of creating beautiful dream or fantasies in which included many lies.
The home of Stanley and Stella, which was on the street Elysian Fields.
Elysian Fields is the name of the street that Stanley and Stella lived on. According to Roman Mythology, Elysium (or Elysian Fields), was a part of the underworld and a place of reward for the "dead". Therefore Elysian Fields was a place where souls went in their journey back to life. This could be very much related to Blanche, and her being put into a mental hospital at the end of the book because of all that she's been through in her life, but most importantly what Stanley did to her.
References to Belle Reve and Elysian Fields
"The exterior of a two-story building on a street in New Orleans which is named Elysian Fields..."
"They told me to take a street-car named Desire, and transfer to one called Cemeteries, and ride six blocks and get off at—Elysian Fields!"
"There are thousands of papers, stretching back over hundreds of years, affecting Belle Reve as, piece by piece,..."
"Oh, I guess he’s just not the type that goes for jasmine perfume, but maybe he’s what we need to mix with our blood now that we’ve lost Belle Reve."
"I know, I know. But you are the one that abandoned Belle Reve, not I! I stayed and fought for it, bled for it, almost died for it!"
"Belle Reve? Lost, it is? No!"
"Why, the Grim Reaper had put up his tent on our doorstep!. . .Stella. Belle Reve was his headquarters!"
"I wasn't so good the last two years or so, after Belle Reve had started to slip through my fingers."
"My sister lived at Belle Reve. " "This is after the home-place had slipped right through her lily-white fingers!"