What do you do with 11,000 tons of garbage per day? That’s the problem — well, one of the problems — plaguing New York City, whose trash disposal system is becoming a political, logistical, and financial headache for the beleaguered metropolis. A $6 billion long-term garbage-management plan devised by the Giuliani administration is stalled and may fail entirely, and a short-term plan to truck waste out of the city is extremely costly. But any move to store the waste within one of the five boroughs — which Giuliani promised not to do — is sure to touch off a firestorm of debate, with concerns likely to be raised about racism and classicism in the siting of the trash. Meanwhile, a plan by the Bloomberg administration to cut the city’s recycling program almost in half for 18 months may be extended indefinitely, much to the dismay of environmentalists. Financial assessments suggest that unless the city’s economy recovers dramatically — or recycling becomes more profitable — the program itself could be trashed.