Mayor Coleman A. Young, who is up for re-election Tuesday, came face to face with his opponent for the first time in the campaign last Thursday morning on the steps of a local library.

"Are you ready to debate?" asked Perry Koslowski. "I haven't got time for this," Young snorted. "Get yourself a platform."

In Minneapolis, Mayor Don Fraser has been debating his four opponents regularly this fall, but they haven't found much on the voters' minds.

"The issues have not grabbed people," Fraser said as he knocked on doors on a recent snowy Saturday morning.

His 90-minute tour through one neighborhood bore out his comment. The longest discussion he had that morning was about hardening of the arteries. "Our biggest opponent is apathy," the mayor said.

Apathy is also an issue in Cleveland, where Mayor George Voinovich, a Republican in a city that is 6-to-1 Democratic, is trying to pretend he is in a serious re-election contest Tuesday.

"As long as there are ballots to be counted, it's a race," he deadpanned the other day in his cavernous city hall office.

When the votes were counted in the nonpartisan primary in September, Voinovich led his Democratic opponent with 68 percent of the vote.

In what may be the most stressful period in local government in decades, the elections in these three cities reflect a puzzling public mood. At a time when city services are being cut because of federal and state budget reductions, when recession has crippled local economies, and when the Midwest continues to lose population to the Southwest and West, there is no debate going on over the future of these cities.

The incumbents, who ordinarily might be considered vulnerable because of their constituents' woes, are frontrunners and not just frontrunners--they are lengths ahead and enormously popular.

Voters in these cities not only appear satisfied with the leadership they are getting, but also do not hold their mayors responsible for what problems exist. "If an individual is doing a halfway decent job, he's a shoo-in," said Bob Hughes, the Republican county chairman in Cleveland. "I think people recognize how difficult it is to be mayor of a large city."

Arthur Naftalin, the former mayor of Minneapolis, put it somewhat differently. "People don't think that any bold new change is going to make a lot of difference," he said. "Challengers are having trouble getting visibility. It's a time of great confusion and perplexity for everyone."

There may be another reason for the lack of interest in the elections this fall. So far, the federal and state budget cuts have been felt only by city officials, who are struggling to make ends meet. The people have yet to feel the pain.

As one person said in Minneapolis, "The ice rinks haven't been flooded yet, so people don't know which ones won't be flooded this year."

These cities all have serious problems. Cleveland has struggled out of bankruptcy but faces continued population loss and deteriorating services. Detroit, with 14.1 percent of its workforce unemployed and the automobile industry still in a recession, is struggling to keep its budget in balance, even though a sharp tax increase was approved by voters last June. Minneapolis, the healthiest of the three cities, faces a reduction in services because of a state budgetary crisis and further cuts in the federal budget.

Minnesota faces a new deficit that could go as high as $1 billion. Michigan's three-week-old budget just underwent a $270 million cut because of the recession in the automobile industry. Ohio is operating on an interim budget and expects to spend 10 percent less than last year.

Fraser told a neighborhood group last Monday that the city "would be devastated" by the state's newest fiscal crisis.

Young said in an interview Thursday that Detroit has few options left for handling its financial problems. "At some point you run out of fat, and you're dealing with muscle and gristle and finally bone," he said. "And we're down there."

Voinovich is carrying on a public campaign against the latest federal budget cuts proposed by his fellow Republican, President Reagan. "I hope the Republicans know that going another step in cutting the budget is suicide, political suicide," he said.

Meanwhile, taxes are going up.

Young put his considerable prestige on the line in June to push through a $94 million tax increase as part of a financial survival plan that also included wage freezes for city workers and the purchase of $113 million in city bonds by local banks and city employe pension funds.

Voinovich won a $35 million tax hike from the voters of Cleveland last February as part of his plan to pay off the city's debts and keep its budget in balance.

In Minneapolis, where dependence on property taxes has declined dramatically over the last 10 years, residents now face property tax increases of more than 10 percent because state aid is being reduced.

Given conditions in the three cities, the re-election campaigns for Young, Fraser and Voinovich are textbook examples of political survival. The men are markedly different, but each in a curious way reflects the character of his city, and each has adapted himself to the city's particular needs.

Young is the flamboyant black mayor of the gritty, predominantly black Motor City, a man who cradles a $20 bill in his hands during an interview, whose posters do not even bear his name but only the slogan "The man who moves Detroit" and whose office passes out an interview with the mayor in the format of the Playboy Interview.

Voinovich is an ethnic in a heavily ethnic city, a man who lowered the temperature of the city in the aftermath of the chaotic reign of Dennis Kucinich and earned the nickname of Mr. Tranquility.

Fraser, who served in the U.S. House of Representatives for 16 years, is a liberal's liberal, quiet and low-key in a city of Scandinavians that has prided itself on clean, progressive government but which has had several stormy mayoral elections in the past decade.

Each man is familiar to the voters and has a solid record of accomplishment, achieved in part by bringing disparate elements of the community together.

Young is the master of coalition politics, the glue that holds together the business-labor partnership that is responsible for moving Detroit forward in the face of severe problems. He is loved in the black community, where he is known simply as Coleman, and he has solid support among the corporate elite.

Among his fund-raisers this fall was a $1,000-a-head cocktail party thrown by Henry Ford II. Said Thomas Murphy, former chairman of General Motors, "This city has problems, but it would have been worse off had Coleman Young not been the political leader..."

Voinovich not only has the support of the Republican corporate community, but also in two years as mayor has won over black voters--he got two-thirds of the black vote in the primary--and many Democrats, especially by separating himself from Reagan on the budget cuts.

"Blacks voted for the mayor because the mayor had answered the needs of the black community," said George Forbes, the black city council president.

Neither Forbes nor Rep. Louis Stokes, the two leading black Democrats in town, has endorsed Voinovich's Democratic opponent. And not one of the 33 Democrats on the city council is actively campaigning against him.

"Voinovich assumed the leadership in Cleveland at a time it was drifting," Forbes said. "We were in default, we couldn't sell our bonds, we didn't have any money to run our city with. He brought in a crew of young guys who've worked to turn it around, and I think the city remembers."

Fraser's record of accomplishment is less, if only because Minneapolis had fewer problems when he was elected in 1979. But he has carried forward the traditions of joint corporate-city development, has appointed a number of task forces to study some of the city's most pressing problems, and has worked not only to make peace with a jealous and powerful city council, but also to push through several reforms that could make the city more efficient in the future. With a 4 percent unemployment rate in Minneapolis and the expectation of 25,000 to 40,000 new jobs in its downtown over the next decade, Fraser's problems are of a different order of magnitude than Young's or Voinovich's.

The three men share two other things. Each is enormously popular with the local press, and each is outspending his principal opponent by a wide margin.

Fraser will spend about $80,000, compared to $20,000 for Mike Barros, his independent-Republican opponent. Voinovich had spent $208,500 as of Oct. 15, compared to $23,700 for Patrick Sweeney, a Democratic state representative. Young has a $1 million campaign war chest and plans to spend about $200,000 of it in his campaign against Koslowski, who will spend no more than $10,000.

What does Young do with $1 million? "That's like asking Chase Manhattan what it does with more money," he said. "You campaign. You play politics."

Heading into Tuesday vote, the three mayors seem to have all the political capital they need for playing politics.