District of Columbia Mayor Marion Barry met yesterday with two of his special constituents -- President Carter and Vice President Mondale -- and received as surances of a close working relationship with the White House in solving several high priority problems in the nation's capital.

"I'm pround of the new administration and we're going to work in close partnership. Isn't that tight?" Carter asked Barry, as the two prepared to begin a 20-minute meeting in the Oval Office at the White House.

"That's right," Barry said with a grin.

The meetings represented the first formal face-to-face contact between the president and the new mayor, who took office Jan. 2. Barry brought his general assistant, Ivanhoe Donaldson, to the oval office session, and six other aides, including city administrator Elijah B. Rogers, to the longer conference with Mondale and other top White House assistants.

But there were no major policy decisions reached, no differences ironed out or commitments made, other than to establish the framework for future relations.

"What I tried to do," Barry said, standing outside the White House at the end of the meeting, "is to indicate that I want the District to be the best, that I'm giving it that kind of direction that won't be an embarrassment to the federal government.

"On the other hand, the president recognized some federal responsivility, so I feel very happy about what happened."

Barry said the meeting laid the foundation for "a new beginning, a new era" in White House-District Building relations.

James Dyke, a special assistant to Mondale, said the sessions were "very productive." Dyke said at one point Carter told Barry that "he considers Marion his mayor because he [Carter] lives here."

Several major issues -- including some on which the city and the Carter administration have disagreed in the past -- were discussed. They included funding for the city's staggering pension dept, completion of the full 100 miles of the Metro subway system and budget autonomy and a fixed federal payment for the District government.

Barry also outlined his administration's concerns about possible transter of St. Elizabeths hospital to the D.C. government, as well as his support for giving the local government aughority to prosecute criminal cases and the power to select its own judges.

Gene Eodenberg, deputy assistant to the president for intergovernmental affairs, said, however, that Carter did more listening than talking. "It was really an opportunity for the mayor to sketch out his concerns," Eidenberg said.

Participants in the meetings agreed that the District would not have a single person to contact at the White House, as has sometimes been the case in the past. District-White House affairs will be initially channeled through the office of Jack H. Watson, the president's principal liaison with state and local governments.

But a special, high-level contact for the District government will also be established in the Office of Management and Budget.

The White House Task Force on District of Columbia Problems, set up under Mondale's leadership in 1977, will not be reactivated, Dyke said.

During much of the two-minute public session that reporters were allowed to attend, Barry and Carter, standing together baside the President's desk, chatted amiably and quietly.

Barry thanked the president for his support of "public education" by enrolling the Carters' 11-year-old daughter Amy in D.C. public schools.

"We have benefited from that, too," Carter said. "It's been good for Amy. It's been good for us, good for public education all over the country."

Carter also nothe that his wife Rosalynn had taken "a very strong interest in the city's hospitals."

"I need that," Barry responded.

Those who attended the meeting from the city government staff, in addition to Donaldson and Rogers, were Barry's five assistant city acministrators -- Judith W. Rogers, Colin F. S. Walters, Carroll B. Harvey, Gladys W. Mack and Jomes O Gibson.

White House aides present, in addition to Dyke and Eidenberg, included OMB Director James T. McIntyre Jr., Stuart Eizenstat, Carter's chief domestic policy adviser and presidential assistant Louis Martin.