Lebanon's new President Amin Gemayel, in a visit to the United States this week, will be seeking American and U.N. support, both military and political, for his fledgling government, whose authority still barely extends to the city limits of the capital.

Today, Gemayel will address the United Nations, where he is expected to ask for a continuation of the U.N. peace-keeping force's mandate in southern Lebanon. Tuesday, he will meet with President Reagan at the White House to plead for a wider role for the U.S., French and Italian troops stationed here as part of another multinational peace-keeping body.

Gemayel arrived in New York Sunday afternoon, Reuter reported.

Catapulted into power by the assassination of his brother, Bashir, last month, the 40-year-old Lebanese leader faces a formidable task in trying to put the pieces of his war-devastated nation back together and reestablish central authority over a land still occupied by more than 100,000 Israeli, Syrian and Palestinian troops.

Even here in Beirut, Gemayel's authority depends largely on the presence of the multinational peace-keeping force, whose 4,000 troops have made the capital a kind of artifical oasis of peace that the government speaks of hopefully as a model of the "new Lebanon."

Under its umbrella, Gemayel's two-week-old government has reunified the capital, long divided between its warring Moslem and Christian sectors, cleaned its streets of mountains of war rubble, largely disarmed the roughly 40 militia groups that once reigned over West Beirut, and reestablished the presence of the regular Lebanese Army and police in most of the city.

But the final test of Gemayel's authority in the capital probably will await his return late this week from his visits to the United States, France and Italy. This involves imposing the Army's will in Christian East Beirut, where the Phalangist private militia, the Lebanese Forces has ruled for seven years.

The Lebanese Army resumed security sweeps yesterday in slum districts of West Beirut after a two-day pause, hunting people who allegedly assaulted troops at a protest Friday in which soldiers fired at the crowd, killing at least one person and possibly as many as five, according to differing accounts.

Army patrols went door to door in the Ouzzai and Burj al Barajinah neighborhoods on the southern edge of West Beirut, and an Army statement said 11 persons were arrested and many weapons seized. The government has been clearing what it calls illegal squatters' shanties in the section, near Beirut airport.

Before leaving for New York, Gemayel presided over an early-morning meeting with leaders of the Druze community and Christian Phalangist militias. The conference agreed on proposals announced by Prime Minister Shafiq Wazzan aimed at establishing an effective truce in the mountain villages 15 miles southeast of the capital where the two sides clashed for four days last week until Israeli troops halted the fighting.

Wazzan said the Lebanese government would prefer to have its Army go into the mountains to keep the peace, and local Druze officials said they would welcome this. But Israel and Lebanon have not agreed yet on a transfer of control.

Optimists are few in this land of seemingly perpetual strife. About the closest to one among West Beirut's Moslem leaders is former prime minister Saeb Salam, 77, who heads the Islamic group that backed Amin Gemayel's candidacy for the president.

He noted that Gemayel, a Christian, has begun his six-year tenure with generally "good support" from both the Moslem and Christian populations and added that he regards him as a "seasoned politician."

Above all, he praises Amin Gemayel for having distanced himself from his younger brother, whose rise to power before his death in a bomb blast Sept. 14 was accomplished over the bodies of many rivals.

Amin Gemayel "has never been involved in violence," Salam said in an interview. "On the contrary, he denounced violence all the way through his career."

The only really optimistic note being struck here, however, about Gemayel's chances of success comes from his own circle of aides and ministers, who represent a new breed of decision-makers on a political landscape long dominated by sectarian warlords and patriarchs.

Aside from Prime Minister Wazzan, none of the 10 ministers in his reduced Cabinet is a member of parliament or really even a politician.

Gemayel's lieutenants speak enthusiatically of the "new mood" and "new spirit" afoot in Lebanon with his ascendency to power.

But the high hopes and optimism seemed somewhat out of place as Salam spoke recently in the light of candles and a gas lamp because of another power blackout in the capital due this time to renewed fighting between Christian and Druze militias in the nearby mountains.

At the same time, only a short distance away, Israeli armored personnel carriers were rumbling through the streets.

The blackout and continuing Israeli military presence on the outskirts of the capital were reminders of some of the problems facing Gemayel in his struggle to reestablish central authority.

Gemayel is counting heavily on U.S. diplomacy to roll back the Israeli and Syrian presence and on U.S. military aid to build up the Lebanese Army to the point where it can fill the vacuum. The 24,000-man Army has just completed its first on-the-ground exercise since the civil war here in 1975-76, which left it hopelessly weak and divided.

With a massive display of tanks, personnel carriers and troops, the Army moved into Moslem West Beirut in early October and carried out for two weeks a house-by-house search for guns, munitions depots, criminals and illegal aliens.

In the process, the Christian-officered Army's rough style and treatment of detainees, notably Palestinians, raised new fears among Moslems and the deep concern of the French, Italians and Americans.

The question being raised by Western embassies and analysts was whether the operation might trigger a backlash discrediting the impartiality of the Army among Moslems--particularly Palestinians.

At midweek, Gemayel rushed to the defense of the Army, praising it for its work and saying a strong military was essential to restoring government legitimacy.

"Today, we are in dire need of a powerful Army," he said, "so that we may dispense with borrowed security and get rid of all the chaos . . . ."