Yes they can, when they are directed to do so. An extreme example is the “noise abatement departure” from Orange County’s John Wayne Airport. The Airport maintains 10 permanent noise monitoring stations located to the north
and south of the Airport. The local aviation noise ordinance specifies noise limits at each monitoring station that vary by time of day. Multiple noise violations can lead to a denial of airport use for the airline.
Essentially the aircraft will takeoff at full power and immediately executes a very steep climb. Upon reaching quickly 1,000 feet, the pilot will push the nose over and reduce power to a very low power and low noise setting.
Then the aircraft limps along slowly and silently until out over the water. It can be quite an experience for both passengers and pilots alike.