WASHINGTON — The Supreme Court raised serious doubts Tuesday about a $399 million judgment against smart phone maker Samsung for illegally copying parts of the patented design of Apple’s iPhone.

Justices hearing arguments in the long-running dispute seemed troubled that Samsung was ordered to pay all the profits it earned from 11 phone models, even though the features at issue are just a tiny part of the devices.

But some justices struggled over how exactly a jury should be told to compute damages if the case is sent back to a lower court.

“If I were a juror, I wouldn’t know what to do,” said Justice Anthony Kennedy.

Justice Stephen Breyer appeared to embrace a test proposed by a group of Internet companies including Facebook and Google that would outline new limits on such damage awards. Other justices seemed to favor a different test proposed by the Obama administration.

The outcome could have ripple effects across the high-tech industry as the court balances the need to encourage innovation against a desire to protect lucrative design patents.

The case is part of series of high-stakes lawsuits between the technology rivals that began in 2011. None of the early generation Samsung phones involved in the lawsuit remains on the market.