Apple is hoping to press its advantage against Samsung Electronics, asking for a court order on a wider U.S. ban on its products and an additional $707 million in damages after scoring a victory over the Korean giant last month.

For the additional request, Apple went back to the same San Jose, Calif., court that ruled last month that Samsung's products copied several key elements of the iPhone, Reuters reported. Samsung, for its part, is seeking a new trial, arguing that there were a number of instructional errors and decisions on evidence that require additional examination.

Apple scored a big victory when the San Jose jury ruled that Samsung had infringed on a number of its key patents and awarded Apple $1.05 billion in damages.

Following the victory, Apple sought to ban the products mentioned during the trial. The company is now seeking to extend that ban to any Samsung products that potentially infringe on its design patents, which could extend to Samsung's current flagship Galaxy S3 .

The case in San Jose was just one of many ongoing trials in the world in a litigation battle that has gone on in multiple courts in multiple countries. Yesterday, a court in Germany ruled that Samsung didn't violate an Apple patent related to how the finger reacts to touch screens.

The latest legal developments come as Apple launched the iPhone 5 on Friday, garnering the usual lines and frenzy for the launch of a new Apple product. Sales estimates for the iPhone 5 on the first weekend alone have gone as high as 10 million. Apple and Samsung are in a tight battle for smartphone supremacy, with Samsung in the lead thanks to the sheer breadth of different phones it makes.

Samsung again warned that if left to stand, the ruling would hurt U.S. consumers and lead to less choice and higher prices down the line. It would also embolden Apple to take other competitors to court, the company said in a statement e-mailed to CNET.

"Indeed, it is unfortunate that patent law can be manipulated to give one company a monopoly over rectangles with rounded corners, or technology that is being improved every day by Samsung and other companies." Samsung said.

Apple has said that it was seeking damages appropriate to Samsung's willful use of Apple patents in its products, which has caused harm to the iPhone maker.

Samsung has already said it would drag the iPhone 5into the legal battle, indicating that there is no likely end to the squabble for a while.