On this page of the guide to Lost Judgment you'll learn what the Gauntlet is and how to unlock additional content in the game.
The Gauntlet is a set of additional challenges in Lost Judgment which let you test your skills.
They can be accessed from the main menu in The Gauntlet tab. You unlock new challenges by progressing through the game's storyline.
Completing the challenges rewards you with valuable items you can redeem from your mailbox in the Yagami Detective Agency or the Yokohama 99 office.