An electronic device is provided with a display such as a liquid crystal display. The display includes a display module having an array of display pixels and a backlight unit configured to provide backlight to the array of display pixels. The array of display pixels includes display pixels in a central region surrounded by display pixels in a border region. To minimize light leakage from the display, display control circuitry drives the display pixels in the central region according to a first gray level mapping function and drives the display pixels in the border region according to a second gray level mapping function. Light leakage reduction structures may be used to reduce the intensity of backlight received by display pixels in the border region relative to the intensity of backlight received by display pixels in the central region.