I've had the trailing (purple) kind of lantana under a live oak for the past 5 years...it's done well and spread nicely. Since I limbed up the tree and there's more light under it, the lantana is blooming better. It dies back if there's much cold weather, but hasn't the past couple of winters.

The best thing about it is I never have to water it.

Lantana might be too scruffy looking, though, if you're visualizing something dense and dark green like Asian jasmine.

From what I have seen here....Asian jasmine could care less about edging and will creep, if not leap, out of bounds......

I have ferns under my large tree--a variety similar to Kimberly Queen.

I wouldn't consider Asian jasmine well behaved. One of the mondo grasses might work - or a liriope. Just be sure your edging goes down into the ground a little, as they will try to go under, but I don't find them invasive, especially in heavy shade.
Neil Sperry always recommends Purple Wintercreeper Euonomous (sp?). I've not used it, but I usually trust what he says.
Ferns would be another good choice. I like the Japanese Painted Ferns. Usually takes them a while to fill in, and they do die down to the ground in the winter, but I really love mine.

I see a lot of Aspidistra used under oak trees. It's been aptly called 'cast iron plant' because it grows and stays evergreen under almost any shady conditions It takes a while to get a good colony established, but it's beautiful when a good stand gets going or is nice with just a few clumps here and there until it fills in.

Here is a link that might be useful: Aspidistra

Here's a photo showing a nice combo of several of the plants mentioned above and then some; aspidistra, mondo grass, holly ferns, giant lirope, aralia. It was taken at Los Patios in San Antonio.

Here is a link that might be useful: Aralia ...

I've thought about planting mondo grass but it's such a large area. Directly under the tree we have Japanese ferns that are doing great. Out from there is the problem area. I cringe at planting Asian jasmine. Even lirope is a monster at spreading.

I've found ground morning glory (Convolvulus sabatius) to be very well behaved. Unlike regular morning glory it doesn't reseed and it's easy to control. The bluish-purple flowers are quite nice too.

I got mine at Northaven Gardens.

Here is a link that might be useful: Ground morning glory

This is very well behaved and has electic blue flowers.

Here is a link that might be useful: leadwort

I have to second the leadwort, it's pretty and if it does grow over your border, it's easy to pull up. Goes a nice reddish purple when it gets cold.

Pam

Hi - I think artemesia ''limelight'' is wonderful!
Goodluck - tina_2

This plant, Bell Harbor Dwarf Nandina, was mentioned on another thread and I thought it might be of interest to those seeking a well behaved ground cover.

Although I have not grown this variety I am a fan of Nandinas. They are easy to grow, offer a variety of year round color and are very full and bushy when given just a little water, but will live with no irrigation.

Here is a link that might be useful: Bell Harbor Dwarf Nandina