When all else fails—pave it. Just kidding, sort of.
Many folks find that using what landscapers call “hardscape” in areas where grass is difficult to grow is an excellent choice for low maintenance and appearance. Let's face it—what will look better, scraggly grass or a well designed constructed deck or deck and water feature combination? How about a shady, vine-covered gazebo with areas to sit and watch the birds play in the naturalized berry shrubs?
Indeed some form of hardscape will permanently and attractively solve the problem of what to grow and how to grow it in that particularly tough area. Think about it.
One common solution for the area against the trunk of a large tree is to use what is called a “paver” in the trade or a paving stone or cobblestone by consumers. A very attractive pattern can be constructed and set relatively easily under the tree to form a permanent mulch around the trunk.
You won't have to mow or cultivate this area (other than to keep the inevitable weed from becoming established), and it will continue to lend a sense of design to the garden. Your local garden center will have pamphlets that show you this kind of installation and give tips on doing it yourself.
Having said all that, of course grass is simply the best ground cover there is.