Answer: Unfortunately, it does not sound like this is an area suitable for growing a lawn. Lawn grasses require a bare minimum of half a day of strong direct sun to stay healthy. Without that, they die out.
Often, in addition to the shade issue, tree roots make it difficult for plants to grow beneath them, resulting in bare ground. You could mulch beneath the trees with an organic mulch two to three inches thick. A larger textured mulch such as bark nuggets will be longer lasting than a fine textured one.
Or, you could try to establish a groundcover planting such as Vinca minor, English ivy, hostas or epimedium. Depending on how thick the tree roots are it may be somewhat difficult to plant there. You would need to dig a generous hole between roots for each plant, enrich it with compost, plant, water thoroughly, and mulch between the plants. Then you would need to water each plant for at least the first year or two until it becomes established.
Another option might be to use an attractive gravel and create a seating area beneath the trees using a good looking bench for all season interest along with some large planters filled with shade tolerant annuals such as impatiens during the summer. You could also add stepstones and a birdbath or fountain to make a delightful seating area in the shade beneath the tree.
I hope this gives you some ideas.
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