If you're planning on creating a lawn in shady conditions, species selection is critical. Not many turf grasses thrive in shade. Fescues (fine fescue, red fescue, hard fescue) have the highest tolerance and do pretty well. Usually used in areas of low maintenance and dry than average soils, fescues are quite tough. Mix with some of the more shade tolerant Kentucky bluegrass cultivars in areas with moderate amount of shade for best results.
Selective pruning of some of the trees creating the shade will allow more sunlight into the area and increase the vigor of the lawn. Allow the grass to grow to a higher height than your other lawn area. More of the plants foliage exposed will increases the process of photosynthesis. Water somewhat less frequently to decrease the potential for lawn disease. Though some grasses are adaptable to shade ornamental groundcovers are often a better choice.