The Chicago legal services market is keeping busy despite the city’s high unemployment, thanks to a stable base of local corporations, an upswing in private-equity work and business from clients located outside the area.

A 7.1 percent unemployment rate in June, compared with 6.1 percent nationally, is a burden for the area, which includes Joliet and Naperville, Ill. But the region’s varied business base continues to churn out corporate work, and firms like Sidley Austin run national practices that rely ever less on local clients, said Larry Barden, a Chicago corporate partner who chairs Sidley’s management committee.