Quasiconcave solutions to elliptic problems in convex rings

We investigate the convexity of level sets of solutions to general elliptic equations in a convex ring. In particular, if u is a classical solution which has constant (distinct) values on the two connected components of the boundary, we consider its quasi-concave envelope v (i.e., the function whose superlevel sets are the convex envelopes of those of u) and we find suitable assumptions which force v to be a subsolution of the equation. If a comparison principle holds, this yields v=u and then u is quasi-concave.