Symmetry and related properties via the maximum principle

We prove symmetry, and some related properties, of positive solutions of second order elliptic equations. Our methods employ various forms of the maximum principle, and a device of moving parallel planes to a critical position, and then showing that the solution is symmetric about the limiting plane. We treat solutions in bounded domains and in the entire space.Supported in part by the National Science Foundation, grant no. PHY-78-08066Partially supported by the U.S. Army Research Office, grant no. DAA 29-78-G-0127