ABSTRACT
We use a variant of the moving hyperplane method of Alexandroff to obtain radial symmetry for solutions of Δu+ f(x, u, ∇u) = 0 on non-convex, annulus-type domains. For an annulus A we prove a symmetry theorem for boundary conditions u = 0 on the outer sphere, u = a > 0 on the inner sphere and 0 ≤ u ≤ a on A. We then drop the restriction 0 ≤ u ≤ a and consider two arbitrary real numbers as boundary values on the limiting spheres. Using the first result we can in this case prove a characterization of the radially symmetric solutions by their sets of extremal points. We also consider an overdetermined BVP (i.e., constant Neumann and Dirichlet boundary values) on a general ring-domain Ω = https://www.w3.org/1998/Math/MathML"> Ω ¯ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math487.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and prove the radial symmetry of the solution and the domain. In a forthcoming paper a corresponding theorem for an exterior domain will be given.
