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A singularly perturbed free boundary problem arising from a real problemassociated with a Radiographic Integrated Test Stand concerns a solution of theequation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundarydata, where the function $f$ in general is not monotone. When the domain$\Omega$ is a perfect ring, we incorporate a new idea of radial correction intothe classical moving plane method to prove the radial symmetry of a solution.When the domain is slightly shifted from a ring, we establish the stability ofthe solution by showing the approximate radial symmetry of the free boundaryand the solution. For this purpose, we complete the proof via an evolutionarypoint of view, as an elliptic comparison principle is false, nevertheless aparabolic one holds.

Symmetry and Approximate Symmetry of a Nonlinear Elliptic Problem over a Ring