Using a new approach due to F. Brock called the Steiner symmetrization, we show first that if
is a solution of an overdetermined problem in the divergence form satisfying the Neumann and non-constant Dirichlet boundary conditions, then is an N-ball. In addition, we show that we can relax the condition on the value of the Dirichlet boundary condition in the case of superharmonicity. Finally, we give an application to positive solutions of some semilinear elliptic problems in symmetric domains for the divergence case.
Submitted October 1, 1999. Published June 12, 2000.
Math Subject Classifications: 28D10, 35B05, 35B50, 35J25, 35J60, 35J65.
Key Words: Moving plane method, Steiner Symmetrization, Overdetermined problems, Local Symmetry.
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