In this paper we study some overdetermined boundary value problems for three classes of fully nonlinear elliptic equations. In each case we prove that the solution exists if and only if the underlying domain is the interior of an ellipsoid (or ellipse in two dimensions). The proofs make use of the uniqueness theorem, maximum principles or some geometric arguments involving the curvatures of the free boundary.
- Free boundary
- Fully nonlinear elliptic equations
- MSC (2010) 35J65
- Overdetermined problems
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