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Characterization of ellipsoids through an overdetermined boundary value problem of Monge-Ampère type. (English) Zbl 1292.35174

Summary: The study of the optimal constant in an Hessian-type Sobolev inequality leads to a fully nonlinear boundary value problem, overdetermined with non-standard boundary conditions. We show that all the solutions have ellipsoidal symmetry. In the proof we use the maximum principle applied to a suitable auxiliary function in conjunction with an entropy estimate from affine curvature flow.

MSC:

35N25 Overdetermined boundary value problems for PDEs and systems of PDEs
35B50 Maximum principles in context of PDEs
35J60 Nonlinear elliptic equations
35J96 Monge-Ampère equations
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