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

a_ne curvature ow. This is a joint work with B. Brandolini, C. Nitsch and C. Trombetti.