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.
Brandolini B., Gavitone N., Nitsch C., Trombetti C. (2014). Characterization of ellipsoids through an overdetermined boundary value problem of Monge-Ampère type. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 101(6), 828-841 [10.1016/j.matpur.2013.10.005].
Characterization of ellipsoids through an overdetermined boundary value problem of Monge-Ampère type
Brandolini B.;
2014-01-01
Abstract
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.
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