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A generalized rotationally symmetric case of the centroaffine Minkowski problem. (English) Zbl 1393.35067

The paper deals with a centroaffine Minkowski problem, that is equivalent to solving the following Monge-Ampère-type equation \[ \det\big(\nabla^2H+HI\big)=\dfrac{f}{H^{n+2}}\quad \text{on}\;S^n, \] where \(f\) is a given positive function, \(H\) is the support function of a bounded convex body \(X\subset\mathbb{R}^{n+1},\) \(I\) is the unit matrix, and \(\nabla^2H=(\nabla_{ij}H)\) stands for the Hessian matrix of covariant derivatives of \(H\) with respect to an orthonormal frame over \(S^n.\)
The author provides two sufficient conditions for existence of generalized rotationally symmetric solutions. The approach relies on the variational structure of the equation and blow-up analysis.

MSC:

35J96 Monge-Ampère equations
35J75 Singular elliptic equations
58J05 Elliptic equations on manifolds, general theory
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