In this paper we deduce a formula for the fractional Laplace operator on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with the fractional laplace operator , and apply it to a problem related to the Hessian inequalityof Sobolev type and the k-Hessian operator on under some restrictions on a k-convex function u. In particular, we show that the class of u for which the above inequality was established in Ferrari et al. [5] contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang (1994) [15]. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest.

F. Ferrari, I. E. Verbitsky (2012). Radial fractional Laplace operators and Hessian inequalities. JOURNAL OF DIFFERENTIAL EQUATIONS, 253, 244-272 [10.1016/j.jde.2012.03.024].

Radial fractional Laplace operators and Hessian inequalities

FERRARI, FAUSTO;
2012

Abstract

In this paper we deduce a formula for the fractional Laplace operator on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with the fractional laplace operator , and apply it to a problem related to the Hessian inequalityof Sobolev type and the k-Hessian operator on under some restrictions on a k-convex function u. In particular, we show that the class of u for which the above inequality was established in Ferrari et al. [5] contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang (1994) [15]. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest.
2012
F. Ferrari, I. E. Verbitsky (2012). Radial fractional Laplace operators and Hessian inequalities. JOURNAL OF DIFFERENTIAL EQUATIONS, 253, 244-272 [10.1016/j.jde.2012.03.024].
F. Ferrari; I. E. Verbitsky
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/117074
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