Radial fractional Laplace operators and Hessian inequalities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:253

Article

Ferrari, Fausto¹  Verbitsky, Igor E.² 

[1] Univ Bologna, Dipartmento Matemat, I-40126 Bologna, Italy

[2] Univ Missouri, Dept Math, Columbia, MO 65211 USA

关键词: Fractional Laplacian;    k-th Hessian operators;    Radially symmetric functions;    Hypergeometric function;    Log-convexity;   

DOI  :  10.1016/j.jde.2012.03.024

来源: Elsevier

【 摘 要 】

In this paper we deduce a formula for the fractional Laplace operator (-Delta)(s) on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with (-Delta)(s), and apply it to a problem related to the Hessian inequality of Sobolev type: integral(Rn) vertical bar(-Delta) (k/k+1 u)vertical bar(k+1) dx <= C integral(Rn) -uFk[u]dx, where F-k is the k-Hessian operator on R-n, 1 <= k < n/2, 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. (C) 2012 Elsevier Inc. All rights reserved.

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