Radial fractional Laplace operators and Hessian inequalities

By Fausto Ferrari, Igor Verbitsky. 2012

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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: $$\int_{\mathbb{R}^n}|(-\Delta)^{\frac{k}{k+1}} u|^{k+1} dx \le C \int_{\mathbb{R}^n} - u \, F_k[u] \, dx, $$ where $F_k$ is the $k$-Hessian operator on $\mathbb{R}^n$, $1\le k < \frac{n}{2}$, 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 \cite{FFV} contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang \cite{W1}. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/386caddd76373dcb5f3fdf619697dbddbdbe60bb

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Optimal results for the fractional heat equation involving the Hardy potential (2014, background): “As in [23], taking into consideration the behavior of K near σ = 1 and at +∞, we can prove that |ΛN,s,γ | <∞.”
Delaunay-type singular solutions for the fractional Yamabe problem (2015, unknown): “Inspired in the computations by Ferrari and Verbitsky in [16], we write s = rs̄, so the radial function v can be expressed as v(r) = (1− s̄−n−2γ 2 )v(r) + s̄−n−2γ 2 v(r).”
On fractional p-Laplacian parabolic problem with general data (2016, methodology): “Following the radial computations in [11] and [12], we get that a self-similar solution u(x, t) = tΥ( r tβ ) with r = |x|, β = 1 ps+N(p−2) , must to solve the following equation β [ NΥ(r) + rΥ(r) ] = 1 rps ∫ ∞ 0 |Υ(r)−Υ(σr)|(Υ(r)−Υ(σr))σKs(σ)dσ where Ks(σ) = ∫”
Entire $s$-harmonic functions are affine (2014, background): “ville theorem holds as well and was proved by K. Bogdan et al. in [4]. Applications of Liouville theorems for nonlocal operators in the study of nonlocal elliptic systems of equations can be found in [1,8,13,14]. The aim of this paper is to classify all s-harmonic functions in RN, thereby obtaining ”
On the fractional p-Laplacian equations with weight and general datum (2016, unknown): “as in [20], we get (23) K(σ) = 2 π N−1 2 β( 2 ) ∫ π”
The effect of the Hardy potential in some Calder\'on-Zygmund properties for the fractional Laplacian (2015, background): “Considering the behavior of D near from 0, 1 and at ∞ (see [22]), we obtain that”
Caffarelli–Kohn–Nirenberg type inequalities of fractional order with applications (2016, methodology): “BENTIFOUR To compute the above integral, we closely follow the arguments used in [17].”