565 research outputs found
Weyl and Marchaud derivatives: a forgotten history
In this paper we recall the contribution given by Hermann Weyl and Andr\'e
Marchaud to the notion of fractional derivative. In addition we discuss some
relationships between the fractional Laplace operator and Marchaud derivative
in the perspective to generalize these objects to different fields of the
mathematics.Comment: arXiv admin note: text overlap with arXiv:1705.00953 by other author
The Soap Bubble Theorem and a -Laplacian overdetermined problem
We consider the -Laplacian equation for , on a
regular bounded domain , with , under
homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap
Bubble Theorem and of Serrin's symmetry result for the overdetermined problems,
we prove that if the mean curvature of is constant, then
is a ball and the unique solution of the Dirichlet -Laplacian
problem is radial. The main tools used are integral identities, the
-function, and the maximum principle.Comment: 18 pages, 0 figure
Radial fractional Laplace operators and Hessian inequalities
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 , and
apply it to a problem related to the Hessian inequality of Sobolev type:
where is the -Hessian
operator on , , under some restrictions on
a -convex function . In particular, we show that the class of 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
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