832 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
Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator
We study the asymptotics of solutions of logistic type equations with
fractional Laplacian as time goes to infinity and as the exponent in nonlinear
part goes to infinity. We prove strong convergence of solutions in the energy
space and uniform convergence to the solution of an obstacle problem. As a
by-product, we also prove the cut-off property for eigenvalues of the Dirichlet
fractional Laplace operator perturbed by exploding potentials
On higher order extensions for the fractional Laplacian
The technique of Caffarelli and Silvestre, characterizing the fractional
Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an
elliptic equation in the upper half space with one extra spatial dimension, is
shown to hold for general positive, non-integer orders of the fractional
Laplace operator, by showing an equivalence between the H^s norm on the
boundary and a suitable higher-order seminorm of U
Optimal rearrangement problem and normalized obstacle problem in the fractional setting
We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) s , 0 0} , which happens to be the fractional analogue of the normalized obstacle problem ∆u = χ{u>0}
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