A H\"older Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians

TL;DR

This paper investigates the asymptotic behavior of solutions to a family of fractional differential problems, including the fractional p-Laplacian as p approaches infinity, leading to a new H"older infinity Laplacian in the limit.

Contribution

It introduces a new limit operator, the H"older infinity Laplacian, as the asymptotic limit of Orlicz fractional Laplacians, extending previous p-Laplacian results.

Findings

Limit equation involves the H"older infinity Laplacian.

Solutions to fractional problems converge to the limit involving the H"older infinity Laplacian.

Generalizes the fractional p-Laplacian to a broader class of operators.

Abstract

This paper concerns with the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $p_n$-Laplacian when $p_n\to\infty$ as a particular case, tough it could be extended to a function of the H\"older quotient of order $s$, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the H\"older infinity Laplacian.