The trace fractional Laplacian and the mid-range fractional Laplacian

TL;DR

This paper introduces two novel fractional Laplacian operators based on eigenvalues and mid-range means, establishing their fundamental properties, regularity, and connection to the classical Laplacian.

Contribution

The paper defines two new fractional Laplacians, proves comparison principles, existence, uniqueness, and regularity results, and links the first operator to the classical Laplacian as s approaches 1.

Findings

Established comparison principles for the new operators

Proved existence and uniqueness of solutions to the Dirichlet problem

Showed boundary regularity and convergence to classical Laplacian as s approaches 1

Abstract

In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem. We also show that solutions are $C^\gamma$ smooth up to the boundary when the exterior datum is also H\"older continuous. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s\nearrow 1$.