The trace fractional Laplacian and the mid-range fractional Laplacian
Julio D. Rossi, Jorge Ruiz-Cases
TL;DR
This work introduces two nonlinear nonlocal operators, the trace fractional Laplacian $(-\Delta)^s_{tr}$ and the mid-range fractional Laplacian $(-\Delta)^s_{mid}$, defined via fractional eigenvalues $\Lambda_i^s u$ and directional 1D fractional derivatives. It develops a viscosity-solution framework, proves a comparison principle, establishes existence/uniqueness for Dirichlet problems, and shows boundary $C^\gamma$ regularity with Hölder exterior data, plus a rigorous $s\nearrow 1$ limit to local Hessian-based problems. Specifically, $(-\Delta)^s_{tr}$ converges to the classical Laplacian $\Delta u$ while $(-\Delta)^s_{mid}$ tends to $\lambda_1(D^2u)+\lambda_N(D^2u)$ (which equals $\Delta u$ in $N=2$). The paper also proves the nonlinearity of these operators and outlines natural extensions to inhomogeneous terms and coefficient-structured variants, broadening the nonlocal PDE landscape with connections to classical local theory.
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 $C^γ$ smooth up to the boundary when the exterior datum is also Hölder continuous. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s\nearrow 1$.