Fractional truncated Laplacians: representation formula, fundamental solutions and applications

TL;DR

This paper introduces nonlinear extremal nonlocal operators approximating truncated Laplacians, constructs their fundamental solutions, and explores their unique properties and applications, revealing significant differences from local and diffused nonlocal cases.

Contribution

The paper defines new nonlocal operators approximating truncated Laplacians, develops their representation formulas, and analyzes their fundamental solutions and applications.

Findings

Constructed representation formulas for the new operators

Established Liouville type results for these operators

Discovered significant differences from local and diffused nonlocal cases

Abstract

In this note we introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called "fundamental solutions". This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to "understand" what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differs significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian.