# Min-max formulas for nonlocal elliptic operators on Euclidean space

## TL;DR

This paper revisits the min-max representation of nonlocal elliptic operators satisfying the Global Comparison Property in Euclidean space, clarifying the proof and extending results for operators with translation invariance or spatial regularity.

## Contribution

It simplifies the proof of min-max formulas for nonlocal elliptic operators in Euclidean space and introduces new results for translation-invariant and spatially regular operators.

## Key findings

- Clarified proof of min-max representation in Euclidean space
- Extended results to translation-invariant operators
- Established new properties for spatially regular operators

## Abstract

An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min-max formula in terms of L\'evy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min-max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.