Min–Max formulas for nonlocal elliptic operators on Euclidean Space. (April 2020)

• Record Type: Journal Article
• Authors: Guillen, Nestor; Schwab, Russell W.
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: 35J99 -- 35R09 -- 45K05 -- 46T99 -- 47G20 -- 49L25 -- 49N70 -- 60J75 -- 93E20; Global comparison principle -- Integro-differential operators -- Isaacs equation -- Whitney extension -- Dirichlet-to-Neumann -- Fully nonlinear equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.02.021
• Languages: English
• ISSNs: 0362-546X

Abstract

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 Lévy 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.