On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

TL;DR

This paper establishes quantitative maximum principles and boundary derivative estimates for solutions to elliptic integro-differential equations with measurable coefficients, extending classical results to nonlocal operators.

Contribution

It provides new quantitative bounds and maximum principles for solutions of elliptic integro-differential equations with non-smooth coefficients, generalizing classical PDE results.

Findings

Quantitative bounds on outward normal derivatives at boundary maxima.

Lower bounds on solutions using principal eigenfunctions.

Proofs of maximum principles and Harnack inequalities for nonlocal operators.

Abstract

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution %on a boundary of a domain in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with $A$ and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.