Maximum Principle for Higher Order Operators in General Domains

TL;DR

This paper extends maximum principle results to higher order elliptic operators in general domains by establishing new integrability and Harnack inequalities, enabling the proof of strong maximum principles beyond classical second-order cases.

Contribution

It introduces a novel approach to higher order operators by proving De Giorgi type estimates and a new Harnack inequality, broadening the scope of maximum principles in PDE theory.

Findings

Established De Giorgi type level estimates for functions in $W^{1,t}( abla)$ with $t>N$.

Proved a new Harnack inequality for functions outside classical De Giorgi classes.

Validated the strong maximum principle for uniformly elliptic operators of any even order in general domains.

Abstract

We first prove De Giorgi type level estimates for functions in $W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi's classes as obtained in [Di Benedetto--Trudinger, AIHP (1984)] for functions in $W^{1,2}$. As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.