Advances in Nonlinear Analysis (Nov 2021)

Maximum principle for higher order operators in general domains

  • Cassani Daniele,
  • Tarsia Antonio

DOI
https://doi.org/10.1515/anona-2021-0210
Journal volume & issue
Vol. 11, no. 1
pp. 655 – 671

Abstract

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We first prove De Giorgi type level estimates for functions in W1,t(Ω), Ω⊂RN$ \Omega\subset{\mathbb R}^N $, with t>N≥2$ t \gt 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 [10] for functions in W1,2(Ω). 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.

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