On the strong maximum principle

ABSTRACT

In this paper we study the strong maximum principle for equations of the form $\mathcal{F}\left[u\right]=H(u,|Du|)$ where $\mathcal{F}$ is either a fully nonlinear elliptic operator or is the p-Laplace operator. We give sufficient conditions on H to ensure that the strong maximum principle (SMP) holds. The condition is also necessary for SMP to hold for the the equation $\mathcal{F}\left[u\right]=g\left(|Du|\right).$

KEYWORDS:

• Strong maximum principle
• fully nonlinear elliptic equation
• p-Laplacian
• viscosity solutions

AMS SUBJECT CLASSIFICATIONS:

• 35J60
• 35J70
• 35B09
• 35B50

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Except for Lemma 2.2 taken from [15].

Additional information

Funding

This work was supported by Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni.