A note on boundary point principles for partial differential inequalities of elliptic type

Abstract

In this note we consider boundary point principles for partial differential inequalities of elliptic type. First, we highlight the difference between the conditions required to establish classical strong maximum principles and classical boundary point lemmas for second-order linear elliptic partial differential inequalities. We highlight this difference by introducing a singular set in the domain where the coefficients of the partial differential inequality need not be defined and, in a neighborhood of which, can blow-up. Secondly, as a consequence, we establish a comparison-type boundary point lemma for classical elliptic solutions to quasilinear partial differential inequalities. Thirdly, we consider tangency principles, for C^{1} elliptic weak solutions to quasilinear divergence structure partial differential inequalities. We highlight the necessity of certain hypotheses in the aforementioned results via simple examples.