Some Zaremba-Hopf-Oleinik Boundary Comparison Principles at Characteristic Points

Abstract

We investigate the so-called Hopf lemma for certain degenerate-elliptic equations at characteristic boundary points of bounded open sets. For such equations, the validity of the Hopf lemma is related to the fact that the boundary of the open set reflects the underlying geometry of the specific operator. We present here some recent results obtained in [21] in collaboration with V. Martino. Our main focus is on conditions on the boundary which are stable by changing our operators in some particular classes, for example in the class of horizontally elliptic operators in non-divergence form. We also study what happens to these conditions for degenerate operators with first order terms.

Keywords

• Hopf lemma
• degenerate elliptic operators