ON THE REGULARITY OF DOMAINS SATISFYING A UNIFORM HOUR-GLASS CONDITION AND A SHARP VERSION OF THE HOPF-OLEINIK BOUNDARY POINT PRINCIPLE

摘要

We prove that an open, proper, nonempty subset of R~n is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The limiting cases are as follows: Lipschitz domains may be characterized by a uniform double cone condition, and domains of class I~(1,1) may be characterized by a uniform two-sided ball condition. We discuss a sharp generalization of the Hopf-Oleinik boundary point principle for domains satisfying an interior pseudoball condition, for semi-elliptic operators with singular drift and obtain a sharp version of the Hopf strong maximum principle for second order, nondivergence form differential operators with singular drift. Bibliography: 66 titles. Illustrations: 7 figures.