A Generalization of the Flat Cone Condition for Regularity of Solutions of Elliptic Equations

Barrier arguments are used to prove regularity of boundary points for a large class of uniformly elliptic operators when the domain satisfies a geometric condition. The condition is that the exterior of the domain contains a suitable lower dimensional set. Although necessary and sufficient conditions are known for the regularity of boundary points for the Dirichlet problem for a uniformly elliptic equation when the coefficients have appropriate global continuity properties (see, e.g., [1]) or when the equation is in divergence form [9], the conditions for general coefficients are not yet fully understood. Counterexamples show that points regular for the Laplacian need not be regular for other uniformly elliptic operators, and vice versa (see [11]), while the recent work of Bauman [3] characterizing regular points for such equations does not seem to present an easily verified condition. However, various simple sufficient conditions for a boundary point to be regular for the class of uniformly elliptic equations are known. We present here a simple sufficient geometric condition (the exterior generalized flat cone condition) for a boundary point to be regular for any uniformly elliptic equation; this condition is satisfied in many cases when the exterior of the domain is only (n — l)-dimensional. Our results include most of the well-known geometric conditions such as the cone condition [10] and the flat cone condition [4, p. 165] (only discussed for the Laplace operator in that work). On the other hand, the well-known condition A of Ladyzhenskaya and Ural'tseva [6, p. 6] is neither contained in nor contains our condition as we show in the third section of this paper. Another sufficient condition for regularity of boundary points is the Wiener criterion of Landis [7]; however, the relation between that criterion and ours is currently unclear. Also to be mentioned here is the different approach to constructing barrier functions, in some ways perhaps more significant than the result itself. One such construction creates higher dimensional barriers from lower dimensional ones (see the proof of Lemma 2 below). We wish to thank Professor David Gilbarg for pointing out this idea, which, although implicit in most studies of the mixed boundary value problem (see [2, 8], etc.), seems not to have been explicitly used before. Another means of obtaining barriers is by using a simple geometric observation (see the proof of Theorem 1). The plan of this work is as follows. In §1 we define the generalized flat cone condition and provide some preliminary discussion. The sufficiency of this condition is demonstrated in §2. As boundary regularity follows in a standard way from the Received by the editors April 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 31B25, 35B45, 35J67; Secondary 35B65, 35J25. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page