Convergent nets of parabolic and generalized superparabolic functions

Abstract

The well-known convergence properties of families of harmonic functions are generalized to functions which satisfy Lu = 0 where L is the weak parabolic operator in divergence form. Properties of superharmonic functions are obtained for generalized superparabolic functions. These results are obtained on any bounded domain in En+l Consider the parabolic operator in divergence form given by Lu = ut-a Uij(x, t)Zu, + d 7(x, t)7u}, -bi(X, t)7I, c(x, t)u. In two preceding papers by the author [4], [5], existence, representation, and a maximum principle were obtained for solutions of Lu = 0 in the cylindrical domain Q = Q x (0, T) for Q C En, and generalized superparabolic functions in Q were introduced. In this article the author will consider convergent nets of parabolic and superparabolic functions on a bounded domain U assumed to be in E x (0, T). Since some of the properties obtained in this article will depend on results for superparabolic functions, it is necessary to restate these results for the domain U. The numbering of definitions and theorems in this article continues from those in [5]. Definition 7. Let z = (x, t), w = (y, s) 6 U. z -<w in U if there is a polygonal path ICZ w(a), 0 < a < 1I such that (1) CZ W(0) = iZ}, CZ W(1)= =W}, (2) if CZ,2U(a) = '(fal ra)I9 then a < f3 implies ra K 7,8 (3) ICZ W(a); O < a < II c U. Note that Definitions 4, 5, and 6 can be generalized to the domain U since all properties are in terms of standard rectangles in the given domain. Accordingly, let Su, ', , and j denote the corresponding spaces. The Received by the editors April 3, 1974. AMS (MOS) subject classifications (1970). Primary 35K10; Secondary 31C99.