Parabolic Green functions in open sets

Abstract

Consider the parabolic operator L defined by LuI = uit{a.ijt, i + dill,i -bit,. -cit in an open set U in En x (0, T). By using the natural abstraction of the notion of a Green function, the author obtains the existence of a unique Green function for Lit = 0 on U. In two preceding papers the author has generalized the notion of superparabolic functions on cylinders and considered nets of superparabolic functions. These results together with the known existence of the fundamental solution in El x (O, T) and the Green function in a cylinder Q = U x (0, T) are used to obtain the Green function for any domain U C En x (0, T). The function G will be called a Green function if it is the minimal, nonnegative, superparabolic function which can be written as a sum of the fundamental solution of Lu = 0 and a superparabolic function. This function will be shown to coincide with the Green function y(z; 4) for the cylindrical domain Q which was obtained by Aronson [1]. For 4((, r) e Q, the fundamental solution F(z; 4) was obtained on (x, t) c El x $[0, r) u (r, T]}. To guarantee that R(; 4) was in S' it was necessary to extend the domain of F so that F = 0 for t = r. Whenever this function is referred to, it will be in this extended sense. Since the results obtained here depend significantly on the author's two preceding papers, definitions, theorems, and lemmas will be numbered sequentially, following those in [2]-[4]. Also, many of the proofs are identical to those for the Laplace operator given in Helms' text [61 and will not be repeated here. Definition 13. Let U be an open subset of En x (0, T) for some T > 0. A Green/unction, if it exists, for U is an extended real valued function Gu on U x U with the following properties: Presented to the Society, January 23, 1975; received by the editors April 3, 1974. AMS (MOS) sitbject classifications (1970). Primary 35K10; Secondary 31B05.