Parabolic equations of the second order

Abstract

from a unified point of view, namely, the extensive use of Green's function. Our main interest is concerned with the first mixed boundary problem (for definition, see ?4) for f linear or nonlinear in u, Harnack theorems, etc. Most of our methods are known. Green's function for the heat equation in rectangular domains was constructed already by E. E. Levi [21] (in particular ?7) in 1907 and was used by him to derive some existence theorems. An extensive use of Green's function for more general parabolic equations was made by Gevrey in his fundamental paper [16] (in particular ??4, 4*, 24, 28, 39, 40, 41). The analogue of the Harnack convergence theorem was first proved for the heat equation in Levi's paper [21, pp. 386-387]. Nonlinear existence problems were considered by Gevrey [16, ?28] by reducing them with the aid of Green's function to integral equations and then applying successive approximations. A more detailed survey on the older literature concerned with Green's function is to be found in the book of Ascoli-BurgattiGiraud [I]. In this paper we use all the above mentioned methods and a few new ones to treat more general problems than those considered in earlier papers. Essential use of Dressel's fundamental solutions for general linear second order parabolic equations [10; 11] enables us to perform this extension. We give below a brief description of our results and their connection to previous papers. In ?1 we construct Green's function for linear second order parabolic equations with smooth coefficients in an (n+1)-dimensional rectangle. In this construction we employ the fundamental solutions constructed by Dressel [11]. Green's function for the heat equation in one dimension was constructed by Levi [21] by reflecting the fundamental solution t-12 exp(_-.2/4t) with respect to the x-variable; our method is an extension of Levi's method.