CHAPTER I

THE DIRICHLET PROBLEM AN D PARABOLIC MEASURE

1. INTRODUCTION

The study of parabolic pde's has a long history and closely parallels the study

of elliptic pde's. To mention a few highlights, the modern theory of weak solu-

tions of elliptic and parabolic pde's in divergence form was developed in the late

1950's and early 1960's by Nash [N], DiGiorgi [DG], Moser [M], [Ml], and oth-

ers. These authors obtained interior estimates (boundedness, Harnack's inequality,

Holder continuity) for weak solutions which initially were assumed to lie only in a

certain Sobolev space and satisfy a certain integral identity. The classical problem

of whether solutions to Laplace's equation in Lipschitz domains had nontangential

limits almost everywhere with respect to surface area and the corresponding LP

Dirichlet problem was not resolved until the late 70's when Dahlberg [D] showed

that in a Lipschitz domain harmonic measure and surface measure, da, are mutually

absolutely continuous, and furthermore, that the Dirichlet problem is solvable with

data in

L2(da).

R. Hunt proposed the problem of finding an analogue of Dahlberg's

result for the heat equation in domains whose boundaries are given locally as graphs

of functions ij){x,t) which are Lipschitz in the space variable. It was conjectured at

one time that I/J should be Lipi in the time variable, but subsequent counterexam-

ples of Kaufmann and Wu [KW] showed that this condition does not suffice. Lewis

and Murray [LM], made significant progress toward a solution of Hunt's question,

by establishing mutual absolute continuity of caloric measure and a certain para-

bolic analogue of surface measure in the case that ip has \ of a time derivative in

BMO(Rn)

on rectangles, a condition only slightly stronger than Lipi. Further-

more these authors obtained solvability of the Dirichlet problem with data in

Lp,

for p sufficiently large, but unspecified. Hofmann and Lewis [HL] obtained, among

other results, the direct analogue of Dahlberg's theorem (i.e,

L2

solvability of the

Dirichlet problem for the heat equation) in graph domains of the type considered

by [LM] but only under the assumption that the above BMO norm was sufficiently

small. They also provided examples to show that this smallness assumption was

necessary for

L2

solvability of the Dirichlet problem.

In this memoir we study the Dirichlet problem and absolute continuity of para-

bolic measure for weak solutions to parabolic pde's of the form,

(1.1) Lu = ut-V- {AVu) - BVu = 0.

Here A — {Aij(X,t)),B — (Bi(X,t)) are n by n and 1 by n matrices, respectively,

satisfying standard ellipticity conditions with X — (xo,xi,..., x

n

_i) = (xo,x) G

Rn, t G R. Also \7u denotes the gradient of u in the space variable X only, written

as an n by 1 matrix, while V- denotes divergence in the space variable. This problem

in the elliptic case has been studied in [JK], [FJK], [Dl] and [FKP]. As a starting

point for these investigations we note that Jerison and Kenig in [JK] gave another

proof of Dahlberg's results (mentioned above). To outline their proof let

0Research

of both authors was supported in part by NSF grants

0Received

by the editor June 16, 1997