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
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