PARABOLIC OPERATORS WITH SINGULAR DRIFT TERMS
7
for some A, 0 A oc, and all rectangles with Q2r(y, s) C Qdipc, t)- Let ||/||
a
(Qd(x,t))
be the infimum of the set of all A such that (2.7) holds.
Next let A= ( ^ ( X , t ) ) , 0 ij n- 1, B = (Bt(X,t)),0 i n - 1, be the
n x n and 1 x n matrices defined in section 1. We assume that A^.Bi : U—+R are
Lebesgue measurable and that A satisfies the standard ellipticity condition,
(2.8) {A{X,t)Z,£) 7i|^|2
for some 71 0, almost every (X, t) G U and all n x 1 matrices £. Here (•, •) denotes
the usual inner product on
Rn.
We also assume that
(2.9) V x0 \Bi\ + y \Aij\ \(X,t) M 00
for almost every (X,t) G U. To simplify matters we shall assume for some large
p 0, that
(2.10) A = constant matrix in £/ \ Qp(0,0).
Following Aronsson [A] we say that u is a weak solution to (1.1) in U if for U
{(XQ, X) : x £ Rn~l, XQ 0}, —00 T 00, we have
(2.11)
ueL2(-T,T,H\oc (U))nL~(-T,T,L2loc
(U))
and
(2.12) /
Ju
n-1
/
J
A{jUXj pXi
ij=0
n-1
-
^2
B*u^
i=0
ujt+
j
AijUXi (\)x% - } Bi ux% (j) dXdt = 0
forall0GCo°°(/7).
In the sequel we shall identify dll with Rn. The continuous Dirichlet problem
for U can be stated as follows: Given g : Rn^R, continuous, and bounded, find u
a bounded weak solution to (1.1) in U with u continuous on U and u = g on dU.
Assume that the Dirichlet problem for a given A, B always has a unique solution.
Under this asumption we define parabolic measure u at (X, t) G U of the Borel
measurable set E C
Rn
by
UJ(X, t, E) - inf {v(X, ^-.vef}
where T denotes the family of all nonnegative solutions to the Dirichlet problem
in U with v 1 on E. Finally let j ^ denote the Radon-Nikodym derivative of uo
with respect to Lebesgue measure on
Rn.
With this notation we are now ready to
state the main theorem in chapter I.
Theorem 2.13. Let A,B, satisfy (2.1)-(2.5) and (2.8)-(2.11). Suppose for some
eo 0 and AQ an n x n matrix that
l l k o 5 | | | i o o
( L 7 )
+ | | | A - A o | | | i c c
( t 7 )
+ ||/il|| + ||/i2|| + ||/i3|| 4
If eo 0 is small enough, then the continuous Dirichlet problem corresponding to
(1.1), A,B, always has a unique solution. If uo denotes the corresponding parabolic
measure, then uj(d,x^t -f 2d2,-) is mutually absolutely continuous with respect to
Lebesgue measure on Qd(x,t) and ^ ^ ( d , x, t + 2d2, •) G @2(Qd(x^)) with
\\^-(d,x,t +
2d2,-)y2{QdiX)t))c*™,
Previous Page Next Page