1. PRELIMINARIES 5
Let
\a : m\ = \
where \a: m\ = al/ml H
\~an/mn
^
e t
'
i e
^-principal part of the operator
^4, and let
.4(x;0 = £ v
H
C
a
be its symbol.
We consider the following Cauchy problem:
(1.2.1) Au = f on £1,
{d/dn)ju
= gj, j m{-l onT,
DaueL2{Q)
if |a: m| 1.
Here T is a part of d£l, T e
CWl
, and T is open in dQ.
We assume the coefficients a are bounded and measurable and the coef-
ficients of the m-principal part of A belong to C (Q).
Let cp
eC2(Q),
y = 0 on dQ.\T, Vqcp ^ 0 on Q, 0 p on ft.
THEOREM
1.2.1. Suppose either a) ^
m
(x ; £) ^ 0 /or all ^ e R " \ {0} or
b) the coefficients of Am are real-valued.
If from the equalities
^
m
(x;C) = 0, C = £ + I T V ^ , T ^ O ,
or
it follows that
(1.2.2) £ {DjfyvidAJdCjHdAJdCJ + r '
1
Im DkAm{dAjaHk)) 0
ybr a/ry X G H //zeft
i) //zere w a constant C such that
(1.2.3)
T
2 W
'
( 1 H Q
: m|)
/"
\Dau\2e2x*
dx Cx f
\Aufe2xf
etc
Jn Jo.
for C x and for all functions u from
H™x'
(Q);
ii) a solution u to the Cauchy problem (1.2.1) satisfies the estimate
(1.2.4) ||/)
a
«||
2
(^)CJI/
1
-
A
(||/||
2
(n)+ Yl H^llK-;)(
r
)
when £l{ is a subdomain of Q satisfying the condition dist(Qj, d£l\T) 0,
both C and A, 0 A 1, depend on Q{ and M is the sum of
||Z)aw||2(Q)
over a with la: ml 1.
Previous Page Next Page