1. PRELIMINARIES 7
So summing up we get the estimate (1.2.6).
We remark that the requirement T e C can be reduced; see, for example,
the book of Lavrent'ev [87], Chapter 2.
In Corollary 1.2.3 we assume that the coefficients
aj
of an operator A
are real-valued and an operator A itself is elliptic on R" , i.e.,
(1.2.7) X y ^ - ^ A -
~eol£|2 f o r
all x G fi and J e R*.
COROLLARY
1.2.3. Let Q. be a domain in
Rn
and T be a (nonempty)
hypersurface lying on dQ.. Then the estimate (1.2.4) is valid for any solution
to the Cauchy problem (1.2.1) with m = (2, ... , 2).
PROOF.
Since it is possible to cover Q,{ by a finite number of C -diffeo-
morphic images of the half-ball {\x\ 1, xn 0} so that images of {\x\ =
1, xn 0} belong to T, we may assume Q is this half-ball and T = dtln
{xn 0} . Recall that the ellipticity is invariant under
C2-diffeomorphisms.
Let
(1.2.8) cp(x) = exp(-axn) - 1.
If C = (M , ••• , ^ _ i , in ~ haexp{-(jxn)) and Am(x; Q = 0, then
(1.2.9) E ^ A =
annT2a2exp(-2axn),
^ ^ . = 0,
where the sums are taken over j , k = 1, ... , n . The left-hand side in (1.2.2)
equals
4 A
2
e x p ( - 3 a x J ( 0
2
+ ( 2 / T ) I m £ ^ y V % C * C
7
.
The last sum consists of terms ^^aexp(-crx
w
) or br a exp(-3axn),
where & are bounded. From the first equality in (1.2.9) and the elliptic-
ity condition (1.2.7) it follows that |£| Crcrexp(-crxA7). So choosing a
large we get strict positivity of the left-hand side mentioned. So Corollary
1.2.3 follows from Theorem 1.2.1 ii).
We note that it is sufficient to suppose
aJ
e Lip; we refer to Hormander
[44]. In the plane case, letting v - Dxu- iD2u and using the well-known
representation v(z) = s(z)w(x{z)) where w is a complex-analytic function
and s , x
a r e
Holder functions (s ^ 0, x is
a
homeomorphism), we can even
assume all coefficients of A are only measurable and bounded (see Miranda
[99], pp. 61,267).
In the next corollary we consider a parabolic operator
A(x: D) =
Dn+l+J2«ikDjDk
+
1£iaiDj
+ a
with the coefficients
aj
e
Cl(Q)
satisfying the ellipticity condition (1.2.7)
and
aJ,
a e L^Sl) and sums taken over j , k = 1, ... , n .
Let x , x" be the projections of x onto R" and R"~ .
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