DIFFERENTIAL OPERATOR

`g(') := ?g; Pe( ; Dx)' 0; for all ' 2 C01( )
dense in i.e. with
sHom01;e1 (co)nsdteannt ecd2linRe+arofnuenchtaios nal
is
bounded
with
respect
to
k
k1;1;
,
?g; Pe( ; Dx)' 0; < ck'k1;1; for all ' 2 C01( ):
kuks;t; :=
X
!1=2
kD uk20; ;
j
1j=
s;1j+
2
2j
t
can2dRHB+e01c;1wa(uitshe)
as the smallest closed of the boundedness which
subspace of H1;1( of the functions a
)
i,nctlhuedriengexCis01ts(
and Dj = with a(ij )
@=@xj
= a(ji )
, .
where Note
a(ij ); b(i );
that then
c( ) 2
there
C1( ) are are constants
X n a(ij )(x ) i j % X n i2
i;j=1
i=1
for all x 2 and ( 1; : : : ; n ) 2 Rn :
tively) with boundaries @ ( = 1; 2) of
ctilhnaesRssunCb11seoqr(uceRf.nnte2.tgh.(enGo1rr;einsmv2as.rd
2, 2,
respecDe ni-
tion 1.2.1.1]). Thus the domains product domain := 1 2
Now we formulate the inhomogeneous Dirichlet problem of classical type for the non-hypoelliptic product operator P ( ; Dx) de ned by
P (x; Dx) := P1(x1; Dx1)P2(x2; Dx2) for x = (x1; x2) 2 1 2:
i;j=1
i=1
1991 Mathematics Subject Classi cation: Primary 35B30; Secondary 35A05.
180
Reinhard Hochmuth
with Dx = (D1;
%give2nRre+alw-viathlued
: : : ; Dn ) functions
which we can write the operator P ( ; Dx) in the form
(2:2)
P ( ; Dx) = X (?1)j jD ?a ( )D :
; 2?
Inhomogeneous Dirichlet problem for non-hypoelliptic di erential operator 181
Rn
satisfy (n =
the uniform cone n1 + n2 ) has the
condition and same property
the (cf.
Hochmuth 3, Satz 3.1]). Furthermore, is a Lipschitz domain (cf. Grisvard 2,
ptoroHdu1c;1t(do)maHin01;1(ha)s.
a Lipschitz boundary @ and because of the continuous embedding H1;1( ) ,!
H1( ) we can introduce the trace operator : H1;1( ) ! H1=2(@ ) (cf. Gris-
For P ( ; Dx) the formal adjoint operator is denoted by Pe( ; Dx).
Problem ( ). For f 2 C( ) and g 2 C(@ ) nd a function u 2 C2;2( ) \
C( ) such that
P (x; Dx)u(x) = f(x) for x 2 ; u(x) = g(x) for x 2 @
for all ' 2 C01( );
w = 0:
For the present it seems that the related homogeneous problem with g = 0 di ers from the weak problem (B) in 1]. However, the two formulations are equivalent (cf. Theorem 3.2). But, by the one given here, one can easily discern that the star-shapedness of the domain assumed in 1] is not necessary for the fact that the solutions of problem ( ) are also solutions of problem (W). Furthermore, it is easier to handle the inhomogeneous problem by this formulation.
2. The inhomogeneous problem
Most of the de nitions used here are taken from 1]. We recapitulate only
some notation necessary for understanding Let 1 and 2 be bounded domains
fn1
+
=( 1; : :
:
1; n; :g:
:; n) and
(anisotropic) Sobolev
s22kpaN=cen00H(fosr;t=(al1l);
Cs+t( ) with respect to the norm
k2()s2;mtfu12l;t:iN-:i:n0;)dnica1egss,tfhroeerspcwoehmcitciphvleeltyi1j,ona=nod0f
By Pe ( ; Dx ) we denote the formal adjoint operators
Pe ( ; Dx ) := ? X n Dj?a(ji )( )Di ? X n Di?b(i )( ) + c( )( ):
i;j=1
i=1
We consider the classical homogeneous elliptic Dirichlet problems:
is valห้องสมุดไป่ตู้d.
To give a Hilbert space formulation of problem ( ) we de ne
(2:1)
? := f 1 2 Nn01 j j 1j 1g f 2 2 Nn02 j j 2j 1g:
A simple calculation shows that there are functions a 2 C1( ) ( ; 2 ?) with
Annales Academi Scientiarum Fennic Mathematica Volumen 21, 1996, 179{187
FAONRINAHNOODMNIOF-HFGYEERPNEOENEOLTULISAIPDLTIOIRCPICELHRINLAEETATORRPRPAORBTLIEAML
vard 2, Theorem 1.5.1.3]).
For the weak formulation of the inhomogeneous Dirichlet problem ( ) we call
an admissible boundary function every g 2 L2( ) for which the through
ear partial di erential operators which are non-hypoelliptic. We prove uniqueness, existence and regularity results for its solutions.
1. Introduction
In 1] K. Doppel and the present author stated a homogeneous Dirichlet problem for non-hypoelliptic linear partial di erential operators, especially for a product of uniformly elliptic di erential operators with smooth coe cient functions. Here we state an inhomogeneous Dirichlet problem for the same class of partial di erential operators. The related homogeneous problem turns out to be an equivalent reformulation of the homogeneous Dirichlet problem in 1]. But this new formulation yields more general results than the former one. For further details of this problem and for some literature relevant in this context we refer to the introduction in 1].