I01
Internat. J. Math. & Math. Sci.
(1988) 101-114
VOL. ii NO.
ON SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
WITH BITSADZESAMARSKI[ CONDITION
J.H. CHABROWSKI
The University of Queensland,
Department of Mathematics,
St. Lucia,
4067,
Old,
Australia.
(Received January 21, 1987)
I,
ABSTRACT.
l,roblem
tiLLs
l,,)l, I,.m
wlue
paper we
1,v a line;-r
with
the solvability of a non-local
investigate
elliptic equation, which is also known as the boundary
the
Lilt sadze- Sm, arski
condLt ion.
We
prove
existence and uniqueness of a classical solution to this problem.
the
In tile
fnal part ,f" this paper we propose an L’-approach which gives a rise to
weak stlutims in a weighted Sobolev space.
the
B
KEY
existence
of
weak
sadze--Samarsk [ condi
WORDS
ANt)
Bi tsa,-tz.-.aar.k
solutions
is
INTRODUCTION.
PHRASES.
modification
of
the
Elliptic
equations,
non-local
problems,
cond ions.
35J, 35C, 35R.
In recent years several
solvability of non-local
[1-9].
The crucial point in proving
suitable
on.
1980 S SUI.JECT CLASSIFICATION CODES.
I.
a
authors
have
studied
the
problems for elliptic and parabolic equations
The importance of non-local problems appears to have been first
noted in the literature by Bitsadze-Samarski.
The problem studied in
these papers constitutes a direct generalization of the classical boundary
value problems. The most significant feature of nonlocal problems is that
the boundary condition relates values of a solution on the boundary to its
values on some part of the interior of the region.
This
type of the
boundary value problem is often referred to as the boundary value problem
with he B[tsadze-Samarskii condition
[7],[8].
The problem (2.1),(2.2)
discussed in this article arises from the mathematical descri.ption of some
processes in a plasma (see paper
of non-],’al problems).
[6] for full account of physical aspects
J.H. CHABROWSKI
102
’rhc
organized
is
pal,er
as
In Section 2 we give the
follows.
uniqueness and existence theorem of the classical solutions of the problem
Our mehd is bnsed on lhe aximm principle developed in
o,.]),I,..
[5].
papers [.] and
Section 3 contains a discussion of the solvability of
th non I(,caI problem for harmonic functions in a disc in
2"
The results
of this section slightly improve the explicit formulae derived by Bisadze
see ’1]
f’r h,rmoni, functions associated with some non-local
[2])
:,,,,I
in a general case of a linear elliptic equation we reduce the
problem,
2.1,’2.’2)
probl,m
lhe solvability of the
integral equation of the
The final sections 4 and 5 are devoted to the study of the
second kt,l.
non--],,::,l
I.,,
p’ohle.m l’,,i" a linear elliptic equlion with a paretcr, whose
This allows us to remove some
princp:] part is in a divergence form.
restri,:-ons on
appearing in the boundary condtion (2.2).
the coefficient
On the olher hand ths also suggests further extensions of the solvability
of the l,roblem (2 l) (o 2) in a weighted Sobolev space
1,2 ().
We adopt
2
here the L-approach to the irichlet problem with L-boundary data from
[13]
2.
a,,d
I0].
UNIOUENESS AND A PRlORl ESTIMATE.
We consider a linear equation of the elliptic type
n
n
a..(x) D .u + Z b (x)D u + c(x)u
Z
i
ij
i,j=l J
i=l
[u
in O, where 0 is a bounded domain in R
p
of
defined on the boundary
(2.1)
The purpose of this paper is to
n
investigate the following non-local problem:
and
f(x)
given continuous functions h
find a solution u
satisfying the boundary condition
u(x)
@
where
p(x)
u(@(x))
(2.2)
h(x) on 00,
is a given continuous mapping of 00 into O.
Throughout this section we make the following assumption
(,%)
The coefficients of the operator L are bounded in Q and there
0 such that
exists a constant
n
i,j=l
aij(x) iCj
for all x e 0 and
R
n
.Moret, w.r we assume that 0 satisfies an
interior sphere condition at
each poknt of o0 (see [11], p. 33-35).
The uniqueness of the problem
strong
mxu,m,
PROPOSITION 1.
(2.1),(2.2) is a consequence of the
principle.
Let
Ifl(x)
1 on DQ and c(x) _< 0 in 0
md suppose that
SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
103
e ther
(a’
-]
(b’
c(x
,(x O)
at some point x
0
e aO
or
0 at some point
e O.
x]
Then the problem (2.1),(2.Z) has at most one solution in
It
Pt/OOF.
02(0)
C().
sufficient to show that if f(x)
0 on Q and h(x) _= 0 on 00
0 is tl,,. only solution of the problem (2.1),(2.2).
It is clear
that u,d--r-ach of the asstmq,tions (a) or (b) any constant solution must be
is
then u
identially equal to O.
/
If u
0 then u must be a non--constant solution
and by the strong mximum principle ([Ii], Theorem 3.5) we may assume that
u
...,:_
It" "xo)
m_ax u(x)
0 with x
0
2
e 00.
0 we get a contradiction.
Therefore it remains to consider
two cases
(i)
(x 2) < i
0
In the first case
#(X 2)
e O.
and
u(x 2)
u(@(x2)
(ii). we
u(x 2)
t,((x 2))
u(x2),
(x2)
In the second case
-I _< /(x 2) < O.
(ii)
which is
impossible since
have
0
P(x 2)
and by the strong maximum principle u takes on a negative minimum at
x e 00, that is
3
u(x 3)
and we may
asstme
min u(x)
that
0
(x3)
0 since otherwise we get a contradiction.
Hence
u(x 3)
u(#(x 3))
>
0.
P(x 3)
Now we distinguish two cases either
u(x 2) _<
[u(x3)
or
u(x 2)
>
[u(x3) I.
We show that both cases lead to a contradiction.
we have
which is [ml, ossible.
In the second case we have
Indeed, in the first case
J.H. CHABROWSKI
104
Since both values u(xo) and
min2mum :I
:.xo)
u(x..,))
are negative u attains its negative
e O and we arrive at a contradiction.
[nsl-,p-tion of the proof of Proposition
shows that the following
version or’ the maximtun principle holds true.
i’ItOI-’OSITION 2.
Let
Lu
(_> O)
0
Then u Cx)
.
Suppose that
0 (_< O) on
0 in Q and
c()
u(x) -p(x)
in Q, and
0
u((x))
0 (< O) on 8Q.
As an immediale consequence we deduce an a priori estimate
THEOIEM 1.
d
(
Suppose that c(x)
0 and 0
1)
o
2)
then
lu(x)[
PROOF.
fl(x) _< a where
If u is a solution of the problem
d in O and 0
are constants.
a
sup[f(x)[
:1
+
suplh(x)[.,
for all x e
o.
Let us define
then we have
Lv
f-
c
:c suplh()
aQ
suplf()
Q
_> f + suplf(x)
Q
in 0
1
1
0
-,
00
(x)
(1
on OQ.
llen,:e by Proposition
uCx3
on O.
_
suplf(x)
+
l-a
sup[h(x)]
2
+
l__ suplhCx)
Similarly we can establish the inequality
u,’x
suplfCx)
l..
0
$:S suplh(x)
aO
on O, ,_-onsderng the auxiliary function
0
_> o
SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
’""’
ItEMA}tE
I.
105
i--A suplh(x)
oo
u":’d’u’lr)l
0
on OO, then any two solutions of the
If c m 0 or 0 and (x)
problem (’.1),(’2.2) differ by a constant.
3.
EX I,’t’I,:NCE
01," CI,ASSICAL
SOLUTIONS.
We commence by considering a particular case of the non-local problem
’2. 1),,’,.’.’2 which consists of finding a harmonic function u on
B(O,I) and
satist’ying the boundary condition
pu((z))
uz’
where B(O,I
on 08(0,1),
h(z’
(3.1)
an open disc in
of radius
centred at O, 1 is a
constant in the interval [-1,1] and h is a continuous function on B(O,1).
The mal,png
is given by (z)
with 0
is a
5
1, where
is
N2
,
,(hz)
],(z)]
univalent analytic function on B(O,1) such thmt
,(0)
O.
By virtue of Schwarz’s lemma we have
Izl ro
I%C.)1
The function
1,2
o(Z)
Letting
z and
(_l(Z))
k(Z)
for
we have
klz
for k
(3.2)
B(O,1).
maps disc B(O,1) conformally and univalently onto certain
set contained in B(O,1).
k
_< 1 in B(O,1) and
in B(0,1),
are harmonic functions we have the
u((z))
Since u and
1,2
following representation formula
uCz)
uC(Z))
Suppose first that -1
u(z)
/l
n
/
n
[Re
I+6
2(I-)
for n
1,2
n
+
z
k=l
.
n
0
]
aB{0, I)
1
t-z
t
]h(t)dt
mF(z)
(3 4)
Iterating (3.4) we get
1.
U(n(Z))
It follows from (3.3) that
[F(n(Z))
Re
pk-1 r([k_ l(z)).
j"
OB(O,1)
t+(z)
t-n(Z)t
h(t) dt
(3.5)
_<
lh(e is) lds
and consequently letting n
in (3.5) we obtain
I06
J.H. CHABROWSKI
r
uz)
n=l
pn-lF(-1 (z))
un for,,,1 y on
@ =-I.
L,’t us now consider the case
h(t) dt
2t
I
"* oB(0, I)
Re---.-
2u(0)
It follows from (3.4) that
(3.6)
and tile functional equation (3.4) can be written in the form
u(#(z))
u(z) +
Re
1
1
Re
2u(0)
i=
i
I
OB(O, 1)
h(t) dt -z
I
dB(0,1) t(t-z)
"t
_1
1
2t
z
1
2t
]h(t)
dt
Re[ (z).z].
Iterating the last equation we obtain
2n-I
u(@on(z))
uCz)
z
+
(-i) J
Re[}(@j(z))
(3.7)
and
2n-2
u(z)
2u(0)
U(@Zn_l(z))
+
Z
j=0
(-])J Re[}(@j(zl) @j(z)].(3.81
It is easy to see that
2
I_(%(z))
for all n
series
1,2
(-
Z
1) J
j =0
n
n
_< =(_s)
0
[h(e s) [ds
B(0,1).
and z
Re[(j(z)),(z)]__.a
Since
u(0)+
z (-i) j
oJ
i. B(0,)
converges uniformly on B(0,1).
(3.7) and (3.8) we obtain the se
u(z,
l#j(z)l <
Letting
lim[ in boh cases
Re[(#j(z)) j(z)]
and invoking (3.6) we get that
u(z)
Finally let 1
u’z)
By Remark
Re
1
4n
h(t) dt
I
z (-I) j
+
OB(O,I)
j=O
Re[(@j(z))@j(z,].
i, then
u((z))
Re
I=
’1
I
OB(O, 1)
{
any two solutions differ by a constant.
h(t) dt
If z
O, then
the
SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
u((O)):
urO)-
0
ICe
_1..
n.
107
f(t) dt
I
6B(0,1)
which g,w".; a necessary and sufficient condition for the solvability of the
Hence
problem (3.l).
a[(z)z].
u(#(z))
()
Iterating this functional equation we obtain
n--1
j:l
we obtain
Letting n
u(O) +
u(z)
Z
j=l
Re[(j(z)) j(z)].
To determine a solution in a unique way we may impose an additional
condition u(0)
0 is a given constant. The case /
C, where C
1 was
considered by Bitsadze in [1] and [2] under the assumption that h is Holder
continuous
In a general case we reduce the problem (2.1),(2.2) to the Fredholm
integral equation of the second kind.
THEOHEM 2.
.
D[j
aij
Suppose that the assumptions of Proposition 1 hold and that
n), D
(i,j
b
i
n) and c are Holder continuous on
(i
Then the pr,blem (2.1),(2.2) admits a unique solution u in
PROOF.
C2(O)
O
C().
We try to find a solution in the form
ux)
-
C(oO)
where v
operator I, and
dG
f
dn
dO
is
v(y)
y
I
dSy
G(x.y) f(y) dy,
(3.9)
O
to be determined, G is the Green function for the
denotes the conormal derivative.
The boundary condition
y
(2.2) ]ud
o
tho Fredholm integral equation of the second kind.
dn
80
+
y
y
I ,tl(x) G((x),y)
f(y) dy.
(3.10)
(&(
a_G,v,x,,y,
is continuous function on
O
Since
(O0)
c
O
the kernel
p(x)
dn
0 x 0.
[;y Proposition
has only trivial solution,
Y
the homogeneous equation corresponding to (3.10)
tlence by the Fredholm alternative there exists
J.H. CHABROWSKI
108
a unqu,. :.,,lution v e
c, C(oO’.
(2.1), (.’.. ’:’.
4.
.
ENERGY
L"’dO) which by the continuity of the kernel belongs
’,nse,luently the formula (3.9) gives a solution to the problem
F.T[MATE.
In tlis se-tlOn we
Mu +
the elliptic equation in the form
co,raider
n
Z I) (a (x)
ij
i,j:l
hu
+ hu
f(x)
Dju)
n
Z b (x) D u + c(x)u +
i
i=l
+
in O,
(4.1)
with the boundary condition (2.2).
Throughout this section we assume that /] is a continuous function on
bO and
assume
:o,O
C1--mapping
O is a
b
ai3 Diai3
t|mt
Further we
with the positive Jacobian.
n) c and f are Holder continuous on
(i,
0 and that O is a bounded domain with the boundary of class C
Th, ,,bje,:tive of this section is to show that
2.
the problem (4.1),(2.2)
hns a unique solution for large values of the parameter A.
For small
6
0 we define
According to Lemma
r(x)
06
14.16
dist(x,O0) belongs to
0
Ix-y[ > ).
n {x" rain
y. OO
[II]
in
C’(0-06.
(p. 355),
o
if
0
the
distance
is sufficiently small.
Denote
by p(x) the extension of the function r(x) into 0 satisfying the following
properties
p.’.’<
-1
I
r(x) for x e
r(x:
6.1for
p(x)
tJ
6 e
6
6
06 o
p e
C2(),
36
p(x)
_< ]
-4
I>
r(x) in 0 for some constant v
and finally o0
(x’p(x)
0}.
TIIEOREM 3.
There exist positive constants A
solutioi
(:"0)rl C() of the problem (4. I),(2.2) for
tn
llVu(x)[
r(x)dx +
0
by
I h(x)
0<_< d
dS
X
+
I
f
(x)2
06
006
0
O,
{x" p(x)
C and d such that if u is a
2
lu(x) r(x)dx + sup
0
C(
Proof.
o
in
A
O
2
I u(x)
o08
dS
then
X
dx).
We follow the proof of Theorem 5 in [13].
Multiplying (4.1)
SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
:u(x)
v(x’l
Ip(x)-6}
on
109
(}6
and integt’at[ng by parts we obtain
n
dO
aij(x) DiP
i,j--I
6
.v
j
u
DiP
dS
.
1
x
n
Z
I
0
Di(aij(x)
i,j:l
6
u2(p-6)dx-
f (c(x) + A)
6)dx +
b.f) Dju.u(p
2
jr
Djp)u2dx+
fu(p 5)dx.
Applying llolder’s inequality we easily obtain
sup
f
O,6<d
006
+
where d and C
dS
u
C
x-
2
[Dul
1
2
p dx + f u dx +
0
f u’ p dx +
f
0
0
dx
(4.2)
are positive constants.
Similarly
I [Iu[
+ h
p
0
I u" p
O.
for some (:,,
+ a
P
I
I
h
dS
+
f
4
hat
f
u(
dO
where C
with
5
dx +
I
dx +
I
u
2
oO
0
dSx]
f2
dx + f
(4.3)
dx +
O
2dSx
(4.4)
p dx +
sup
O<<d
I
ao
I
dS
+
0
C
for some C
2
P dx <_ C 3
u(())
0
obvious
u
The estimates (4.2) and (4..4) yield that
O.
(:
u
0
+
[
It follows from (2.2) that
0
where
<_ C 2
dx
0
>
4
u" dS x
O.
Since
{ x. )2 dSx
0 and C
6
dist(,o(})50.
C5
>
I
dx +
0
is a
(dO)$
dO
cl-mapping
u
2
X-
I u((x)) 2
dS
,O
(4.5)
x
with the positive Jacobian it is
dSx < c6
0 are constants and
x
dS
[0
Dul2 d +
O@ is
I u2
dx},
a domain containing
(4.6)
@(dO)
Consequently by virtue of the Caccioppoli inequality
110
J.H. CHABROWSKI
we have
x-
00
for some C
O.
7
I u"
I
Ul O
dx
5-
i!!f p(x).
d]
0
We now observe that
0
where
(4.7)
[0
u2
p dx + d
sup
I
O<6<d
dO
u2 d S
x’
Choosing d sufficiently small and k sufficiently
0
d
large we easily derive the desired estimate from
(4.5),(4.S),(4.7) and
(4.8).
[ep,_-atlng the argument of Theorem 2 we deduce the following
TI1EOREM ,1.
There exists a positive constant A
0
such that for every
c 2 ()
the problem (4.1),(2.2) admits a unique solution in
5.
WEAl(
C().
SOLUTIONS.
The energy estimate from Section 4 shows that one can expect solutions
of the problem (4.1),(2.2) in a weighted Sobolev space defined by
wl"(o>
{u
Wl’oc
(o);
llDu(x)
2
r(x) dx +
I
u(x)
2
dx
}
and equtl,l,e’d with the norm
I [{0u(x)
2 r(x) + u(x
)2
dx.
We r,,-al] briefly that a function u is said to be a weak (generalized)
solution of (4.1) if u e
n
Z
I
for each v e W
]’2(0)
W1,2
loc(O)
and it satisifies
n
a.. D.uD.v +
Z
b.D.u.v + (c+A)uv]dx
I
f v dx (5.1)
with compact support in O.
To proceed further we need some terminology.
It follows from the
such that for
0
regularity of the botmdary dO that there exists a number 6
6 e
(0,o)
the domain 0
(defined in Section 4) with the boundary dO
6
6
to each x e 0 there exists a unique
o
such that
e
6v(x
where v(x
ls the
possesses the following property:
point
x6(xo,
of class C
I,
xs(Xo),
dO6
outward normal to dO at x
of dO onto
O.
Xo
o),
o)
The above relation gives a one-to-one mapping
SOLVABILITY
c)F
BOUNDARY VALUE PROBLF FOR ELLIPTIC EQUATIONS
It is known that elements of the space
have trc:es on lhe boundary O0 (see
do not
in general,
llowever, by Theorem 4 in [13],
[12]).
";0) is a solution of (4.1) then there exists a function
if u e
e
1’2(0),
Iii
L2(oO)
such that
O.
(x)]" dS x
I [u(x 6)
lim
60
aQ
Thc.r-I’,re, as in the paper [1], we adopt the following L"-approach
the l,r,l.,
I.,’l. ), (’2.2).
,.m
Let h e L(OO).
1,2
Wloc( O)
A weak solution u
of (4 I) is a solution of
the non-local problem with the boundary condition (2.2) if
h(x)]2
(x) u((x))
I [u(x 8)
]im
dS
oO
50
in []3] that if u
It f,,ll,w. from Theorem
the problem (,1.1),(2.’2)
the sens of (5.2)
O.
x
W1,2
lot(O)
i5.2)
is a solution of
(with the boundary condition (2.2) understood in
’(O).
then u e
We mention also that
.
u((x))
understood n the sense of trace, which is well defined since u e
1-1 ], ,’h,i,. G
(see
We now
ar’"
is
1,2(0
Wlo
c
in a position to establish the existence result in
l,2
of tlc proll,,m (4.1), :2.2).
Let
THEOREM 5
that
for"
|,
Then there exists a positive constant
e L(6Q)
o
2 2)
(4 1)
the problem
in
1,2
(Q)
loc
admits
such
^
unique
a
solution.
PROOF
Let
_>
m
b’t h
ii!
be a sequence in
cl(oo)
I
such that lim
m- OO
A
le a constant from Theorem 4 and assume that
0
0
Thcor,.m 4
2
u
m
e C ((J) 0
2- dS
(hm-h)
x
O.
For each
^
guarantees the existence of the unique solution
C(O) of the problem (4.1),(2.2) with h
h
Moreover we have
m
for ea,’h u
I
O
IDUm 12l
rdx +
I
u
2
dx
C(l
f2
dx +
O
0
where C
0 is a constant independent of m.
bounded in
1,2(,
convering ealy in
’wl’2(O)
is
f
hm2
dS
x)
Since the sequence u
is
there exists a subsequence, which we relabel as u m
I,2()
to a function u.
compac’tJy embedded in
L2(Q)
By Theorem 4.11 in [15],
and therefore we may assume that u
m
i12
J.H. CHABROWSKI
converg,:. i u in
By
’rheore,.
.I
12 (0).
It is obvious that u is a weak solution of (4.1).
[13]
in
has
u
trace
a
L-(OO)
e
in
the
sense
of
L 2- con,-, g,.nc,, that is
2
I [u
(x)] dS
x
oO
To complete the proof we show that (x)
Jim
Let
C
(x)
]O).
It is easy to show
(x)
O.
u((x))
fl(x)
+ h(x) a.e. on 0.
p(x) is a legitimate test function
in (5.1) a,] inlegrating l,y parts we obtain
n
I1
DiP Ijp
aij
i,j=l
40
I
dS
D (a
0 i,j:l
i ij
Dip {)
u dx +
n
o
i=l
O
O
5.3)
Similarly
[
:,0
[hm(x)
n
Z
i, j
p(X)Um({(x))]
+
aij Dip I}jp
dSx F(um).
(5.4)
ApI, lyng the estimate (4.6) and the obvious analogue of the energy
estimate t, u
u
P
r
[u
,’O
q
we obtain
P (<x))
Ilu
C
Uqi
I
O0
where k
u
1;u)
m
(x)
fl(x)u((x))
C
dx]
of
as m
as
q f(x))]
[hp- hq["
,mtinuity
(@(x))u((x))
F(u
2
inf p,x’), and C
the
by
u
m
dS
C[ I
x
O
IDuP
Du
q
12
dx
2
j.
dS x,
0 is a constant independent of p and q.
solutions
weak
2
on
0
we
may
assume
Hence
that
Combining this with the fact that
in L (aO).
we deduce from
(5.3) and (5.4) that
+ h(x) a.e. on O0 and this completes the proof.
REMARK 2.
(4.1),(2.2)
It is worth noting that the non-local problem of the type
equations.
The corresponding boundary datum h in [9] belongs to the space
H(bO).
has
been
Since H(80)
studied
in
[9]
for
is a proper subspace
deduced from the results of the paper [9].
the
higher
order
elliptic
L’(aO), Theorem 5 cannot be
SOLVABILITY OF BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS
113
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