On Non-Closure of Range of Values of Elliptic Operator for a Plane Angle. V. N. MASLENNIKOVA(*) - M. E. BOGOVSKII(**). We consider the following system of ...
Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Vol. XXXIX, 65-75 (1993)
On Non-Closure of Range of Values of Elliptic Operator for a Plane Angle. V. N. MASLENNIKOVA(*) - M. E. BOGOVSKII(**)
We consider the following system of the 1st order, which is elliptic in sense of Douglis-Nirenberg v + Vu = F(x),
(1)
divv = 0,
x e~ cR 2 .
where v = v ( x ) = (Vl, v2) and F ( x ) = ($'1, F2) are two-dimensional vector fields, u = u ( x ) is a scalar function, satisfying either the Dirichlet problem
(2)
ula,~ = O ,
or the Neumann problem (3)
av I au
= (F, v)laa,
where v is an external unit normal to am. We define as Jp (D) the closure of the subspace of 2-dimensional vector fields o
o
J| (~) = {v(x) ~ C |
(D; R2): divv = 0},
in the norm of Lp(~; R2). Let Gp be a subspace of all potential vector fields in Lp(D; R2), i.e. Gp(l)) = {v(x) e Lp(~; R2): v = V~b, ~b(x) e L I ( ~ ) } ,
(*) Indirizzo dell'A.: Via Garibaldi 23-1-51, Moscow 13-335, 117335 Russia. (**) Indirizzo dell'A.: Department of Differential Equation and Functional Analysis, Russian People's Friendship University, Mikluho-Maklaya str. 6, Moscow 117198, Russia.
66
v.N.
M A S L E N N I K O V A - M. E . BOGOVSKII
where L~ (~9) is a Sobolev space with a semi-norm =
D a
o
We defme Gp(D) the following subspace of Gp(D): o
Gp(~9) = {v(x) 9 Gp(D): v = V~b, ~la~ = 0}. Let Jp (~9) be the space constituted by all solenoidal vector fields belonging to Lp(t~; R2), i.e.
{
o}
Jp(~) -- v ~ Lp(~; R2): (v, V ~ ) ~ = 0 V~(x) 9 C | (~) , 17
where (-, .) is a scalar product in R 2. The following elliptic operator corresponds to the Diricblet problem (1), (2): o
(4)
L: Jp(tg) x Gp(D)-*Lp(D; R2).
This operator is defined by the equality
(5)
L { v , Vu} = v + Vu.
The elliptic operator o
(6)
L: Jp(D) x Gp(D)---)Lp(tg; R~),
is defined by the same identity (5) and corresponds to the Neumann problem (1), (3). The elliptic operator (4) is naturally related to the Dirichlet problem for the Poisson equation (7)
Au = divF(x), ula~ = o,
x e t~,
in the sense of the following defmition. DEFINITION 1. We shall call the function u(x) e L~ (D) a generalized solution of problem (7) if u la~ = 0 and the integral identity
(8)
I(vu,
= I (F(x),
holds true for all test functions ~(x) 9 {~(x): ~ 9 L~,(~9), ~la~ = 0}, where p' = p / ( p - 1), 1 < p < ~ .
ON NON-CLOSURE OF RANGE OF VALUES ETC.
67
The elliptic operator (6) is naturally related to the Neumann problem through the Poisson equation
{
Au = divF(x),
(9)
dv
a19
= (F,
xeD,
v)]a19,
in the sense of the following definition. DEFINITION 2. We shall call the function u(x) 9 L~ (D) a generalized solution of problem (7) ff u(x) satisfies the integral identity
(lo)
f(vu, V+)dx= f (r(x), V+)dx, 19
19
for all test functions ~(x) e L~ (~), 1 < p < oo. We denote by F~ an angle with the opening a e (=; 2=] which belongs to the plane R 2 , in other words it is a domain which may be defined in polar coordinates as follows:
r ~ = { ( r , ~ ) : r>O, 0 < ~ < ~ } . The objective of the present work is to prove that the range of values Rp (L) of the elliptic operator (4) or (6) is not closed in Lp(~; R 2) when ~ = F~ for p = 2/(1 _+ =/~). THEOREM. Let = < a 4,
4
in case p = 2/(1 + r:/a) and
(13)
v(r)=
in case p = 2/(1 - r:/a). It is easy to see that
F~
0
0
r
ON NON-CLOSUREOF RANGEOF VALUESETC.
69
for p = 2/(1 --+ =/a). The function divF(x) has the following form in polar coordinates: 7~
(14)
divF = f ( r ) sin --
where the function f ( r ) is defined as follows:
(15)
f(r) =
In r
v(r) 1
-In r
p
+ rv'(r) .
We look for the solution in the class L~ (r~) by the use of Fourier method. Namely, proceeding from the assumption that the solution u ( x ) 9 LI(F~) exists, we decompose it into Fourier-series in polar coordinates (16)
u = ~ R . (r) sin rcn ~, n=l
with the coefficients (17)
R , ( r ) = -2 ] u . s i n m7~n ~d~, 6f
n~>l.
6C 0
Now, choosing in (8) the test functions 7rn
= ~(r) s i n - - ~,
n ~> 1,
o
where ~(r)e C | (0, oo) is arbitrary, we obtain the following identity, due to (17):
;
, , Rn(r)~ (r)-
7rn 2
r)~(r)rdr, Rn(r)~(r
n = 1,
rdr= [0,
n~>2,
o
for all ~(r) E C ~ (0, oo) which actually represents the definition of a generalized solution R~(r) of the equation
(18)
r
[0,
n~>2,
70
V. N. M A S L E N N I K O V A - M. E . BOGOVSKII
in the following class of functions: (19)
/(
R" (r)IP +
I "2, r > 0 ,
from which, remembering (19), we obtain that an = b~ = 0 for n I> 2, i.e. R~ (r) - 0 for n I> 2. In case n = 1 we find the general form of solution from (18): (20)
R l ( r ) = a l r=/= + blr -=1= -
r>0.
2=
1
1
To define the constant al, bl we have the only condition (19). The cases p = 2 / ( 1 + r:/a) and p = 2 / ( 1 - r:/a) are investigated below separately. We note
g(r) = - v(r)( ~-
r>O.
--lnl+ r)~ ' ( r ) ' r
Then, for p = 2/(1 + 7:/=) we have (21)
f(r) =
g(r) , r 2+~/=.In r
r > O.
Substituting (21) into (20), we get (22)
R 1( r) = ar ~1=+ br -'/~ + F
1/4
= Ir_=l ~ f
g(p) d plni~
with some constants a, b.
1/4
r~l= f
"]
g(p)dp
,- P'+2"/
