q (x) for x 2 ;
where p (x) := nnp(x;x) the fractional critical variable Sobolev exsp(x;x) ponent. The variable exponent Lebesgue space Lq(:) ( ) is de…ned by Z q(:) L ( ) := u : ! R measurable, q(:) (u) = ju (x)jq(x) dx < 1 . The function q(:) : Lq(:) ( ) ! [0; 1) is called the modular of the space Lq(:) ( ). We de…ne a norm, the so called Luxembourg norm, in this space by ( ) Z q(x) u (x) dx 1 . kukLq(:) ( ) := inf > 0; It is needed to pass between norm and modular very often. In [12], the authors prove the following useful result: Theorem 1.1 Let u 2 Lq(:) ( ). Then 1. If kukLq(:) (
)
> 1, then kukqLq(:) (
2. If kukLq(:) (
)
< 1, then kukqLq(:) (
+
)
q(:)
(u)
kukqLq(:) ( ) ;
)
q(:)
(u)
kukqLq(:) ( ) :
+
0
As in the classical case, the dual variable exponent function q of q 0 1 is given by q(x) + q0 1(x) = 1, and dual space for Lq(:) ( ) is Lq (:) ( ). If
2
0
u 2 Lq(:) ( ) and v 2 Lq (:) ( ) then the following Hölder’s inequality holds: Z juvj dx 2 kukLq(:) ( ) kvkLq0 (:) ( ) .
We de…ne the fractional Sobolev space with variable exponents via the Gagliardo approach as follows: ) ( R R ju(x) u(y)jp(x;y) q(:) dxdy < 1, u 2 L ( ) ; p(x;y) jx yjn+sp(x;y) . W s;q(:);p(:;:) ( ) := for some > 0 Let [u]
s;p(:;:)
( ) := inf
(
> 0;
Z Z
ju (x) p(x;y)
u (y)jp(x;y)
jx
yjn+sp(x;y)
)
dxdy < 1 ;
be the corresponding variable exponent Gagliardo seminorm. It is easy to see that W s;q(:);p(:;:) ( ) is a Banach space with the norm kukW s;q(:);p(:;:) (
)
:= kukLq(:) (
)
+ [u]s;p(:;:) ( ) :
It is clear that W s;q(:);p(:;:) ( ) can be seen as a natural extension of the classical fractional Sobolev space; however, fractional Sobolev spaces with variable exponents are also used to study some nonlocal problems as described in [3],[18],[8]. In [18], the authors prove the following theorem: Theorem 1.2 Let Rn be a bounded domain of class C 0;1 and s 2 (0; 1). Let q (x) ; p (x; y) be continuous variable exponents with sp (x; y) < n for (x; y) 2 and q (x) > p (x; x) for x 2 . Assume that r : ! (1; 1) is a continuous function such that p (x) > r (x) r > 1, for x 2 . Then there exists a constant C = C (n; s; p; q; r; ) such that for every f 2 W s;q(:);p(:;:) ( ), it is held that kf kLr(:) (
)
C kf kW s;q(:);p(:;:) ( ) :
Thus, the space W s;q(:);p(:;:) ( ) is continuously embedded in Lr(:) ( ) for any r 2 (1; p ). Moreover, this embedding is compact. According to the Theorem 1.2 and the Hölder’s inequality with 1 1 = p (x) p (x; x)
s and p (x) > q (x) a.e. x 2 , n
we get the Sobolev-type inequality stated in the following theorem. 3
Theorem 1.3 Let Rn be a bounded domain of class C 0;1 and s 2 (0; 1). Let q (x) ; p (x; y) be continuous variable exponents with sp (x; y) < n for (x; y) 2 and q (x) > p (x; x) for x 2 . Then there exists a constant C such that for every f 2 W s;q(:);p(:;:) ( ), it is held that kf kLp
(:) (
)
C kf kW s;q(:);p(:;:) ( ) :
For general theory of Lebesgue and Sobolev spaces with variable exponent we refer the reader to [19],[18],[9],[8],[7],[3],[12]. We say that Rn is a W s;q(:);p(:;:) extension domain if there exists a continuous linear extension operator E : W s;q(:);p(:;:) ( ) ! W s;q(:);p(:;:) (Rn ) such that Euj = u for each u 2 W s;q(:);p(:;:) ( ). In [4], the authors prove the following theorem: Theorem 1.4 Suppose that extension domain. Let m :
is of class C 0;1 . Then
is a W s;q(:);p(:;:)
! [1; 1) be a variable exponent such that
1 < m := essinf m (x)
m (x)
x2
m+ := esssup m (x) < +1: x2
Assume that p (x; :) m (x) q (x) a.e. x 2 . There are two main compactness results for W s;q(:);p(:;:) extension domain. The …rst result is a criterion for the relative compactness in Lebesgue spaces with variable exponent. Theorem 1.5 Let be a bounded domain of class C 0;1 , and F be a bounded subset of Lq(:) ( ). Suppose that Z Z jf (x) f (y)jp(x;y) sup dxdy < +1: f 2F jx yjn+sp(x;y) Then F is pre-compact in Lm(:) ( ). The second result is a compact embedding theorem for fractional Sobolev spaces with variable exponent. As we will see, the theorem is a fairly elementary consequence of the above criterion for compactness. Theorem 1.6 Let be a bounded domain of class C 0;1 , p (x; :) m (x) < q (x) a.e. x 2 , and F be a bounded subset of Lq(:) ( ). Suppose that Z Z jf (x) f (y)jp(x;y) sup dxdy < +1: f 2F jx yjn+sp(x;y) Then F is pre-compact in Lm(:) ( ). 4
Notice that Theorem 1.6 can be seen as a simple variant of the Rellich–Kondrachov theorem. R ! R is a Carathéodory function, that is, Suppose that f : f (:; ) : R ! R is measurable for all 2 R, and f (x; :) 2 C (R) for almost all x 2 . Further suppose that there exist a positive measurable function g : ! [0; 1) lying in Lm(:) ( ) and constant b 0 such that q(x) jf (x; )j g (x) + b j j m(x) for all (x; ) 2 R. We de…ne the Nemytskii (superposition) operator Nf acting on the measurable function u : ! R by Nf (u) (x) := f (x; u (x)) for all x 2 . Let X; Y be two Banach spaces and T : D X ! Y . The operator T is said to be completely continuous if it is continuous and maps any bounded subset of D into a relatively compact subset of Y . Our next result relates to the Nemytsky operator. More precisely, we have the following theorem: R ! R be a Carathéodory function as above. Theorem 1.7 Let f : Then the Nemytsky operator Nf is completely continuous from Lq(:) ( ) into Lm(:) ( ). Notation used in the paper is standard. The symbol C will be used to designate a general constant whose value may change even within a single string of estimates. This paper is organized as follows. In Section 2 we prove the main Compactness result involving the fractional Sobolev spaces. As immediate consequence, we get that any bounded sequence in fractional Sobolev spaces with variable exponent has a subsequence that converges strongly in Lebesgue spaces with variable exponent. In Section 3 we apply our result to prove that the Nemytsky operator is completely continuous in Lebesgue spaces with variable exponent.
2
Compact embeddings
Question about characterization of compact or precompact sets is extremely important in mathematical analysis. Hence, it is important to be able to decide whether a family of functions in variable exponent spaces is precompact. We recall that the Ascoli-Arzelà theorem answers the same quesion in C (K), the space of continuous functions over a compact metric space K with value in R, and the KolmogorovM. Riesz-Frèchet compactness theorem answers the same question in 5
classical Lebesgue spaces. In this section, we prove some compactness results for the strong topology involving the fractional Sobolev spaces in bounded domains. Theorem 2.1 Let be a bounded domain of class C 0;1 , and F be a bounded subset of Lq(:) ( ). Suppose that Z Z jf (x) f (y)jp(x;y) dxdy < +1: sup f 2F jx yjn+sp(x;y) Then F is pre-compact in Lm(:) ( ). It is well known that a metric space is compact if and only if it is complete and totally bounded. Since we are interested in compactness results for subsets of Banach spaces, we may, and shall, concentrate our attention in the proof on total boundedness. In [15], the authors prove the following useful lemma: Lemma 2.1 Let X be a metric space. Assume that, for every > 0, there exists some > 0, a metric space W , and a mapping : X ! W so that [X] is totally bounded, and whenever x; y 2 X are such that d ( (x) ; (y)) < , then d (x; y) < . Then X is totally bounded. Proof. of Theorem 2.1. Since Lm(:) ( ) is complete, it is enough to show that F is totally bounded in Lm(:) ( ). Since is of class C 0;1 , then by Theorem 1.4, there exists a function fe 2 W s;q(:);p(:;:) (Rn ) such that fe
C kf kW s;q(:);p(:;:) ( ) :
W s;q(:);p(:;:) (Rn )
Thus, for any cube Q containing in fe
fe
W s;q(:);p(:;:) (Q)
, we have
W s;q(:);p(:;:) (Rn )
C kf kW s;q(:);p(:;:) ( ) :
Note that, since Q is a bounded open set, and m (x) then the following injection hold
q (x) a.e. x 2 ,
Lq(:) ( ) ,! Lm(:) ( ) , hence fe belongs also to Lm(:) ( ). For any f 2 F, let 1 P (f ) (x) := jQi j
Z
Qi
fe(y) dy, for x 2 Qi and i 2 1; 2; ::::; N , 6
and take a collection of disjoints cubes Q1 ; :::::; QN of side N [
Q=
such that
Qi :
i=1
Observe that P is linear and constant. Furthermore, it is a vector of R . From the de…nition of P f we have, for f 2 F, N
Z
m(x)
jf (x)
P (f ) (x)j
dx = = =
N Z X
i=1 Qi \ N Z X
where
1
Qi \
i=1
1 jQi j Z
f (x)
i=1 Qi \ N Z X i=1 Qi \ N Z X
P (f ) (x)jm(x) dx
jf (x)
jQi jm(x) Qi Z f (x)
Z
m(x)
Qi
fe(y) dy
dx m(x)
fe(y) dy
f (x)
dx
m(x)
fe(y) dy
Qi
1
dx,
. ) Now for any …xed i 2 1; 2; ::::; N and x 2 , by the Hölder inequality with p (x; y) and 1 p(x;y) we get p(x;y) Z
=
min(
nm+ ; nm
m(x)
f (x)
Qi
fe(y) dy
jQi j n(
m(x) (p(x;y) p(x;y)
Z
1)
fe(y)
f (x)
Qi
n+sp(x;y) 2
m(x) ) p(x;y)
dy
2 Z m(x) (n+sp(x;y)) 6 p(x;y) 4
f (x) jx
Qi
2 Z 6 4
f (x)
Qi
where
= max n
n+sp+ 2
(n+sp+ ) ; n
7
jx n+sp 2
fe(y)
p(x;y)
n+sp(x;y)
yj
(n+sp )
m(x) 3 p(x;y)
7 dy 5
!max
m(x) p(x;y)
p(x;y)
,
m+ m ; p+ p
.
fe(y)
p(x;y)
n+sp(x;y)
yj
m(x) 3 p(x;y)
7 dy 5
Now by using the Jensens’s inequality, we obtain that m(x) 3 p(x;y) 2 p(x;y) Z Z Z fe(x) fe(y) 7 6 m(x) dxdy 5 jf (x) P (f ) (x)j dx C 4 n+sp(x;y) jx yj Q Q m+ m ; p+ p
max
CC1
.
Since m+ < +1, this implies by Theorem 1.1 that kf moreover kf kLm(:) ( ) f; g 2 F and kP (f )
P (f )kLm(:) (
)
C,
C + kP (f )kLm(:) ( ) . By the linearity of P , if P (g)kLm(:) ( ) < then kf
gkLm(:) (
)
(1)