Sobolev spaces with variable exponents on Riemannian manifolds

Nonlinear Analysis 92 (2013) 47–59

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Sobolev spaces with variable exponents on Riemannian manifolds Michał Gaczkowski, Przemysław Górka ∗ Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland

article

info

Article history: Received 4 March 2013 Accepted 25 June 2013 Communicated by Enzo Mitidieri MSC: 46E35 53B21

abstract In this article we study variable exponent Sobolev spaces on Riemannian manifolds. The spaces are examined in the case of compact manifolds. Continuous and compact embeddings are discussed. The paper contains an example of the application of the theory to elliptic equations on compact manifolds. © 2013 Elsevier Ltd. All rights reserved.

Keywords: Lebesgue spaces with variable exponents Sobolev spaces Riemannian manifolds

1. Introduction Variable exponent Lebesgue and Sobolev spaces are natural extensions of classical constant exponent Lp -spaces. Such kind of theory finds many applications for example in nonlinear elastic mechanics [1], electrorheological fluids [2] or image restoration [3]. During the last decade Lebesgue and Sobolev spaces with variable exponents have been intensively studied; see for instance the following surveying papers [4,5]. In particular, the Sobolev inequalities have been shown for variable exponent spaces on Euclidean spaces (see [6–8]). Recently, the theory of variable exponent spaces has been extended on metric measure spaces; see e.g. [9–11]. In this article we investigate variable Sobolev spaces on Riemannian manifolds. The theory of Sobolev spaces with constant exponents on Riemannian manifolds has been studied very intensively for more than last fifty years; see [12–14] or [15]. The theory has been applied for example, to the Yamabe problem [16] or to show isoperimetric inequalities [15]. This paper is dedicated to the theory of Sobolev spaces with variable exponents on compact Riemannian manifolds. In the next two subsections we review some definitions and present the theory of variable exponent spaces. In Section 2 we introduce the theory of variable Sobolev spaces on Riemannian manifolds. Moreover we recall the most important facts about Riemannian geometry. In Section 3 we prove embeddings of our spaces into Lebesgue and Hölder spaces. We are able to show the Sobolev embeddings with critical exponents. In order to obtain such kind of results we need to assume that the exponent is log-Hölder continuous. Moreover, we prove higher order embeddings and we construct a counterexample for which the Sobolev embedding does not hold. In Section 4 we show compact embeddings. Let us mention that a similar result can be found in [17]. As an application of compactness we obtain a Poincaré type inequality. Moreover, we study a PDE problem involving p(x)- Laplacian (see [18]). In the Appendix one can find the theorem crucial for the compact embedding into variable Hölder spaces.

∗

Corresponding author. Tel.: +48 22 621 93 12. E-mail address: [email protected] (P. Górka).

0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.06.012

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1.1. Variable exponent Lebesgue spaces In this subsection we recall some basic facts and notation about variable exponent Lebesgue and Sobolev spaces. Most of the properties of these spaces can be found in the book of Diening, Harjulehto, Hästö and Růˆziˆcka [19]. Let (Ω , µ) be a σ -finite, complete measure space. By a variable exponent we shall mean a bounded measurable function p : Ω → [1, ∞). For U ⊂ Ω , we put p+ (U ) = ess sup p(x),

p− (U ) = ess inf p(x).

x∈U

x∈U

If U = Ω , we shall write p+ , p− . The variable exponent Lebesgue space Lp(·) (Ω ) consists of those µ-measurable functions f : Ω → R for which the following semimodular

ρp(·) (f ) =

 Ω

|f (x)|p(x) dµ(x)

is finite. This is a Banach space with respect to the following Luxemburg norm

∥f ∥p(·)

    f = inf λ > 0 : ρp(·) ≤1 , λ

where f ∈ Lp(·) (Ω ). Variable Lebesgue space is a special case of the Musielak–Orlicz spaces. When the variable exponent p is constant, then Lp(·) (Ω ) is an ordinary Lebesgue space. Moreover, the Hölder inequality

 Ω

fgdµ(x) ≤ 2∥f ∥Lp(·) (Ω ) ∥g ∥Lp′ (·) (Ω ) p(x)

holds, where as usual, p′ denotes the conjugate exponent given by p′ (x) = p(x)−1 . If p− > 1, then the dual space of Lp(·) (Ω ) ′ is Lp (·) (Ω ). Moreover Lp(·) (Ω ) is reflexive, see [20]. It is needed very often to pass between norm and semimodular. An important property of the variable Lebesgue spaces is the so-called ball property: ∥f ∥Lp(·) (Ω ) ≤ 1 if and only if ρp(·) (f ) ≤ 1. Moreover, the following inequality

∥f ∥Lp(·) (Ω ) ≤ ρp(·) (f ) + 1 holds. 1.2. Variable exponent Sobolev and Hölder spaces in the Euclidean setting Let µ be the n-dimensional Lebesgue measure and Ω be a µ-measurable subset of Rn . The variable Sobolev space W k,p(·) (Ω ) consists of all functions f ∈ Lp(·) (Ω ) for which distributional derivatives of order less than k + 1 belong to Lp(·) (Ω ). W k,p(·) (Ω ) is a Banach space equipped with the norm

∥u∥W k,p(·) (Ω ) =



∥Dα u∥p(·) .

|α|≤k

For more basic properties of spaces Lp(·) (Ω ) and W k,p(·) (Ω ) we refer to [20]. As in the classical case, one also defines the Hölder space of variable exponent C 0,α(·) (Ω ). For a measurable function α : Ω → (0, 1] and a bounded continuous function f we introduce the following seminorm [f ]α(·),Ω = sup

x,y∈Ω

|f (x) − f (y)| . |x − y|α(x)

By C 0,α(·) (Ω ) we denote the space of all bounded continuous functions f for which [f ]α(·),Ω is finite. This is a Banach space with respect to the following norm

∥f ∥C 0,α(·) (Ω ) = ∥f ∥∞ + [f ]α(·),Ω , see [21] for more details. Let us stress that some basic properties of the standard Lebesgue spaces are not valid in the variable exponent case. For instance, Zhikov [22] observed that in general smooth functions are not dense in W k,p(·) (Ω ). Density of smooth functions in the space W k,p(·) (Ω ) was proved under the condition of the so-called log-Hölder continuity: c |p(x) − p(y)| ≤ − log(|x − y|) for all x, y ∈ Ω such that |x − y| ≤ 12 . If the variable exponent p satisfies the above condition, we shall write p ∈ P log (Ω ). With assumption of log-Hölder continuity Diening [6] showed the Sobolev embedding for all p with p− > 1. Moreover. Harjulehto and Hästö [8] showed the following result.

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

49

Theorem 1.1. Suppose that p ∈ P log (Ω ) and 1 ≤ p(x) < n. If Ω has a Lipschitz boundary, then

  f − f¯  p∗ (·) ≤ ∥∇ f ∥Lp(·) (Ω ) , L (Ω )  where p1∗ = 1p − 1n and f¯ = |Ω1 | Ω fdx. From the above theorem one can easily obtain the following theorem. Theorem 1.2. Suppose that p ∈ P log (Ω ) and 1 ≤ p(x) < n. If Ω has a Lipschitz boundary, then ∗ W 1,p(·) (Ω ) ↩→ Lp (·) (Ω ).

Moreover, Sobolev embedding theorems are valid also for p(x) > n, see [23,24]. Almeida and Samko showed the following Morrey type theorem. Theorem 1.3. Let Ω be a bounded domain with Lipschitz boundary, p ∈ P log (Ω ) and p(x) > n. Then, the following embedding holds W 1,p(·) (Ω ) ↩→ C

n 0,1− p(·)

(Ω ).

2. Variable exponent Sobolev spaces on Riemannian manifolds For the convenience of the reader we present relevant material about Riemannian geometry (see e.g. [25,15]). Let (M , g ) be a smooth Riemannian n-manifold. Let u be a smooth function on (M , g ). By ∇ k u we shall denote the k-th covariant derivative of u. The norm of k-th covariant derivative in local chart is given by the following formula

  |∇ k u| = g i1 j1 · · · g ik jk ∇ k u i

1 ···ik

 k  ∇ u j

1 ···jk

,

where Einstein’s summation convention is adopted. Next, we also recall the notion of Riemannian measure on Riemannian manifolds. Let {Ui , φi } be any atlas of M. There exists a partition of unity {Ui , φi , ηi } subordinate to {Ui , φi }. Given a continuous function f : M → R we define the integral as follows

 fdVg = M

 φi (Ui )

i

   ηi det gf ◦ φi−1 dx,

where dx is the Lebesgue measure on Rn . Now, we are in position to define variable Sobolev spaces. Given k ≥ 0 and p variable exponent we set p(·)

Ck (M ) = u ∈ C ∞ (M ) : ∀0≤j≤k |∇ j u| ∈ Lp(·) (M ) .





p(·)

On Ck (M ) we define the following norm k 

∥u∥H p(·) (M ) = k

|||∇ j u|||Lp(·) (M ) .

j =0

Thus, we can define variable Sobolev spaces in the following way. Definition 2.1. Let (M , g ) be a smooth Riemannian n-manifold, k a integer and p a variable exponent. The Sobolev space p(·) p(·) Hk (M ) is the completion of Ck (M ) with respect to ∥ · ∥H p(·) (M ) . k

Subsequently, we introduce variable Hölder space on Riemannian manifolds [26]. For this purpose, we need to recall the notion of geodesic distance. For any smooth curve γ : [a, b] → M we define the length of γ by l(γ ) =

b



 g (γ (t ))

a



dγ dt

,

dγ



dt

dt .

For x, y ∈ M we define a distance dg by dg (x, y) = inf {l(γ ) : γ : [0, 1] → M , γ (0) = x, γ (1) = y} . For instance, by the Hopf–Rinow theorem we get that if the Riemannian manifold is compact, then any two points x, y in M can be jointed by a minimizing curve γ , i.e. l(γ ) = dg (x, y). Now, we are in position to define Hölder spaces. For a measurable function α : M → (0, 1] and a bounded continuous function u we introduce the seminorm [u]α(·),M = sup

x,y∈M

|u(x) − u(y)| . dg (x, y)α(x)

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M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

By C k,α(·) (M ) we denote the space of functions u : M → R of class C k for which the norm

∥u∥C k (M ) =

k 

sup |∇ j u(x)|

j=0 x∈M





and the seminorm |∇ k u| α(·),M are finite. This is a Banach space with respect to the following norm

  ∥u∥C k,α(·) (M ) = ∥u∥C k (M ) + |∇ k u| α(·),M . Finally, we introduce the notion of log-Hölder continuity on Riemannian manifolds. Definition 2.2. We say that u : M → R is log-Hölder continuous if there exists C > 0 such that for each x, y ∈ M the following inequality holds C

|u(x) − u(y)| ≤



log e +

1 dg (x,y)

.

Subsequently, we define the following class of variable exponents. Definition 2.3. We say that a measurable map p belongs to P log (M ) if p ≥ 1 and p is log-Hölder continuous. In the next lemma we explain the connection between the well known case of P log (Rn ) and P log (M ), where M is a compact Riemannian manifold. Namely, Lemma 2.1. Let p ∈ P log (M ), and (U , φ) be a chart such that 1 2

δij ≤ gij ≤ 2δij

as a bilinear forms, where δij is a standard delta Kronecker symbol. Then p ◦ φ −1 ∈ P log (φ(U )). Proof. Let us fix x, y ∈ φ(U ). It is not hard to see that

  √ 1 √ |x − y| ≤ dg φ −1 (x), φ −1 (y) ≤ 2 |x − y| . 2

Hence,

  p ◦ φ −1 (x) − p ◦ φ −1 (y) ≤

C



log e +

≤

1 dg φ −1 (x),φ −1 (y)

(

C



1 2|x−y|

log e + √



)

 ≤

2C



1 log e + |x− y|

. 

3. Sobolev embedding In this section we show the so-called Sobolev embedding theorem which is the core of the Sobolev theory. We start by proving that the continuous embedding theorem for the first order Sobolev spaces. Subsequently, we show the theorem for higher order spaces. Theorem 3.1. Let (M , g ) be a compact Riemannian n-manifold and q ∈ P log (M ). q(·)

(i) If q(x) < n, then H1 (M ) ↩→ Lp(·) (M ), where p(1x) = q(1x) − 1n . q(·) (ii) If q(x) > n, then H1 (M ) ↩→ C 0,α(·) (M ), where α(x) = 1 − q(nx) . Proof. Our proof starts with the following observation. Since M is compact, it can be covered by finite number of charts {(Ul , φl )}Nl=1 , such that for any l the components gil,j of g in (Ul , φl ) satisfy 1 2

δij ≤ gijl ≤ 2δij

as a bilinear forms. Let {ηj } be a smooth partition of unity subordinate to covering {Ul }Nl=1 .

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

51

(i) For any u ∈ C ∞ (M ) and any i = 1, . . . , N we have



|ηi u|p dVg =



M

|ηi u|p dVg

Ui





= φi (Ui )

n

det gij |ηi u|p ◦ φi−1 (x)dx ≤ 2 2

 φi (Ui )

|ηi u|p ◦ φi−1 (x)dx.

(1)

On the other hand, we can estimate the gradient in the following manner



|∇(ηi u)| dVg = q

 φi (Ui )

M

 q◦φ −1 n   i  −1 −1  det gij  gkj Dk ((ηi u) ◦ φi )Dj ((ηi u) ◦ φi ) dx k,j=1     −1 ∇((ηi u) ◦ φ −1 )q◦φi dx. i



≥ 2−

n+q+ 2

(2)

φi (Ui )

Next, let us assume that

∥ηi u∥H q(·) (M ) ≤ 1. 1

Hence, by the definition of Sobolev norm we get

∥∇ (ηi u) ∥Lq(·) (M ) ≤ 1. Then by (2) we obtain the estimate

 φi (Ui )

+  −1  ∇((ηi u) ◦ φ −1 )q◦φi dx ≤ 2 n+2q i



|∇(ηi u)|q dVg ≤ 2

n+q+ 2

.

(3)

M

By Lemma 2.1 we have that p ◦ φi−1 ∈ P log (φ(Ui )). Thus we can use the Euclidean version of the Sobolev embedding theorem (see [19]) on φ(Ui ) with constant Ci

∥(ηi u) ◦ φ −1 ∥Lp◦φ−1 (·) (φ(U )) ≤ Ci ∥∇((ηi u) ◦ φ −1 )∥Lq◦φ−1 (·) (φ (U )) . i

i

i

(4)

Combining (3) and (4) with the well known inequality for semimodulars ∥x∥ρ ≤ ρ(x) + 1 yields

∥(ηi u) ◦ φ −1 ∥Lp◦φ−1 (φ (U )) ≤ Ci (2 i

n+q+ 2

i

+ 1).

Hence, by basic properties of variable Lebesgue spaces we have



 −1   (η u) ◦ φ −1 p◦φi i   dx ≤ 1.   n+q+  φi (Ui ) C (2 2 + 1)  i

That gives us

 φi (Ui )

 p+ +   −1 (ηi u) ◦ φ −1 p◦φi dx ≤ Ci (2 n+2q + 1) .

Taking into account (1), (5) and once again the inequality ∥x∥ρ ≤ ρ(x) + 1, we get

∥ηi u∥Lp(·) (M ) ≤ 2

n 2



Ci ( 2

n+q+ 2

 p+ + 1) +1 .

Introducing the following constant C = max 2 1≤i≤N

n 2



Ci (2

n+q+ 2

 p+ + 1) +1 ,

we obtain the following inequality

∥ηi u∥Lp(·) (M ) ≤ C ∥ηi u∥H q(·) (M ) . 1

(5)

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M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

Finally, using the above estimate and the Hölder inequality, we are able to compute

∥u∥Lp(·) (M ) ≤

N 

∥ηi u∥Lp(·) (M ) ≤ C

N    ∥∇ηi u∥Lq(·) (M ) + ∥ηi ∇ u∥Lq(·) (M ) + ∥ηi u∥Lp(·) (M )

i=1

i=1

≤ CN max ∥∇ηi ∥∞ ∥u∥Lq(·) (M ) + CN ∥∇ u∥Lq(·) (M ) + CN ∥u∥Lp(·) (M ) ≤  C ∥u∥H q(·) (M ) . 1≤i≤N

1

This completes the proof of the first part of the theorem. (ii) In this case q > n. Let us assume as before that

∥u∥H q(·) (M ) ≤ 1. 1

Then, we obtain the following string of inequalities

 φi (Ui )

≤2

  −1 (ηi u) ◦ φ −1 q◦φi dx +



i

n+q+



φi (Ui ) +

|ηi u|q dVg + 2q

2

+   −1 ∇(ηi u) ◦ φ −1 q◦φi dx ≤ 2 n+2q i

|ηi u|q dVg +



M



M



|∇ηi u|q dVg +



M

|ηi ∇ u|q dVg

|∇(ηi u)|q dVg



M



M

     n+3q+ n+3q+ + + |u|q dVg + |∇ u|q dVg ≤ 2 2 + max ∥∇ηi ∥q∞ . ≤ 2 2 + max ∥∇ηi ∥q∞ i

M

i

M

Hence, we can now proceed as in the proof of case (i) obtaining the following estimate

∥(ηi u) ◦ φi−1 ∥

−1 (·)

q◦φ H1 i

n+3q+ 2

≤2

+

+ max ∥∇ηi ∥q∞ + 1. i

(φi (Ui ))

From Lemma 2.1 we have that q ◦ φi−1 ∈ P log (φi (Ui )). Therefore, by the Morrey type theorem for variable spaces (see [23]): −1 q◦φi (·)

H1

(φi (Ui )) ↩→ C

0,1−

n q◦φ

−1 (·)

i

(φi (Ui )).

So, there exists a constant Ci such that

∥(ηi u) ◦ φi−1 ∥

0,1−

C

≤ Ci ∥(ηi u) ◦ φi−1 ∥

n

−1 (·)

q◦φ i

(φi (Ui ))



n+3q+ 2

≤ Ci 2

−1 (·)

q◦φ i

H1

(φi (Ui ))

 + + max ∥∇ηi ∥q∞ + 1 .

(6)

i

By the definition of variable Hölder spaces we can compute

∥ u∥ C

0,1− n q(·) (M )

= sup |u(x)| + sup

x,y∈M

x∈M

≤

N 

|u(x) − u(y)|

sup |u(x)ηi (x)| +

i=1 x∈M

≤

N 

1− q(nx)

dg (x, y) N 

sup

sup |(uηi ) ◦ φi−1 (x)| +

N √ 

2∥(ηi u) ◦ φi−1 ∥

i =1

dg (x, y)

N √ 

i=1 x∈φi (Ui )

≤

|u(x)ηi (x) − u(y)ηi (y)|

i=1 x,y∈M

0,1−

C

2

−1

sup

i=1 x,y∈φi (Ui )

.

n q◦φ i

1− q(nx)

(φi (Ui ))

The above inequality and (6) yield

√ ∥ u∥ C

0,1− n q(·) (M )



≤ N 2 max Ci 2

This completes the proof.

1≤i≤N



n+3q+ 2

q+



+ max ∥∇ηi ∥∞ + 1 . i

|(uηi ) ◦ φi−1 (x) − (uηi ) ◦ φi−1 (y)| 1−

|x − y|

n

−1 (x)

q◦φ i

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

53

Now, we are able to formulate and show Sobolev type theorems for higher order spaces. This theorem can be formulated in the following way. Theorem 3.2. Let (M , g ) be a compact Riemannian n-manifold, q ∈ P log (M ) and k, l ∈ N ∪ {0} with k > l. q(·)

p(·)

n l (i) If q(x) < k− , then Hk (M ) ↩→ Hl (M ), where p(1x) = q(1x) − k− . l n q(·) n (ii) If q(x) > k−l−α(x) , for some log-Hölder continuous function α defined on M with 0 < α(x) < 1, then Hk (M ) ↩→ C l,α(·) (M ). q(·)

Proof. (i) Let us fix u ∈ Hk (M ). Put ql (x) =

q(x)n n − q(x)(k − l)

.

Next, for m ∈ {0, . . . , k − 1}, we define the map ψm = |∇ m u|. Consequently, by Theorem 3.1 we have

  ∥ψm ∥Lqk−1 (·) (M ) = ∥∇ m u∥Lqk−1 (·) (M ) ≤ C ∥∇ m u∥Lqk (·) (M ) + ∥∇|∇ m u|∥Lqk (·) (M ) . By Kato’s inequality (see [14, Proposition 2.11]):

|∇|∇ m u|| ≤ |∇ m+1 u|, we obtain the estimate

∥ψm ∥Lqk−1 (·) (M ) ≤ C ∥u∥H q(·) (M ) = C ∥u∥H qk (·) (M ) . k

k

This gives us

∥ u∥

≤ C ∥u∥H qk (·) (M ) .

qk−1 (·) Hk−1 (M )

(7)

k

Finally, using inequality (7) k − l times we get

∥u∥H p(·) (M ) = ∥u∥H ql (·) (M ) ≤ C ∥u∥H qk (·) (M ) = C ∥u∥H q(·) (M ) , l

l

k

k

and this is precisely the assertion of the first part of the theorem. (ii) Let us define the exponent  q(x) by 1

 q(x)

=

k − l − α(x) n

.

Since α ∈ P log (M ), it follows that  q ∈ P log (M ). Taking into account the conditions on exponent α , we obtain q ( x) <

n k−l−1

.

Now, we can easily see that the assumptions of the first part of the theorem are satisfied. Therefore, we obtain the embedding  q(·)

Hk (M ) ↩→

1

1 k−l−1 n

−  q(·) H l +1

(M ).

Moreover, 1

 q(x)

−

k−l−1 n

=

k − l − α(x) n

−

k−l−1 n

1 − α(x)

=

n

.

Thus, by the Hölder inequality we have the following embedding 1

1 k−l−1 n

−  q(·)

Hl+1

n

1−α(·) (M ) ↩→ Hl+ (M ). 1

n Since 1−α( > n, we can apply the second part of Theorem 3.1. Therefore, by Kato’s inequality we have for each m ∈ x) {0, . . . , l} the following estimate n n ∥∇ m u∥C α(·) (M ) ≤ C ∥∇|∇ m u|∥ 1−α(·) + ∥∇ m u∥ 1−α(·) L (M ) L (M )   m + 1 m ≤ C ∥∇ u∥ n + ∥∇ u∥ n . 1−α(·) 1−α(·)

L

(M )

L

(M )

54

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

This gives us the embedding n 1−α(·)

Hl+1

(M ) ↩→ C l,α(·) (M ).

One can see that  q(x) ≤ q(x) from the definition of  q. Thus, by the Hölder inequality we have the chain of embeddings q(·)

 q(·)

Hk (M ) ↩→ Hk (M ) ↩→ C l,α(·) (M ). This is the desired conclusion.



We finish this section with the example of a Riemannian manifold (M , g ) and a bounded exponent q : M → R such that q(·) the embedding H1 (M ) ↩→ Lp(·) (M ) does not hold. The example is a slight modification of Proposition 8.3.7 in [19]. Example 1. Let us take (S 2 , g ), where g is a standard metric tensor induced from R3 . Let us consider the spherical coordinates system (φ, θ ), where φ ∈ (0, 2π )and θ ∈ (0, π ). We fix t and s such that 1 < t < s < 2 and for τ ∈ [t , 2] we define the following function f (τ ) = 2 τt − 1 . Next, we define variable exponent q by if θ ≤ 1 and φ ≥ θ f (t ) = 1 for τ ∈ (t , s) satisfying θ ≤ 1 and φ = θ f (τ ) if θ ≤ 1 and u ≤ φ f (s) if θ > 1.

  t,   τ, q(φ, θ ) =  s,   1,

Let us consider function u : M → R given by u(φ, θ ) = θ µ , where µ := s−t 2 . We claim that u ̸∈ Lp(·) (S 2 ), where Indeed, we have



|u|p dVg ≥ M

1

 0

f (s)



2sµ

θ 2−s sin θ dφ dθ =

1



0

2sµ

θ 2−s +f (s)+1

0

sin θ

θ

dθ =

1



θ −1

sin θ

θ

0

1 p

=

1 q

− 1n .

dθ = ∞.

By straightforward calculation we can see that

|∇ f |(φ, θ ) = θ µ−1 . We claim that θ µ−1 ∈ Lq(·) (M ). First of all, we consider the part where q = t. Then, 1



2π



0

θ

(µ−1)t

sin θ dφ dθ = (2π − 1)

1



1

θ (µ−1)t +1

sin θ

θ

0

dθ < ∞,

since (µ − 1)t + 1 > −1. Subsequently, if q = s, we have 1



θ f (s)



0

θ

(µ−1)s

sin θ dφ dθ =

0

1



θ (µ−1)s+f (s)+1

0

sin θ

θ

dθ < ∞.

f (τ ) Next, let us   observe that on the remaining part we have q(φ, θ ) = τ and φ = θ . Simple computation gives q = ln φ + 1 t. Thus, we get 2 ln θ 1



θ f (t )



θ f (s)

0

θ

(µ−1)q

sin θ dφ dθ ≤

1



1



0

θ

  1 (µ−1) 12 ln φ+ln θ t

=

e

0

Finally, π

 1



2π

θ µ−1 dφ dθ ≤ 2π (π − 1).

0

This gives that |∇ u| ∈ Lq(·) (M ). Moreover, u(θ , φ) = θ µ = θ µ−1 θ ≤ π θ µ−1 . Hence, u ∈ Lq(·) (M ).

sin θ dφ dθ

0 1



  (µ−1) 2lnlnφθ +1 t

0

sin θ dφ dθ =

1



φ 0

µ−1 2

t

dφ

1

 0

θ (µ−1)t +1

sin θ

θ

dθ < ∞.

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

55

4. Compact embedding In this section we present some compactness theorems for Sobolev spaces with variable exponent on Riemannian manifolds. Results of this type are often called Rellich–Kondrachov theorems. Our result can be formulated as follows Theorem 4.1. Let (M , g ) be a compact Riemannian n-manifold and q ∈ P log (M ). q(·)

(i) If q(x) < n, then H1 (M ) ↩→↩→ Lp(·) (M ), where infx∈M (ii) If q(x) > n, then

q(·) H1

(M ) ↩→↩→ C

0,α(·)



1 p(x)

−





(M ), where infx∈M 1 −

n q(x)



> 0.  − α(x) > 0.

1 q(x)

−

1 n

Before we prove the theorem we stress that in the theorem we do not assume that the exponents α, p are continuous. Proof. (i) Put r (x) =

nq(x) n − q(x)

.

By Theorem 3.1 and by the Rellich–Kondrachov theorem (see [15]) we have q(·)

H1 (M ) ↩→ Lr (·) (M ), q(·)

H1 (M ) ↩→ H11 (M ) ↩→↩→ L1 (M ). Next, using complex interpolation theory (see [19] Corollary 7.1.6), we obtain for θ ∈ (0, 1) the following embeddings q(·)

  ↩→↩→ Lr (·) (M ), L1 (M ) θ = Lrθ (·) (M ),    > 0, there exists ϵ > 0 such that the inequality where r 1(x) = r (θx) + 1 − θ . Since infx∈M p(1x) − q(1x) − 1n θ H1

1 p(x)

>

1 r (x)

+ϵ

n −1 holds for each x ∈ M. Moreover, since nr ⇒ r (1x) , we have (x)

1 p(x)

−

1 r n−1 (x)

=

n

1 p(x)

−

n−1 nr (x)

−

1 n

>

ϵ 2

1 for sufficiently large n. Let us fix such n that the above inequality holds and set θ = n− . Then, the Hölder inequality gives n the embeddings q(·)

H1 (M ) ↩→↩→ Lrθ (·) (M ) ↩→ Lp(·) (M ), which proves the part (i). (ii) First of all we show the following lemma. Lemma 4.1. Let M be a compact Riemannian manifold, γ , β : M → (0, 1] with infx∈M (β(x) − γ (x)) > 0. Then C 0,β(·) (M ) ↩→↩→ C 0,γ (·) (M ). Proof. Let us note that the identity operator is a continuous map from C 0,β(·) (M ) to C 0,γ (·) (M ). Indeed, we have the sequence of inequalities

∥u∥C 0,γ (·) (M ) = sup |u(x)| + sup x∈M

x,y∈M

|u(x) − u(y)| dg (x, y)γ (x)

|u(x) − u(y)| dg (x, y)β(x)   |u(x) − u(y)| ≤ (diam (M ) + 1) sup |u(x)| + sup = (diam (M ) + 1) ∥u∥C 0,β(·) (M ) . β(x) x∈M x,y∈M dg (x, y) ≤ sup |u(x)| + sup (diam (M ) + 1)β(x)−γ (x) sup x∈M

x∈M

x,y∈M

Now, we show the compactness. For this purpose we take a bounded sequence {un } ⊂ C 0,β(·) (M ). From the Arzela–Ascoli theorem there exists a subsequence {unk } which uniformly converges to some continuous function u. Then, putting wk = unk − u we get

|wk (x) − wk (y)| sup |wk (x)| + sup ≤ sup |wk (x)| + sup dg (x, y)γ (x) x∈M x,y∈M x∈M x,y∈M



|wk (x) − wk (y)| dg (x, y)β(x)

 γ ( x) β(x)

γ (x) 1− β(x)

sup(2∥wk ∥C 0 (M ) ) x∈M

.

56

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

Since, the quantity

 C = sup

x,y∈M

|wk (x) − wk (y)| dg (x, y)β(x)

 γ ( x) β(x)

is finite, we obtain

∥wk ∥C 0,γ (·) ≤ ∥wk ∥C 0 (M ) + C sup(2∥wk ∥C 0 (M ) )

γ (x) 1− β(x)

.

x∈M

Since wk converges to 0 in C 0 (M ), the above inequality implies that wk converges to 0 in C 0,γ (·) (M ), and the lemma follows.  Now, we are able to finish the proof of Theorem 4.1. Combining Theorem 3.1 with Lemma 4.1 and with the fact that n we get the embedding q(x)

α(x) < 1 − q(·)

H1 (M ) ↩→ C

n 0,1− q(·)

which completes the proof.

(M ) ↩→↩→ C 0,α(·) (M ), 

In order to finish this section, let us make some remarks. Using Theorem 2.3 of [7] and Theorem A.1 from the Appendix of our paper, we can improve slightly the Theorem 4.1 in the following way. Theorem 4.2. Let (M , g ) be a compact Riemannian n-manifold and q ∈ P (M ) ∩ C (M ). q(·)

(i) If q(x) < n, then H1 (M ) ↩→↩→ Lp(·) (M ), where infx∈M



1 p(x)

−



1 q(x)

−

  q(·) (ii) If q(x) > n, then H1 (M ) ↩→↩→ C 0,α(·) (M ), where infx∈M α(x) − 1 −



> 0.  n < 0. q(x)

1 n

Combining Theorem 4.1 with Theorem 3.2 we get the compact embedding theorem for higher order derivatives. Theorem 4.3. Let (M , g ) be a compact Riemannian n-manifold and q ∈ P log (M ) and k, l ∈ N ∪ {0} such that k > l. q(·)

p(·)

n , then Hk (M ) ↩→↩→ Hl (i) If q(x) < k− l

(ii) If q(x) > C l,α(·) (M ).

n , k−l−α(x)

(M ), where

1 p(x)

>

1 q(x)

−

k−l . n q(·)

for some log-Hölder continuous function α defined on M with 0 < α(x) < 1, then Hk (M ) ↩→↩→

4.1. The Poincaré inequality As an application of the compactness Theorem 4.1 we show the so-called Poincaré inequality. Theorem 4.4. Let (M , g ) be a compact Riemannian n-manifold and p ∈ P log (M ). Then, there exists C > 0 such that for each p(·) u ∈ H1 the inequality holds

∥u − u¯ ∥Lp(·) (M ) ≤ C ∥∇ u∥Lp(·) (M ) ,  where u¯ = Vol1(M ) M u dVg . Proof. It is enough to show the inequality for u such that u¯ = 0. Suppose the assertion of the theorem is false. Hence, there exists a sequence un such that un = 0, ∥un ∥Lp(·) (M ) = 1 and

∥∇ un ∥Lp(·) (M ) ≤

1 n

. p(·)

Thus, un is a bounded sequence in H1 (M ) and by Theorem 4.1 there exists a subsequence unk which converges to some p(·)

function u in Lp(·) (Ω ). Since |∇ un | converges in Lp(·) (M ), we can see that unk is a Cauchy sequence in H1 (M ). Thus, p(·)

u ∈ H1 (M ). Moreover,

∥∇ u∥Lp(·) (M ) = 0,

∥u∥Lp(·) (M ) = 1.

This contradicts the fact that u¯ = 0 and completes the proof.



M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

57

4.2. The Poisson equation In this section we examine the following Poisson equation

− div(|∇ u|p(x)−2 ∇ u) = f .

(8)

Namely, we show that above equation possesses a unique solution if and only if the average of f is 0 and p(.) satisfies some technical hypotheses. Theorem 4.5. Let (M , g ) be a compact Riemannian n-manifold, p ∈ P log (M ), p− > 1 and f ∈ Lp (·) (M ). Then, there exists a unique weak solution u of Eq. (8) such that u¯ = 0 if and only if f¯ = 0. ′

Proof. Let us assume that f¯ = 0. We define the set



p(·)

M = u ∈ H1 (M ) :





udVg = 0 . M

Next, we consider the following functional E (u) =



1 p(x)

M

|∇ u|p(x) dVg −



fudVg . M

We claim that there exists u ∈ M which minimizes E in M . In order to do so we shall need the following observation. Lemma 4.2. If ∥u∥Lp(·) (M ) > 1, then



|u|p(x) dVg ≥ ∥u∥Lp−(·) (M ) . p

M

Proof. Since ∥u∥Lp(·) (M ) > 1, for 1 < λ < ∥u∥Lp(·) (M ) we have

 M

|u|p(x) dVg ≥ λp−

 M

|u|p(x) dVg > 1. λp

This gives us



|u|p dVg > λp− . M

Finally, taking the supremum on λ with 1 < λ < ∥u∥Lp(·) (M ) , we finish the proof.



Theorem 4.4 implies that the norms ∥u∥H p(·) (M ) and ∥∇ u∥Lp(·) (M ) are equivalent on M . Combining this fact with Lemma 4.2, 1

we get for ∥∇ u∥Lp(·) (M ) > 1 the following inequality E (u) ≥

≥



1 p+ C p+

|∇ u|p(x) dVg −

M

∥u∥

 fudVg ≥ M

p− p(·) H1 (M )

1 p+

p

∥∇ u∥Lp−(·) (M ) − 2∥u∥Lp(·) (M ) ∥f ∥Lp′ (·) (M )

− 2∥u∥H p(·) (M ) ∥f ∥Lp′ (·) (M ) . 1

p(·)

Now, let us take a minimizing sequence {un } ⊂ M . From above inequality we see that {un } is bounded in H1 (M ). Since p(·) H1

(M ) is reflexive, there exists a subsequence {unk } such that un k ⇀ u

p(·)

weakly in H1 (M ). p(·)

Clearly E is convex and sequentially lower semicontinuous in H1 (M ). Therefore, E is weakly sequentially lower semicontinuous. This gives us E (u) ≤ lim inf E (unk ) = inf E (u). n→∞

u∈M

Moreover, M is closed and convex. Hence, by Mazur’s Theorem M is weakly closed. Thus, u ∈ M . This two facts imply that  u is a minimizer of E with constraint M udVg = 0. Since M is a C 1 -Banach submanifold, we can apply Ljusternik Theorem. Hence, there exists a Lagrange multiplier λ ∈ R such that the following equality



|∇ u|p−2 ∇ u∇φ dVg − M



f φ dVg − λ M



φ dVg = 0 M

(9)

58

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

holds for all φ ∈ C0∞ (M ). Since M is compact, the map φ = 1 is a good test function, i.e. φ ∈ C0∞ (M ). Hence, putting φ = 1 in (9) we get λ = 0. This show that u satisfy (8). Now let us show that solution is unique. Suppose that u1 and u2 satisfy (8). Then we have



|∇ u1 |p−2 ∇ u1 (∇ u1 − ∇ u2 )dVg =

M

|∇ u2 |p−2 ∇ u2 (∇ u2 − ∇ u1 )dVg =

M



f (u1 − u2 )dVg

M

f (u2 − u1 )dVg .

M

Adding the above two equations yields



(|∇ u1 |p−2 ∇ u1 − |∇ u2 |p−2 ∇ u2 )(∇ u1 − ∇ u2 )dVg = 0. M

But



(|∇ u1 |p−2 ∇ u1 − |∇ u2 |p−2 ∇ u2 )(∇ u1 − ∇ u2 )dVg M  = |∇ u1 |p − |∇ u2 |p−2 ∇ u1 ∇ u2 − |∇ u1 |p−2 ∇ u1 ∇ u2 + |∇ u2 |p dVg M  ≥ |∇ u1 |p − |∇ u2 |p−1 |∇ u1 | − |∇ u1 |p−1 |∇ u2 | + |∇ u2 |p dVg M    = |∇ u1 |p−1 − |∇ u2 |p−1 (|∇ u1 | − |∇ u2 |) dVg . M

Hence, we obtain that u1 = u2 almost everywhere. This completes the proof of the existence and of the uniqueness. Finally, let us assume that the equation has a unique weak solution u. Thus, taking φ = 1 as a test function, we just get  fdVg = 0.  M Acknowledgments The authors wish to thank Daniel Pons for his collaboration in proving Lemma 2.1. Moreover, they wish to thank Enrique Reyes for his fruitful comments and suggestions. Some part of the research for this paper was performed during the visit of MG and PG to the Universidad de Santiago de Chile (partially supported by FONDECYT grant 1111042). The work of the second author was partially supported by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Program. Special thanks are due to Campanario Sour for having created special feelings of collaboration on this topic. Both authors want to thank the referees for their careful reading of the paper and remarks which helped make the paper better. Appendix In this section we give the proof of the following theorem.

¯ ) ∩ C (Ω ¯ ), α ∈ C (Ω ¯ ) such that p(x) > n and 0 ≤ α(x) Theorem A.1. Let Ω ⊂ Rn be a precompact open set. If p ∈ P (Ω < 1 − p(nx) , then W 1,p(·) (Ω ) ↩→↩→ C 0,α(·) (Ω ). The principal significance of the theorem is that we do not assume that p is log-Hölder continuous. To the best of our knowledge, this theorem seems to be new. Proof. The proof is based on the concept of Fan and Zhao (see [7]). Since Ω is a precompact set, there exists an finite open covering {Ui }Ni=1 of Ω such that

 − n α + (Ui ) < 1 − (Ui ). p

By the Sobolev embedding theorem for Sobolev spaces with constant exponent we have: 0,1− −n − 0, 1− np p (Ui ) W 1,p(·) (Ui ) ↩→ W 1,p (Ui ) (Ui ) ↩→↩→ C (Ui ) = C



−

(Ui )

(Ui ) ↩→ C 0,α

+ (U ) i

(Ui ) ↩→ C 0,α(·) (Ui ).

M. Gaczkowski, P. Górka / Nonlinear Analysis 92 (2013) 47–59

59

Now, let us denote by δ < 1 a Lebesgue number of cover Ui . If x, y are such that dg (x, y) ≤ δ , then there exists i ∈ {1, . . . , N } such that x, y ∈ Ui . This allows us to write the following estimation

|u(x) − u(y)| |u(x) − u(y)| ≤ ≤ + α( x ) dg (x, y) dg (x, y)α (Ui )

|u(x) − u(y)| dg (x, y)



1− np

−

(Ui )

≤ ∥u∥ 0,(1− n )− (U ) ≤ C ∥u∥W 1,p(·) (Ui ) ≤ C ∥u∥W 1,p(·) (Ω ) . i (U ) p C i If dg (x, y) > δ , then we get

|u(x) − u(y)| 2 2 ≤ α+ ∥u∥∞ ≤ max α+ ∥u∥ 0,(1− n )− (U ) α( x ) i (U ) p i dg (x, y) C δ δ i ≤ max i

2C

δ α+

∥u∥W 1,p− (Ui ) (U ) ≤ i

2C

δ α+

∥u∥W 1,p(·) (Ω ) .

Therefore, we conclude from the above inequalities that there exists C˜ such that the following inequality holds

∥u∥C 0,α(·) (Ω ) ≤ C˜ ∥u∥W 1,p(·) (Ω ) .

(10)

This completes the proof of the continuity of the embedding. Now, we show the compactness. For this purpose we define the following map

α( ¯ x) =

α(x) + 1 −

n p(x)

2

.

¯ ), 0 ≤ α( Let us stress that α¯ ∈ C (Ω ¯ x) < 1 − p(nx) and moreover α¯ > α . Hence, from inequality (10) and Lemma 4.1 we obtain ¯ W 1,p(·) (Ω ) ↩→ C 0,α(·) (Ω ) ↩→↩→ C 0,α(·) (Ω ).

This proves the theorem.



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