Capacities in Generalized Orlicz Spaces
In this paper basic properties of both Sobolev and relative capacities are studied in generalized Orlicz spaces. The capacities are compared with each other and the Hausdorff measure. As an application, the existence of quasicontinuous representative of generalized Orlicz functions is proved.
In the calculus of variations one studies existence and properties of solutions to minimization problems such as Classical techniques, by De Giorgi and Moser, cover both the linear case and the -growth case, where . Marcellini  developed the theory of -growth, which is based on the growth assumption . Zhikov  studied such minimizers as models of anisotropic materials and also observed that they exhibit the so-called Lavrentiev phenomenon whereby minimizers do not have improved regularity and may even be discontinuous.
In the variable exponent case, , the change in the anisotropy (growth rate) is gradual owing to the continuity of . For instance, in electrorheological fluid dynamics, where the anisotropy depends on the smooth electrical field, this is a reasonable assumption . In other situations, such as composite materials, a more clear-cut transition is better. To this end, Baroni, Colombo, and Mingione [4–7] have developed a regularity theory of the double phase functional , , which has the property that the growth rate changes abruptly from to in the sets and (see also [8–12]). Recently, we were able to generalize their first step, showing Hölder continuity, to the general -growth case . Also other results have recently been obtained for partial differential equations with generalized Orlicz growth, cf. [13–16].
A different approach to differential equations is based on (nonlinear) potential theory. The foundation of nonlinear potential theory includes general notions of a Radon measure, a capacity and generalized functions. The sets of capacity zero are the exceptional sets for representatives of the function. In this paper we give basic properties of both Sobolev capacity and relative capacity in the generalized Orlicz setting. We follow the general framework of  and its previous adaption to the variable exponent setting [18, Chapter 10]. The results can be applied, e.g., in the study of boundary behavior of solutions to PDE.
We consider general -functions which need not be convex. Many of the proofs in this paper follow a standard pattern, since they do not depend on the exact form of the integrand used in the definition of capacity. We have therefore omitted or abbreviated several proofs (e.g., Section 6). Nevertheless, it is necessary to check these basic building blocks in order to proceed with constructing the theory, since the results are not covered by earlier results, merely similar. There are also some proofs which are new, namely, Theorem 9, Example 13, and Proposition 20. Furthermore, this general setting clarifies the necessity of various assumptions for different properties (e.g., Theorem 8). In particular, we find that the relative capacity is Choquet also for nonconvex -function, whereas convexity is needed for the Sobolev capacity.
It should be noted that Ohno and Shimomura  have recently studied (Sobolev) capacity in the generalized Orlicz case. However, they consider the capacity in a metric measure space setting with Hajłasz gradients. These results therefore work in the Euclidean setting only when the maximal operator is bounded, since the Hajłasz gradient corresponds to .
The outline of the paper is as follows. We start by introducing our notation and basic definitions. Then we study Sobolev capacity and compare it with the Hausdorff measure. Next we derive existence of a quasicontinuous representative of generalized Orlicz function. Finally, we study relative capacity and compare it with the Sobolev capacity.
We study spaces of functions defined in or opens sets .
A real-valued function is -almost increasing, , if for . So a -almost increasing function is increasing. -almost decreasing is defined analogously.
Definition 1. We say that is a weak -function, and write , if the following conditions hold: (i)For every the function is measurable and for every the function is non-decreasing and left-continuous. (ii) and for every . (iii)There exists such that is -almost increasing in , for every . If is convex with respect to the second variable, then it is called a convex -function, and we write .
If there exists such that and for all , then we say that (A0) holds. Further, we say that satisfiesif there exists such that is -almost increasing in for every ,if there exist such that holds,if there exists such that is -almost decreasing in for every ,if there exist such that holds.
The corresponding conditions with are denoted by (Inc) and (Dec). Note that the definition of weak -function includes assumption .
Definition 2. Let and define the modular for by The generalized Orlicz space, also called Musielak–Orlicz space, is defined as the set equipped with the (Luxemburg) norm For , is a Banach space [18, Theorem 2.3.13.].
Definition 3. A function belongs to Sobolev space , if its weak partial derivatives exist and belong to , that is, We define a semimodular on by which induces a quasinorm by .
Lemma 4. Let satisfy (aInc) and (aDec), and let for Assume further that the sequence is bounded. If as , then
Proof. Since is convex and satisfies (aDec) it satisfies (Dec) for some possible larger exponent by Lemma 2.6 of . Let . By convexity and (Dec) we obtain Next we subtract from both sides and integrate over . Hence Note that the choice implies that is a bounded sequence.
Swapping and gives a similar inequality, and, combining the inequalities, we find that The same argument can be applied also to the weak gradient. Hence Let be given. Since , we can choose so small that We can then choose so large that when and it follows that .
The first claim of the following lemma has been proved in [16, Lemma 4.4]. The proof for the second claim is similar.
Lemma 5. Let satisfy (A0) and let be bounded. (a)If satisfies (aInc)p, then . (b)If satisfies (aDec)q, then .
Lemma 6. Let . Let satisfy (aInc)p and (aDec)q. Then where is the maximum of the constants from (aInc) and (aDec).
Proof. If , then and the claim holds.
Let and assume first that .Then (aDec) gives that Integrating over , we find that , which yields . If we similarly use (aInc) to conclude that . The claim follows from these two cases.
3. Sobolev Capacity
We define a set of test-functions for the capacity of the set by The generalized Orlicz -capacity of is defined by
We now prove the following properties for the set function . We emphasize that these do not require the convexity of .
Proposition 7. Let . (S1).(S2)If , then .(S3)If , then .(S4)If , then .(S5)If are compact, then .
Proof. (S1) follows from testing with . Since a test-function for is also a test-function for , (S2) follows from the infimum in the definition of capacity.
If , then by (S2). Thus, For the opposite inequality, let and choose such that Denote . Then , and so This implies (S3) as .
Let . Choose such that Since in an open set containing , it follows that . Similarly, .
The lattice property of Sobolev functions [18, Proposition 8.1.9] implies that and almost everywhere in . Therefore The same argument also gives the equality in the complement of , i.e. . Hencealmost everywhere. An analogous results holds without the derivative .
Since , it follows from the definition that , and similarly . Combining this with the conclusion of the previous paragraph, we find that (S4) follows from this as .
It remains to prove (S5). Since for any , the “”-inequality follows from (S2). To prove the opposite inequality, we choose an open . Since is compact and is decreasing, there is a positive integer such that, for all . Then, again by (S2), we have Taking the infimum of this inequality over such sets , we obtain the claim by (S3).
Notice that we need convexity for the next property.
Theorem 8. Let satisfy (aInc) and (aDec) and . Then (S6).
Proof. Let us denote . By (S2), .
Now to prove the opposite inequality, we may assume that . Let and for every .
The space is uniformly convex and reflexive [20, Theorem 1.3]. The same holds for , which is a closed subspace of . By reflexivity, the bounded sequence has a subsequence which converges weakly to a function . It follows from the Banach-Saks theorem that Let . Then from which we get, by the triangle inequality, that as , so that in .
By the convexity of the modular and the choice of , we obtain that Now increases in whereas decreases. HenceBy considering a subsequence if necessary, we may assume that . Then . By definition of a test-function, there exists an open set such that in . As is an increasing sequence, it follows that for every . In , for all relevant , so that in the same (open) set. Since , we have Taking the union over of the previous inclusion, we find that in the open set , so that . HenceFurthermore, and, hence, by [18, Corollary 2.1.15], as . Then, using Lemma 4, we also have as , which we apply in (31) to obtain and further (29) implies
In the next result we use a trick to get back to weak -functions despite an application of (S6).
Theorem 9. Let satisfy (aInc) and (aDec) and . Then (S7)
Proof. Let with [21, Lemma 3.1]. Since satisfies (aDec) yields . Denote for By induction on (S4), we obtain that The same inequality holds also for . Now, using (S6) for , we have Furthermore, for , by (S4) and , By the two estimates in the previous paragraph, we obtain that Since the sum is finite (otherwise, there is nothing to prove), the second term on the right hand side tends to zero as . This gives the claim.
Remark 10. A set function satisfying the properties (S1), (S2), and (S7) is called an outer measure. This holds if is satisfies (aInc) and (aDec). If is convex and satisfies (aInc) and (aDec), then it is a Choquet capacity, , i.e., it satisfies (S1), (S2), (S5), and (S6). Then, for every Borel set ,
4. Sobolev Capacity and Hausdorff Measure
In this section, we discuss simple relations between the generalized Orlicz capacity and the Lebesgue and Hausdorff measures.
Lemma 11. Let satisfy (aDec) and (A0). Then every measurable set satisfies , where depends on (aDec) and (A0).
Proof. Let . By (aDec), with exponent and constant , we conclude that . We have for by (A0). Therefore Taking infimum over , we obtain the claim.
Proposition 12. Let satisfy (A0), (aInc)p, , and (aDec). If , then .
Proof. Let and be a cut-off function with in and . Let . By Lemma 5, we obtain where c depends on (aInc) and . The same inequality holds for . Now , and, thus by the assumptions on . Since we obtain that Since , we can choose with . Since satisfies (aDec), [18, Lemma 2.1.11]. Thus the above inequality implies that . Since is a test-function for the -capacity of , we get , for every . Since , the claim follows by the subadditivity of the -capacity.
In the previous result the assumption is natural, since gives the capacity to compare with. However, the assumption (aDec) is surprising. The next example shows that it is nevertheless needed.
Example 13. Let when and when . Then satisfies (aInc) with and (A0) with . Let be an open ball with a radius one. Let be a Lipschitz-continuous function that is one in , zero in and linear in . Then in and zero elsewhere. We obtain and thus . On the other hand .
The -dimensional Hausdorff measure of a set is denoted by .
Corollary 14. Let satisfy (A0), (aInc)p with and (aDec). (1)If and with , then for all .(2)If , then if and only if , where .
Proof. If , we then obtain by Proposition 12. This leads to the first claim by [23, Theorem 4, p. 156].
If , then we may choose in (1) and so . Since is a counting measure, must be an empty set. On the other hand, , by (S1).
Corollary 15. Let satisfy (A0), (aInc), and (aDec)q. Let be bounded. If or , then .
Proof. Let . By Lemma 5, . The remaining proof follows the same procedure as in the proof of Proposition 12. If , it follows from [23, Theorem 3, p. 154] that and thus the claim follows from the first part.
In this section, we prove the existence of -quasicontinuous representatives of generalized Orlicz functions. A function is -quasicontinuous if for every there exists an open set with such that restricted to is continuous. We say that a claim holds -quasieverywhere if it holds except in a set of Sobolev -capacity zero.
In this section we assume the density of continuous functions. A sufficient condition for this can be found in Theorem 6.5 of .
Lemma 16. Let satisfy (aInc) and (aDec). Then, for each Cauchy sequence , there is a subsequence which converges pointwise -quasieverywhere in . Moreover, the convergence is uniform outside a set of arbitrarily small Sobolev -capacity.
Proof. Let be a Cauchy sequence in . We assume without loss of generality, by considering a subsequence if necessary, that for every For we denote , and . Note that . We obtain by Corollary 2.1.15(a) of  that The subadditivity (S7) further implies Since is decreasing and , the limit exists. Since for all , we obtain by (S2) that Now converges pointwise in and , so the first claim of the lemma is proved. Moreover, for and , Therefore, the convergence is uniform in , and the second claim follows.
Theorem 17. Let satisfy (aInc) and (aDec). Assume that is dense in . Then for each , there exists a -quasicontinuous function such that almost everywhere in .
Lemma 18. Let satisfy (aDec). Suppose that is a nonnegative -quasicontinuous function with in . Then for every there exists such that .
Proof. Let and , and let be an open set such that restricted to is continuous and . Moreover, let us take such that , and denote . Then . The set is open and contains . Since in , . It remains to estimate .
Now, . By and we have , and so . Therefore by the assumption on . An analogous inequality holds for the gradient, and so the claim follows.
6. Relative Capacity
In this section, we introduce relative -capacity, analogous to the relative -capacity of variable exponent spaces in , taken with respect to an open set, , in .
Definition 19. Let , be compact, and . Denote We define . Further, for open, we set and, for an arbitrary set , we define The number is called the relative -capacity of with respect to .
In the next result we offer two alternate set of assumptions. It seems that (aDec) alone is not sufficient, although we have not been able to prove this.
Proposition 20. If satisfies either (Dec) or (A0), then where .
Proof. Since , the inequality “” is clear. Now, to prove the opposite inequality, we fix and let , be such that If (Dec) holds, we set . Then in , so that where we used (Dec) for the last inequality. The claim follows as .
If, on the other hand, (A0) holds, then we fix with in . Let . We set . Since the supports of and are disjoint, we observe that where has been used in the last inequality. The claim follows as .
Next, we show that and are the same; that is, the relative capacity is well defined on compact sets.
Proposition 21. Let . Then for every compact .
Proof. The inequality follows directly from the definition.
Now, to prove the opposite inequality, fix and let be such that . Then, is greater than one in , which is open since , and contains . Thus, is also a valid test-function for every compact set , so that The result follows from this as .
Next, we show that the relative capacity has the same basic properties as the Sobolev capacity.
Proposition 22. Let be open and . (R1).(R2)If , then .(R3)If , then .(R4)If