Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.
In their seminal paper  Brézis and Lieb prove a result about the decoupling of certain integral expressions, which has been used extensively in the calculus of variations. Using concentration compactness arguments in the spirit of Lions [16, 14, 15], we prove a variant of this lemma under weaker assumptions on the nonlinearity than known before. To describe a special case of the Brézis–Lieb lemma, suppose that Ω is an unbounded domain in , , for , and is a bounded sequence in that converges pointwise almost everywhere to some function u. If one denotes by the superposition operator induced by f, i.e., , then the result in  implies that and
The same conclusion is obtained in that paper for more general functions, imposing conditions that are satisfied for continuous convex f with and imposing additional conditions on the sequence .
A different approach to the decoupling of superposition operators along sequences of functions rests on certain regularity assumptions on f. For example, assume that satisfies
Our aim is to give a decoupling result under a different set of hypotheses that applies to a much larger class of functions f than considered above, within a certain range of exponents p. In particular, we do neither impose any convexity type assumptions on f as was done in , nor any regularity assumptions as in [19, 7] apart from continuity. The price we pay for relaxing the hypotheses on f is that we need to restrict the range of allowed growth exponents p in comparison with , that we need to assume some type of translation invariance for Ω, and that the decoupling result only applies to a smaller set of admissible sequences, namely sequences that converge weakly in . Nevertheless, the numerous applications in the calculus of variations for PDEs where these extra assumptions are satisfied justify the new set of hypotheses.
To keep the presentation simple and highlight the main idea, we only treat the case . From here on, function spaces are taken over unless otherwise noted. It would be possible to consider other domains or superposition operators between other spaces, and we plan to do so in forthcoming work. Nevertheless, we do allow a periodic dependency of f on the space variable.
To explain our results we formalize the notion of decoupling.
Suppose that X and Y are Banach spaces. Consider a map , a sequence and . We say that BL-splits along with respect to u (BL being an abbreviation for Brézis–Lieb) if
We say that almost BL-splits along with respect to u if, starting with any subsequence of , we can pass to a subsequence such that there is a sequence such that and
If u is a limit of in some unambiguous sense then we frequently omit to mention that (almost) BL-splitting is with respect to u.
By , the map induces a map that BL-splits along pointwise a.e. converging bounded sequences in with respect to their pointwise a.e. limits. On the other hand, the technique used to prove [1, Lemma 3.2] (and the related results in [10, 11]) yields the following: if satisfies
then the induced superposition operator almost BL-splits along any -converging bounded sequence in with respect to its limit in , see Theorem 2.1 (a) below. This result is basically Lion’s approach, with a simplifying twist. If in addition is uniformly continuous on bounded subsets of , then it is easy to see that it BL-splits along any -converging bounded sequence in with respect to its limit in , see [2, Lemma 6.3]. For example, this holds true if (1.2) is satisfied.
We illustrate the distinction between BL-splitting and almost BL-splitting by the following examples.
If and if either or , then there is a bounded sequence in that converges in and pointwise a.e. to a function u such that the induced continuous superposition operator does not BL-split along any subsequence of with respect to u. On the other hand, almost BL-splits along any -converging bounded sequence in with respect to its limit in . Hence is not uniformly continuous on bounded subsets of and neither the general conditions used in  nor (1.2) are satisfied for f in these examples.
The sequences mentioned in the example are provided in Section 4 below.
Our main interest is to avoid condition (1.2), or any other conditions on f that ensure uniform continuity on bounded subsets of (e.g., a local Hölder condition, together with an appropriate growth bound on the Hölder constants on bounded intervals). Our result below states that it is sufficient to restrict to bounded subsets of instead.
In this context we now formulate our main theorem, in a slightly more general setting than what we considered above. A function is a Caratheodory function if f is measurable and if is continuous for almost every . The induced superposition operator on functions is then given by . If A is a real invertible -matrix, then f is said to be A-periodic in its first argument if for all , , and .
Denote by if and if or the critical Sobolev exponent for . Recall the continuous and compact embedding of the Sobolev space in for if is a bounded domain.
Consider , , and such that . Suppose that is a Caratheodory function that satisfies
and which is A-periodic in its first argument, for some invertible matrix . Denote by the continuous superposition operator induced by f. Then is uniformly continuous on bounded subsets of with respect to the --norms and hence also with respect to the --norms. Moreover, BL-splits along weakly convergent sequences in with respect to their weak limit.
Our proof of Theorem 1.3 has similarities with the proof of [18, Theorem 3.1] but involves an intermediate cut-off step in the proof of Theorem 2.1. Essentially, we first prove almost BL-splitting of along weakly converging sequences in with respect to their weak limit, using the concentration function and the compactness of the Sobolev embedding for and for a bounded domain U. Then we collect the possible mass loss at infinity along subsequences with the help of Lions’ Vanishing Lemma, employing the assumption .
Theorem 1.3 applies in particular to the functions considered in Example 1.2 when and . On the other hand, for there is a sequence in that converges weakly but possesses no subsequence along which BL-splits with respect to the weak limit. The same is true for . In this sense, Theorem 1.3 is optimal, that is, it cannot be extended in this generality to include the cases and . The existence of these counterexamples is proved in Section 4.
Of course, by Sobolev’s embedding theorem, a map that BL-splits along -converging bounded sequences in with respect to their limits in also BL-splits in along weakly convergent sequences with respect to their weak limits. Therefore, Theorem 2.1 (a), together with (1.2) (or the weaker Hölder condition with growth bound), yields BL-splitting maps along weakly convergent sequences in with respect to weak limits even for and .
The result also holds true in a slightly restricted sense for functions f that are sums of functions as in Theorem 1.3, i.e., functions that satisfy merely
where for . In that case, is uniformly continuous on bounded subsets of with respect to the - norms, and BL-splits along weakly convergent sequences in with respect to their weak limits.
The uniform continuity of operators on bounded subsets of has been used, for example, in the proof of [18, Lemma 3.4]. Nevertheless, we are not aware of a published proof of this fact, which is nontrivial in the generality stated in Theorem 1.3. Note that the uniform continuity of on bounded subsets of is trivial if U is bounded, by the compact Sobolev embedding .
We now discuss additional aspects and applications of the results presented above. To this end we return to a simple setting on . Suppose that satisfies (1.3) with and consider the functional given by
To prove the existence of a minimizer in typical variational problems involving Φ, Lions [14, 15] introduces the concentration-compactness principle. It is a tool to exclude the possibility of vanishing and of dichotomy along a minimizing sequence , in order to obtain compactness of the sequence. Here we are only concerned with dichotomy. In this case, the sequence is approximated by , where . For local functionals like Φ it then follows easily that is approximated by , a fact that yields, together with a hypothesis about energy levels, a contradiction. Clearly, the same can be achieved if Φ BL-splits along in a suitable way. Before our Theorem 1.3, Lions’ approach to concentration compactness was more general, in that, besides continuity and appropriate growth bounds, no extra regularity hypotheses need to be placed on f. On the other hand, the arguments are more involved than when using BL-splitting because one has to insert cut-off functions to obtain sequences and with disjoint supports. As a consequence, it is difficult to give a purely functional (abstract) presentation of Lions’ approach.
To explain the advantage of an abstract presentation using BL-splitting, we note that to treat nonlocal functionals of convolution type, e.g.,
the property of disjoint supports is not as effective anymore. In the convolution, the supports get “smeared out” and one has to control the interaction with more involved estimates, see [14, p. 123]. This is aggravated when one also has to consider the decoupling of derivatives of Ψ. We have shown in  that using BL-splitting is effective in situations involving nonlocal functionals. Moreover, BL-splitting even survives certain nonlocal operations, like the saddle point reduction; see [2, Theorem 5.1].
For particular cases there are other approaches to avoid conditions on f besides continuity and growth bounds. We reformulate and simplify the following cited results slightly to adapt them to our setting and notation. In  we proved, for satisfying (1.4) with , and setting , that the map , given by
BL-splits along a weakly convergent sequence if the weak limit is a function tending to 0 as . Another result was given in [13, Lemma 7.2], when satisfies (1.4) with and , and is as follows: The map (here denotes the space of bounded bilinear maps from into ), given by
is uniformly continuous on bounded subsets of . Together with the almost BL-splitting of Λ given by Theorem 2.1 below this yields BL-splitting for Λ along weakly convergent sequences. Note that the idea of the proof of the latter result does not apply for the maps Φ and Γ defined above (under the respective growth bounds on f). In both cases our result here is stronger, since we show uniform continuity and BL-splitting into the spaces , which are continuously embedded in and , respectively.
A different application of Theorem 1.3, that is independent of variational methods, is the general study of maps that are uniformly continuous on a subset of an infinite dimensional Hilbert space. These play a role in infinite dimensional potential theory [12, 5] or, more generally, in the theory of stochastic equations in infinite dimensions [17, 9, 8].
The paper is structured as follows: In Section 2 we treat almost BL-splitting of along bounded sequences in that converge in and along weakly convergent sequences in . In Section 3 we prove the uniform continuity of on bounded subsets of and BL-splitting of along weakly convergent sequences. In Section 4 we prove the claims made in Example 1.2.
2 Almost BL-splitting
In this section we prove a result on the almost BL-splitting of superposition operators in along bounded sequences that converge in and in along weakly convergent sequences. This is a variation on Lions’ approach in . Note that here the periodicity assumption in x is not needed.
If , then we denote by the norm of .
Consider , , and such that . Suppose that is a Caratheodory function that satisfies (1.4). Denote by the superposition operator on real functions induced by f. Then we have the following results:
If is bounded and converges in to a function u, then and almost BL-splits along with respect to u.
If and in , then almost BL-splits along with respect to u.
In (b), if in addition converges weakly and as , then in and almost BL-splits along and with respect to u, preserving subsequences and the auxiliary sequence in the following sense: for any subsequence there are a subsequence and such that in and, writing and , we have
For the proof, let denote, for , the open ball in with center 0 and radius R.
The functions are uniformly bounded and nondecreasing. We may assume that converges pointwise almost everywhere to a bounded nondecreasing function Q (see ). It is easy to build a sequence such that for every there is , arbitrarily large, with
Hence, for all there exists such that
Consider a smooth cut off function such that on and on . Set . Then
From the continuity of on , on , in , and for a.e. we obtain
Since in , this in turn yields for any and R chosen accordingly, as in (2.1),
where C is independent of ε. Letting ε tend to 0 and using (2.2), we obtain
and almost BL-splits along with respect to u by (a).
3 Uniform continuity
Here we prove uniform continuity on bounded subsets of , making use of the periodicity of f in x. As a consequence, we also obtain BL-splitting along weakly convergent sequences in .
For simplicity we will only prove the case (the identity transformation). The general case follows in an analogous manner. Denote the respective translation action of the additive group on functions by
Let denote the standard scalar product in , defined by
and let denote the associated norm. Also denote by the weak limit of a weakly convergent sequence.
We first recall a functional consequence of Lions’ Vanishing Lemma [15, Lemma I.1.].
Suppose for a sequence that in for every sequence . Then in for all .
Note first that is bounded in since in . We claim that
If the claim were not true there would exist and a sequence such that, after passing to a subsequence of , we would have
Pick such that for all n. With it follows that and hence
for all n. We reach a contradiction since in and hence in by the theorem of Rellich and Kondrakov. Therefore, (3.1) holds true.
Proof of Theorem 1.3.
We start by proving the uniform continuity. Let be bounded sequences in for and set . Suppose for a contradiction that
and that there is such that
Successively we will define infinitely many sequences and , , indexed by and strictly increasing functions with the following properties:
We need to say something about the extraction of subsequences. In order to obtain , , and from , we first pass to a subsequence of and then use its terms in the construction. Once the new sequences and are built we may remove a finite number of terms at their start, modifying accordingly, with the goal of obtaining additional properties. Beginning with the following iteration there are no more retrospective changes to the sequences already built. This is to assure a well defined infinite sequence of sequences, from which eventually we take the diagonal sequence. In this setting it seems clearer to make the selection of subsequences explicit, contrary to what is usually done when using concentration compactness methods [16, 14, 15] or when proving a variational splitting lemma.
For properties (3.4)–(3.8) are fulfilled by the definition of and by (3.2) and (3.3). Assume now that (3.4)–(3.8) hold for some . Denote by the set of such that there are a sequence and a subsequence of with
Pick such that
There are and a strictly increasing function such that
By the equivariance of and the invariance of the involved norms under the -action,
and, since by (3.11) the map BL-splits along with respect to , we have
By the definition of , we have
This proves (3.7) for .
for all . Property (3.8) (for ) implies that
Since BL-splits along weakly convergent sequences, this yields, together with (3.10), that
for all . For large enough m this implies
We fix m with these properties, we write , , and instead of , , and , respectively, and we write instead of , thus all properties proved above remain valid, and, in addition, the following hold true:
for all and
Now we consider the process of constructing sequences as finished and proceed to prove properties of the whole set. By induction, (3.13) leads to
We claim that the diagonal sequence satisfies
Note that by construction, for all we have
Hence we have the representation
First we show that
Fix . For every and there is such that
It is easy to see that the sequence is strictly increasing. Hence the first term in the last expression tends to 0 as by (3.7), and the second term tends to 0 by (3.10) and (3.16). Since and were arbitrary, this proves (3.23).
as , as above. Hence,
with . Since
is a subsequence of , this contradicts the definition of and proves (3.21).
It only remains to prove BL-splitting for along weakly convergent sequences in with respect to their weak limits. Suppose that in . By Theorem 2.1 (b) there is a sequence such that in and, after passing to a subsequence of , we have
as . Since and are bounded in , and by the uniform continuity of on bounded subsets of with respect to the -norm (and hence also with respect to the -norm), it follows that we may replace by v in (3.26). Using this, a standard reasoning by contradiction yields the claim. ∎
4 Construction of examples
Proof of Example 1.2.
We first treat the case . Set and fix a sequence such that and
for . Define real functions u and on by setting
for each . It is straightforward to show that , that is a bounded sequence in , and that pointwise and in . On the other hand, denoting by the volume of the unit ball in , we obtain
as . Since , this implies the claim.
For the other example, , we set and fix a sequence such that and for . We define
for each . Then again, , is a bounded sequence in , and pointwise and in . For we obtain
as . This yields the claim. ∎
Proof of Remark 1.4.
The construction of these counterexamples is closely related to Example 1.2. First consider the function . We define the Lipschitz-continuous cut-off function by
introduce , pick a sequence such that and for , and define
and for each and . It is straightforward to check that and that is bounded in . Since a.e., in . Using (4.2), we estimate
as . Hence by the calculation in (4.3), we have
and the claim follows. Note that the example above has no simple analogue in the case for , using as in the proof of the second case of Example 1.2. The reason is that the analogously defined sequence is not bounded in in that case.
Now we treat the function . To this end put , fix a sequence such that and for , and choose small enough such that
and for each and . It follows that for each n, where denotes the closed ball in with radius r and center z. Again, it is straightforward to check that , that is bounded in , and that in . Using (4.4), we estimate
for all n. Hence by the calculation in (4.1), we have
and the claim follows. Note that this example has no simple analogue in the case for , using as in the proof of the first case of Example 1.2. Here the reason is that for the analogously defined sequence , the sequence is not bounded in . ∎
Funding source: Consejo Nacional de Ciencia y Tecnología
Award Identifier / Grant number: 237661
Funding source: Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México
Award Identifier / Grant number: IN104315
Funding statement: This research was partially supported by CONACYT grant 237661 and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico).
I would like to thank Kyril Tintarev for an informative exchange on this subject. Moreover, I thank the referee for suggesting to analyze the limiting cases in Theorem 1.3.
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