Uniform continuity and Brézis–Lieb-type splitting for superposition operators in Sobolev space

Nils Ackermann

doi:10.1515/anona-2016-0123

Open Access Published by De Gruyter October 11, 2016

Uniform continuity and Brézis–Lieb-type splitting for superposition operators in Sobolev space

Nils Ackermann

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2016-0123

Abstract

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.

Keywords: Superposition operator; Sobolev space; uniform continuity; Brézis–Lieb splitting

MSC 2010: 47H30; 58E40

1 Introduction

In their seminal paper [6] 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 ℝN, p>1, f⁢(t):=|t|p for t∈ℝ, and (un) is a bounded sequence in Lp⁢(Ω) that converges pointwise almost everywhere to some function u. If one denotes by ℱ:Lp⁢(Ω)→L1⁢(Ω) the superposition operator induced by f, i.e., ℱ⁢(v)⁢(x):=f⁢(v⁢(x)), then the result in [6] implies that u∈Lp⁢(Ω) and

(1.1)ℱ⁢(un)-ℱ⁢(un-u)→ℱ⁢(u) in L1⁢(Ω) as n→∞.

The same conclusion is obtained in that paper for more general functions, imposing conditions that are satisfied for continuous convex f with f⁢(0)=0 and imposing additional conditions on the sequence (un).

A different approach to the decoupling of superposition operators along sequences of functions rests on certain regularity assumptions on f. For example, assume that f∈C1⁢(ℝ) satisfies

(1.2)supt∈ℝ⁡|f′⁢(t)||t|p-1<∞.

Then the proof of [19, Lemma 8.1] can easily be extended to obtain (1.1). See also the slightly more general [7, Lemma 1.3], where f is allowed to depend on x explicitly.

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 [6], 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 [6], 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 H1⁢(Ω). 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 Ω=ℝN. From here on, function spaces are taken over ℝN 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.

Definition 1.1.

Suppose that X and Y are Banach spaces. Consider a map ℱ:X→Y, a sequence (un)⊆X and u∈X. We say that ℱBL-splits along (un) with respect to u (BL being an abbreviation for Brézis–Lieb) if

∥ℱ⁢(un)-ℱ⁢(un-u)-ℱ⁢(u)∥Y→0.

We say that ℱalmost BL-splits along (un) with respect to u if, starting with any subsequence of (un), we can pass to a subsequence such that there is a sequence (vn)⊆X such that ∥vn-u∥X→0 and

∥ℱ⁢(un)-ℱ⁢(un-vn)-ℱ⁢(u)∥Y→0.

If u is a limit of (un) in some unambiguous sense then we frequently omit to mention that (almost) BL-splitting is with respect to u.

By [6], the map f⁢(u)=|u|p induces a map ℱ:Lp→L1 that BL-splits along pointwise a.e. converging bounded sequences in Lp 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 f∈C⁢(ℝ) satisfies

(1.3)supt∈ℝ⁡|f⁢(t)||t|p<∞,

then the induced superposition operator ℱ:Lp→L1 almost BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp, see Theorem 2.1 (a) below. This result is basically Lion’s approach, with a simplifying twist. If in addition ℱ is uniformly continuous on bounded subsets of Lp, then it is easy to see that it BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp, 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.

Example 1.2.

If p>1 and if either f⁢(t):=cos⁡(π⁢t)⁢|t|p or f⁢(t):=cos⁡(π/t)⁢|t|p, then there is a bounded sequence (un) in Lp that converges in Llocp and pointwise a.e. to a function u such that the induced continuous superposition operator ℱ:=Lp→L1 does not BL-split along any subsequence of (un) with respect to u. On the other hand, ℱ almost BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp. Hence ℱ is not uniformly continuous on bounded subsets of Lp and neither the general conditions used in [6] 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 Lp (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 H1 instead.

In this context we now formulate our main theorem, in a slightly more general setting than what we considered above. A function f:ℝN×ℝ→ℝ is a Caratheodory function if f is measurable and if f⁢(x,⋅) is continuous for almost every x∈ℝN. The induced superposition operator on functions u:ℝN→ℝ is then given by ℱ⁢(u)⁢(x):=f⁢(x,u⁢(x)). If A is a real invertible N×N-matrix, then f is said to be A-periodic in its first argument if f⁢(x+A⁢k,t)=f⁢(x,t) for all x∈ℝN, k∈ℤN, and t∈ℝ.

Denote by 2*:=2⁢N/(N-2) if N≥3 and 2*:=∞ if N=1 or N=2 the critical Sobolev exponent for H1. Recall the continuous and compact embedding of the Sobolev space H1⁢(U) in Lp⁢(U) for p∈[2,2*) if U⊆ℝN is a bounded domain.

Theorem 1.3.

Consider μ>0, ν≥1, and C0>0 such that p:=μ⁢ν∈(2,2*). Suppose that f:RN×R→R is a Caratheodory function that satisfies

(1.4)|f⁢(x,t)|≤C0⁢|t|μ for all ⁢x∈ℝN,t∈ℝ,

and which is A-periodic in its first argument, for some invertible matrix A∈RN×N. Denote by F:Lp→Lν the continuous superposition operator induced by f. Then F is uniformly continuous on bounded subsets of H1 with respect to the Lp-Lν-norms and hence also with respect to the H1-Lν-norms. Moreover, F:H1→Lν BL-splits along weakly convergent sequences in H1 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 ℱ along weakly converging sequences in H1 with respect to their weak limit, using the concentration function and the compactness of the Sobolev embedding H1⁢(U)↪Lp⁢(U) for p∈[2,2*) 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 p>2.

Remark 1.4.

Theorem 1.3 applies in particular to the functions considered in Example 1.2 when ν=1 and μ=p∈(2,2*). On the other hand, for f⁢(t):=cos⁡(π/t)⁢t2 there is a sequence in H1 that converges weakly but possesses no subsequence along which f:H1→L1 BL-splits with respect to the weak limit. The same is true for f⁢(t):=cos⁡(π⁢t)⁢t2*. In this sense, Theorem 1.3 is optimal, that is, it cannot be extended in this generality to include the cases p=μ⁢ν=2 and p=μ⁢ν=2*. The existence of these counterexamples is proved in Section 4.

Of course, by Sobolev’s embedding theorem, a map Lp→Lν that BL-splits along Llocp-converging bounded sequences in Lp with respect to their limits in Llocp also BL-splits in H1 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 H1 with respect to weak limits even for p=2 and p=2*.

Remark 1.5.

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

|f⁢(x,t)|≤C0⁢(|t|μ1+|t|μ2) for all ⁢x∈ℝN,t∈ℝ,

where μi⁢ν∈(2,2*) for i=1,2. In that case, ℱ:H1→Lν is uniformly continuous on bounded subsets of H1 with respect to the H1-Lν norms, and ℱ BL-splits along weakly convergent sequences in H1 with respect to their weak limits.

Remark 1.6.

The uniform continuity of operators ℱ on bounded subsets of H1 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 ℱ:H1⁢(U)→Lν⁢(U) on bounded subsets of H1⁢(U) is trivial if U is bounded, by the compact Sobolev embedding H1⁢(U)⊆Lp⁢(U).

We now discuss additional aspects and applications of the results presented above. To this end we return to a simple setting on ℝN. Suppose that f∈C⁢(ℝ) satisfies (1.3) with p∈(2,2*) and consider the functional Φ:H1→ℝ given by

Φ⁢(u):=∫ℝNf⁢(u).

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 (un), in order to obtain compactness of the sequence. Here we are only concerned with dichotomy. In this case, the sequence (un) is approximated by (un1+un2), where dist⁡(supp⁡(un1),supp⁡(un2))→∞. For local functionals like Φ it then follows easily that Φ⁢(un) is approximated by Φ⁢(un1)+Φ⁢(un2), a fact that yields, together with a hypothesis about energy levels, a contradiction. Clearly, the same can be achieved if Φ BL-splits along (un) 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 un1 and un2 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.,

Ψ⁢(u):=∫ℝN(f*h⁢(u))⁢h⁢(u),

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 [2] 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 [3] we proved, for f∈C⁢(ℝ) satisfying (1.4) with μ:=p-1, and setting ν:=p/(p-1), that the map Γ:H1→H-1, given by

Γ⁢(u)⁢v:=∫ℝNf⁢(u)⁢v,

BL-splits along a weakly convergent sequence if the weak limit is a function tending to 0 as |x|→∞. Another result was given in [13, Lemma 7.2], when f∈C⁢(ℝ) satisfies (1.4) with μ:=p-2 and ν:=p/(p-2), and is as follows: The map Λ:H1→ℒ2⁢(H1,ℝ) (here ℒ2⁢(H1,ℝ) denotes the space of bounded bilinear maps from H1 into ℝ), given by

Λ⁢(u)⁢[v,w]:=∫ℝNf⁢(u)⁢v⁢w,

is uniformly continuous on bounded subsets of H1. 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 Lν, which are continuously embedded in H-1 and ℒ2⁢(H1,ℝ), 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 ℱ along bounded sequences in Lp that converge in Llocp and along weakly convergent sequences in H1. In Section 3 we prove the uniform continuity of ℱ on bounded subsets of H1 and BL-splitting of ℱ 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 Lp along bounded sequences that converge in Llocp and in H1 along weakly convergent sequences. This is a variation on Lions’ approach in [14]. Note that here the periodicity assumption in x is not needed.

If r∈[1,∞], then we denote by |⋅|r the norm of Lr.

Theorem 2.1.

Consider μ>0, ν≥1, and C0>0 such that p:=μ⁢ν≥1. Suppose that f:RN×R→R is a Caratheodory function that satisfies (1.4). Denote by F the superposition operator on real functions induced by f. Then we have the following results:

If (un)⊆Lp is bounded and converges in Llocp to a function u, then u∈Lp and ℱ:Lp→Lν almost BL-splits along (un) with respect to u.
If p∈[2,2*) and un⇀u in H1, then ℱ:H1→Lν almost BL-splits along (un) with respect to u.
In (b), if in addition (u¯n)⊆H1 converges weakly and |un-u¯n|p→0 as n→∞, then u¯n⇀u in H1 and ℱ almost BL-splits along (un) and (u¯n) with respect to u, preserving subsequences and the auxiliary sequence (vn) in the following sense: for any subsequence nkthere are a subsequence nkℓ and (vℓ) such that vℓ→u in H1 and, writing uℓ:=unkℓ and u¯ℓ:=u¯nkℓ, we have
ℱ⁢(uℓ)-ℱ⁢(uℓ-vℓ)→ℱ⁢(u)
and
ℱ⁢(u¯ℓ)-ℱ⁢(u¯ℓ-vℓ)→ℱ⁢(u).

For the proof, let BR denote, for R>0, the open ball in ℝN with center 0 and radius R.

Proof.

We prove the assertions one by one. (a): From (1.4) and from the theory of superposition operators [4] it follows that ℱ:Lp⁢(U)→Lν⁢(U) is continuous for any open subset U of ℝN. For n∈ℕ define Qn:[0,∞)→[0,∞) by

Qn⁢(R):=∫BR|un|p.

The functions Qn are uniformly bounded and nondecreasing. We may assume that (Qn) converges pointwise almost everywhere to a bounded nondecreasing function Q (see [14]). It is easy to build a sequence Rn→∞ such that for every ε>0 there is R>0, arbitrarily large, with

lim supn→∞⁡(Qn⁢(Rn)-Qn⁢(R))≤ε.

Hence, for all ε>0 there exists R>0 such that

(2.1)lim supn→∞⁡∫BRn\BR|un|p≤ε and ∫ℝN\BR|u|p≤ε.

Consider a smooth cut off function η:[0,∞)→[0,1] such that η≡1 on [0,1] and η≡0 on [2,∞). Set vn⁢(x):=η⁢(2⁢|x|/Rn)⁢u⁢(x). Then

(2.2)limn→∞⁡vn=u in Lp.

From the continuity of ℱ on Lp⁢(BR), vn=u on BR, limn→∞⁡un=u in Lp⁢(BR), and f⁢(x,0)=0 for a.e. x∈ℝN we obtain

limn→∞⁡∫BR|f⁢(x,un)-f⁢(x,un-vn)-f⁢(x,vn)|ν⁢dx=limn→∞⁡∫BR|f⁢(x,un)-f⁢(x,un-u)-f⁢(x,u)|ν⁢dx=0.

Since vn≡0 in ℝN\BRn, this in turn yields for any ε>0 and R chosen accordingly, as in (2.1),

lim supn→∞⁡∫ℝN|f⁢(x,un)-f⁢(x,un-vn)-f⁢(x,vn)|ν⁢dx=lim supn→∞⁡∫BRn\BR|f⁢(x,un)-f⁢(x,un-vn)-f⁢(x,vn)|ν⁢dx

≤C⁢lim supn→∞⁡∫BRn\BR(|un|μ+|un-vn|μ+|vn|μ)ν

≤C⁢lim supn→∞⁡∫BRn\BR(|un|p+|u|p)

≤C⁢ε,

where C is independent of ε. Letting ε tend to 0 and using (2.2), we obtain

limn→∞⁡|ℱ⁢(un)-ℱ⁢(un-vn)-ℱ⁢(u)|ν=0.

(b): The continuous embedding H1↪Lp implies that (un) is bounded in Lp, and the compact embedding H1⁢(U)↪Lp⁢(U) for bounded U implies that un→u in Llocp. Defining vn as in (a), we therefore obtain that

vn→u in ⁢H1,

and ℱ almost BL-splits along (un) with respect to u by (a).

(c): Since |un-u¯n|p→0 and un→v in Llocp, it follows that u¯n⇀v in H1. Taking R large enough, (2.1) also holds true if we replace un by u¯n. Therefore, after passing to a subsequence for (un), and using the same subsequence for (u¯n), we obtain

limn→∞⁡|ℱ⁢(u¯n)-ℱ⁢(u¯n-vn)-ℱ⁢(u)|ν=0.∎

3 Uniform continuity

Here we prove uniform continuity on bounded subsets of H1, making use of the periodicity of f in x. As a consequence, we also obtain BL-splitting along weakly convergent sequences in H1.

For simplicity we will only prove the case A=I (the identity transformation). The general case follows in an analogous manner. Denote the respective translation action of the additive group ℤN on functions u:ℝN→ℝ by

(a⋆u)⁢(x):=u⁢(x-a),a∈ℤN,x∈ℝN.

Let 〈⋅,⋅〉 denote the standard scalar product in H1, defined by

〈u,v〉:=∫ℝN(∇⁡u⋅∇⁡v+u⁢v),

and let ∥⋅∥ denote the associated norm. Also denote by w-lim the weak limit of a weakly convergent sequence.

We first recall a functional consequence of Lions’ Vanishing Lemma [15, Lemma I.1.].

Lemma 3.1.

Suppose for a sequence (un)⊆H1 that an⋆un⇀0 in H1 for every sequence (an)⊆ZN. Then un→0 in Lp for all p∈(2,2*).

Proof.

Note first that (un) is bounded in H1 since un⇀0 in H1. We claim that

(3.1)supy∈ℝN⁡∫y+B1|un|2→0 as ⁢n→∞.

If the claim were not true there would exist ε>0 and a sequence (yn)⊆ℝN such that, after passing to a subsequence of (un), we would have

∫yn+B1|un|2≥ε.

Pick (an)⊆ℤN such that |an+yn|∞<1 for all n. With R:=N+1 it follows that an+yn+B1⊆BR and hence

∫BR|an⋆un|2≥ε

for all n. We reach a contradiction since an⋆un⇀0 in H1 and hence an⋆un→0 in L2⁢(BR) by the theorem of Rellich and Kondrakov. Therefore, (3.1) holds true.

The claim of the theorem now follows from [15, Lemma I.1.] with p=q=2. Compare also with [18, Lemma 3.3]. ∎

Proof of Theorem 1.3.

We start by proving the uniform continuity. Let (ui,n0)n∈ℕ0 be bounded sequences in H1 for i=1,2 and set C1:=maxi=1,2⁢lim supn→∞⁡∥ui,n0∥. Suppose for a contradiction that

(3.2)|u1,n0-u2,n0|p→0 as ⁢n→∞

and that there is C2>0 such that

(3.3)|ℱ⁢(u1,n0)-ℱ⁢(u2,n0)|ν≥C2 for all ⁢n.

Successively we will define infinitely many sequences (ank)n⊆ℤN and (ui,nk)n⊆H1, i=1,2, indexed by k∈ℕ0 and strictly increasing functions φk:ℕ→ℕ with the following properties:

(3.4)maxi=1,2⁢lim supn→∞⁡∥ui,nk∥≤C1,

(3.5)limn→∞⁡|u1,nk-u2,nk|p=0,

(3.6)lim infn→∞⁡|ℱ⁢(u1,nk)-ℱ⁢(u2,nk)|ν≥C2,

(3.7)w-limn→∞⁡(-aψℓk-1⁢(n)ℓ)⋆ui,nk=0 in H1, if 0≤ℓ<k, for i=1,2,

and

(3.8)limn→∞⁡|aψmℓ⁢(n)m-anℓ|=∞ if ⁢0≤m<ℓ<k.

Here,

ψℓk:=φℓ+1∘φℓ+2∘…∘φk if ⁢ℓ=-1,0,1,…,k-1

ψkk:=idℕ.

We need to say something about the extraction of subsequences. In order to obtain φk, (ank)n, and (ui,nk+1)n from (ui,nk), we first pass to a subsequence (ui,φk⁢(n)k)n of (ui,nk)n and then use its terms in the construction. Once the new sequences (ank)n and (ui,nk+1)n are built we may remove a finite number of terms at their start, modifying φk 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 k=0 properties (3.4)–(3.8) are fulfilled by the definition of C1 and by (3.2) and (3.3). Assume now that (3.4)–(3.8) hold for some k∈ℕ0. Denote by Wk the set of v∈H1 such that there are a sequence (an)⊆ℤN and a subsequence of (u1,nk) with

w-limn→∞⁡an⋆u1,nk=v in ⁢H1.

If w-limn→∞⁡an⋆u1,nk=0 in H1 were true for all sequences (an)⊆ℤN, by Lemma 3.1 it would follow that limn→∞⁡u1,nk=0 in Lp. Equation (3.5) and the continuity of ℱ on Lp would lead to a contradiction with (3.6). Therefore,

qk:=supv∈Wk⁡∥v∥∈(0,C1].

Pick vk∈Wk such that

(3.9)∥vk∥≥qk2>0.

There are (ank)n⊆ℤN and a strictly increasing function φk:ℕ→ℕ such that

w-limn→∞⁡(-ank)⋆u1,φk⁢(n)k=vk in ⁢H1.

By (3.5) and by Theorem 2.1 (b) and (c) there exists a sequence (vnk)n⊆H1 such that

(3.10)limn→∞⁡vnk=vk in H1,

(3.11)w-limn→∞⁡(-ank)⋆ui,φk⁢(n)k=vk in H1, for ⁢i=1,2,

and

limn→∞⁡|ℱ⁢((-ank)⋆ui,φk⁢(n)k)-ℱ⁢((-ank)⋆ui,φk⁢(n)k-vnk)-ℱ⁢(vk)|ν=0,i=1,2.

Set

ui,nk+1:=ui,φk⁢(n)k-ank⋆vnk.

By the equivariance of ℱ and the invariance of the involved norms under the ℤN-action,

(3.12)limn→∞⁡|ℱ⁢(ui,φk⁢(n)k)-ℱ⁢(ui,nk+1)-ℱ⁢(ank⋆vk)|ν=0 for i=1,2,

and, since by (3.11) the map ∥⋅∥2 BL-splits along (-ank)⋆ui,φk⁢(n)k with respect to vk, we have

(3.13)limn→∞⁡|∥ui,φk⁢(n)k∥2-∥ui,nk+1∥2-∥vk∥2|=0 for i=1,2.

Equations (3.13) and (3.4) (for k) imply that

maxi=1,2⁢lim supn→∞⁡∥ui,nk+1∥≤C1,

hence (3.4) for k+1. The definition of the sequences ui,nk+1 and (3.5) (for k) imply that

(3.14)limn→∞⁡|u1,nk+1-u2,nk+1|p=limn→∞⁡|u1,φk⁢(n)k-u2,φk⁢(n)k|p=0,

hence (3.5) for k+1. It follows from (3.12) and (3.6) (for k) that

(3.15)lim infn→∞⁡|ℱ⁢(u1,nk+1)-ℱ⁢(u2,nk+1)|ν=lim infn→∞⁡|ℱ⁢(u1,φk⁢(n)k)-ℱ⁢(u2,φk⁢(n)k)|ν≥C2,

hence (3.6) for k+1. Last but not least, from (3.7) (for k), (3.9), and (3.11) it follows that

(3.16)limn→∞⁡|aψmk⁢(n)m-ank|=∞ if ⁢m<k.

Since (3.8) is true for k, together with (3.16) we obtain (3.8) for k+1. Moreover, (3.16), (3.7) (for k) and (3.10) yield

w-limn→∞⁡(-aψℓk⁢(n)ℓ)⋆ui,nk+1=w-limn→∞⁡((-aψℓk-1⁢(φk⁢(n))ℓ)⋆ui,φk⁢(n)k-(ank-aψℓk⁢(n)ℓ)⋆vnk)=0 in H1 if ⁢ℓ<k.

By the definition of ank, we have

w-limn→∞⁡(-ank)⋆ui,nk+1=w-limn→∞⁡((-ank)⋆ui,φk⁢(n)k-vnk)=0 in H1.

This proves (3.7) for k+1.

We now skip a finite number of elements of the sequences constructed in this induction step and adapt φk accordingly. Choosing m∈ℕ large enough, by (3.14) and (3.15) we obtain

|u1,m+nk+1-u2,m+nk+1|p≤1k+1

and

|ℱ⁢(u1,m+nk+1)-ℱ⁢(u2,m+nk+1)|ν≥C2-1k+1

for all n∈ℕ. Property (3.8) (for k+1) implies that

limn→∞⁡|aψmk⁢(n)m-aψℓk⁢(n)ℓ|=limn→∞⁡|aψmℓ⁢(ψℓk⁢(n))m-aψℓk⁢(n)ℓ|=∞ if ⁢m<ℓ≤k.

Since ∥⋅∥2 BL-splits along weakly convergent sequences, this yields, together with (3.10), that

limn→∞⁡∥∑j=ℓkaψjk⁢(n)j⋆vψjk⁢(n)j∥2=∑j=ℓk∥vj∥2

for all ℓ≤k. For large enough m this implies

∥∑j=ℓkaψjk-1⁢(φk⁢(m+n))j⋆vψjk-1⁢(φk⁢(m+n))j∥2≤2⁢∑j=ℓk∥vj∥2 for all ⁢n∈ℕ⁢ and ⁢ℓ≤k.

We fix m with these properties, we write ui,nk+1, ank, and vnk instead of ui,m+nk+1, am+nk, and vm+nk, respectively, and we write φk⁢(n) instead of φk⁢(m+n), thus all properties proved above remain valid, and, in addition, the following hold true:

(3.17)|u1,nk+1-u2,nk+1|p≤1k+1

and

(3.18)|ℱ⁢(u1,nk+1)-ℱ⁢(u2,nk+1)|ν≥C2-1k+1

for all n∈ℕ and

(3.19)∥∑j=ℓkaψjk⁢(n)j⋆vψjk⁢(n)j∥2≤2⁢∑j=ℓk∥vj∥2 for all ⁢n∈ℕ⁢ and ⁢ℓ≤k.

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

∥u1,nk+1∥2=∥u1,ψ-1k⁢(n)0∥2-∑j=0k∥vj∥2+o⁢(1) as ⁢n→∞,

and hence ∑j=0∞∥vj∥2≤C1 by (3.4). In view of (3.9) this yields

(3.20)qk→0 as ⁢k→∞.

We claim that the diagonal sequence (u1,nn) satisfies

(3.21)bn⋆u1,nn⇀0 in H1 as n→∞, for every sequence (bn)⊆ℤ.

Note that by construction, for all ℓ≤k we have

u1,nk=u1,ψℓ-1k-1⁢(n)ℓ-∑j=ℓk-1aψjk-1⁢(n)j⋆vψjk-1⁢(n)j.

Hence we have the representation

(3.22)u1,nn=u1,ψk-1n-1⁢(n)k-∑j=kn-1aψjn-1⁢(n)j⋆vψjn-1⁢(n)j if ⁢n≥k.

First we show that

(3.23)w-limn→∞⁡(-aψkn-1⁢(n)k)⋆u1,nn=0 in H1, for all k∈ℕ0.

Fix k∈ℕ0. For every w∈H1 and ε>0 there is ℓ0≥k+1 such that

∥w∥2⁢∑j=ℓ0∞∥vj∥2≤ε2/2.

Then (3.19), (3.22), and the translation invariance of the norm yield for n≥ℓ0 the following:

+∥w∥⁢∥∑j=ℓ0n-1aψjn-1⁢(n)j⋆vψjn-1⁢(n)j∥

≤|〈(-aψkn-1⁢(n)k)⋆u1,ψkn-1⁢(n)k+1,w〉|+|〈∑j=k+1ℓ0-1(aψjn-1⁢(n)j-aψkn-1⁢(n)k)⋆vψjn-1⁢(n)j,w〉|+ε.

It is easy to see that the sequence (ψkn-1⁢(n))n is strictly increasing. Hence the first term in the last expression tends to 0 as n→∞ by (3.7), and the second term tends to 0 by (3.10) and (3.16). Since ε>0 and w∈H1 were arbitrary, this proves (3.23).

To finish the proof of (3.21), suppose for a contradiction that w-limn→∞⁡bn⋆u1,nn=v≠0 in H1, for a subsequence. Equation (3.23) implies that

limn→∞⁡|bn+aψkn-1⁢(n)k|=∞

for every k∈ℕ0. Pick k∈ℕ0 such that qk<∥v∥. This is possible by (3.20). Then, for every w∈H1, it follows from (3.19) and (3.22) that

|〈bn⋆u1,ψk-1n-1⁢(n)k-v,w〉|≤|〈bn⋆u1,nn-v,w〉|+|〈∑j=kn-1(bn+aψjn-1⁢(n)j)⋆vψjn-1⁢(n)j,w〉|→0

as n→∞, as above. Hence,

w-limn→∞⁡(bn⋆u1,ψk-1n-1⁢(n)k)=v

with ∥v∥>qk. Since

(u1,ψk-1n-1⁢(n)k)n

is a subsequence of (u1,nk)n, this contradicts the definition of qk and proves (3.21).

We are now in the position to finish the proof of uniform continuity of ℱ. Equations (3.17) and (3.18) imply that

(3.24)limn→∞⁡|u1,nn-u2,nn|p=0

and

(3.25)lim infn→∞⁡|ℱ⁢(u1,nn)-ℱ⁢(u2,nn)|ν≥C2.

By Lemma 3.1 and (3.21), we have u1,nn→0 in Lp. Together with (3.24) and (3.25) this contradicts the continuity of ℱ on Lp and therefore proves the assertion about uniform continuity.

It only remains to prove BL-splitting for ℱ along weakly convergent sequences in H1 with respect to their weak limits. Suppose that un⇀v in H1. By Theorem 2.1 (b) there is a sequence (vn)⊆H1 such that vn→v in H1 and, after passing to a subsequence of (un), we have

(3.26)ℱ⁢(un)-ℱ⁢(un-vn)→ℱ⁢(v) in ⁢Lν

as n→∞. Since (un) and (vn) are bounded in H1, and by the uniform continuity of ℱ on bounded subsets of H1 with respect to the Lp-norm (and hence also with respect to the H1-norm), it follows that we may replace vn 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 f⁢(t):=cos⁡(π⁢t)⁢|t|p. Set Rn:=n-p/N and fix a sequence (xn)⊆ℝN such that |xn|→∞ and

BRm⁢(xm)∩BRn⁢(xn)=∅

for m≠n. Define real functions u and un on ℝN by setting

u:=∑k=1∞χBRk⁢(xk),wn:=2⁢n⁢χBRn⁢(xn),and un:=u+wn

for each n∈ℕ. It is straightforward to show that u∈Lp, that (un) is a bounded sequence in Lp, and that un→u pointwise and in Llocp. On the other hand, denoting by ωN the volume of the unit ball in ℝN, we obtain

|∫ℝNf⁢(un)-f⁢(un-u)-f⁢(u)|=|∫BRn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|

=|∫BRn⁢(xn)cos⁡((2⁢n+1)⁢π)⁢(2⁢n+1)p-cos⁡(2⁢n⁢π)⁢(2⁢n)p-cos⁡π|

=|∫BRn⁢(xn)(-(2⁢n+1)p-(2⁢n)p+1)|

(4.1)=ωN⁢((2+1n)p+2p-(1n)p)→2p+1⁢ωN

as n→∞. Since 2p+1⁢ωN>0, this implies the claim.

For the other example, f⁢(t):=cos⁡(π/t)⁢|t|p, we set Rn:=np/N and fix a sequence (xn)⊆ℝN such that |xn|/Rn→∞ and BRm⁢(xm)∩BRn⁢(xn)=∅ for m≠n. We define

u:=∑k=1∞12⁢n⁢(2⁢n-1)⁢χBRk⁢(xk),wn:=12⁢n⁢χBRn⁢(xn),and un:=u+wn

for each n∈ℕ. Then again, u∈Lp, (un) is a bounded sequence in Lp, and un→u pointwise and in Llocp. For x∈BRn⁢(xn) we obtain

(4.2)u⁢(x)+wn⁢(x)=12⁢n⁢(2⁢n-1)+12⁢n=12⁢n-1

and hence

|∫ℝNf⁢(un)-f⁢(un-u)-f⁢(u)|=|∫BRn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|

≥|∫BRn⁢(xn)cos⁡((2⁢n-1)⁢π)⁢(12⁢n-1)p-cos⁡(2⁢n⁢π)⁢(12⁢n)p|-∫BRn⁢(xn)(12⁢n⁢(2⁢n-1))p

=|∫BRn⁢(xn)-(12⁢n-1)p-(12⁢n)p|-∫BRn⁢(xn)(12⁢n⁢(2⁢n-1))p

(4.3)=ωN⁢((12-1n)p+(12)p-(12⁢(2⁢n-1))p)→ωN2p-1

as n→∞. 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 f⁢(t):=cos⁡(π/t)⁢t2. We define the Lipschitz-continuous cut-off function η:ℝ→ℝ by

η⁢(t):={1,t≤0,1-t,0<t<1,0,1≤t,

introduce Rn:=n2/N, pick a sequence (xn)⊆ℝN such that |xn|/Rn→∞ and BRm+1⁢(xm)∩BRn+1⁢(xn)=∅ for m≠n, and define

u⁢(x):=∑k=1∞12⁢n⁢(2⁢n-1)⁢η⁢(|x-xk|-Rk),wn⁢(x):=12⁢n⁢η⁢(|x-xn|-Rn),

and un:=u+wn for each n∈ℕ and x∈ℝN. It is straightforward to check that u,wn∈H1 and that (wn) is bounded in H1. Since wn→0 a.e., wn⇀0 in H1. Using (4.2), we estimate

∫BRn+1⁢(xn)\BRn⁢(xn)|f⁢(u+wn)-f⁢(wn)-f⁢(u)|≤∫BRn+1⁢(xn)\BRn⁢(xn)((12⁢n-1)2+(12⁢n)2+(12⁢n⁢(2⁢n-1))2)

≤3⁢ωNn2⁢((Rn+1)N-RnN)=3⁢ωN⁢((1+n-2/N)N-1)→0

as n→∞. Hence by the calculation in (4.3), we have

|∫ℝNf⁢(un)-f⁢(un-u)-f⁢(u)|=|∫BRn+1⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|

≥|∫BRn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|-∫BRn+1⁢(xn)\BRn⁢(xn)|f⁢(u+wn)-f⁢(wn)-f⁢(u)|

→ωN2p-1

and the claim follows. Note that the example above has no simple analogue in the case f⁢(t):=cos⁡(π/t)⁢|t|p for p>2, using Rn:=np/N as in the proof of the second case of Example 1.2. The reason is that the analogously defined sequence (wn) is not bounded in L2 in that case.

Now we treat the function f⁢(t):=cos⁡(π⁢t)⁢|t|2*. To this end put Rn:=n-2*/N, fix a sequence (xn)⊆ℝN such that |xn|→∞ and B2⁢Rm⁢(xm)∩B2⁢Rn⁢(xn)=∅ for m≠n, and choose γ∈(0,1) small enough such that

(4.4)(32*+22*+1)⁢((1+γ)N-1)≤22*+12.

Define

u⁢(x):=∑k=1∞η⁢(|x-xk|-Rkγ⁢Rk),wn⁢(x):=2⁢n⁢η⁢(|x-xk|-Rnγ⁢Rn),

and un:=u+wn for each n∈ℕ and x∈ℝN. It follows that supp⁡(wn)=B¯(1+γ)⁢Rn⁢(xn) for each n, where B¯r⁢(z) denotes the closed ball in ℝN with radius r and center z. Again, it is straightforward to check that u,wn∈H1, that (wn) is bounded in H1, and that wn⇀0 in H1. Using (4.4), we estimate

∫B(1+γ)⁢Rn⁢(xn)\BRn⁢(xn)|f⁢(u+wn)-f⁢(wn)-f⁢(u)|≤∫B(1+γ)⁢Rn⁢(xn)\BRn⁢(xn)((1+2⁢n)2*+(2⁢n)2*+1)

≤ωN⁢((3⁢n)2*+(2⁢n)2*+n2*)⁢(((1+γ)⁢Rn)N-RnN)

=ωN⁢(32*+22*+1)⁢((1+γ)N-1)

≤22*+1⁢ωN2

for all n. Hence by the calculation in (4.1), we have

|∫ℝNf⁢(un)-f⁢(un-u)-f⁢(u)|=|∫B(1+γ)⁢Rn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|

≥|∫BRn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|-∫B(1+γ)⁢Rn⁢(xn)\BRn⁢(xn)|f⁢(u+wn)-f⁢(wn)-f⁢(u)|

≥|∫BRn⁢(xn)f⁢(u+wn)-f⁢(wn)-f⁢(u)|-22*+1⁢ωN2

→22*+1⁢ωN2

and the claim follows. Note that this example has no simple analogue in the case f⁢(t):=cos⁡(π⁢t)⁢|t|p for p<2*, using Rn:=n-p/N as in the proof of the first case of Example 1.2. Here the reason is that for the analogously defined sequence (wn), the sequence (∇⁡wn) is not bounded in L2. ∎

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).

Acknowledgements

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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Received: 2016-06-02

Revised: 2016-08-15

Accepted: 2016-08-16

Published Online: 2016-10-11

Published in Print: 2018-11-01

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Uniform continuity and Brézis–Lieb-type splitting for superposition operators in Sobolev space

Abstract

1 Introduction

Definition 1.1.

Example 1.2.

Theorem 1.3.

Remark 1.4.

Remark 1.5.

Remark 1.6.

2 Almost BL-splitting

Theorem 2.1.

Proof.

3 Uniform continuity

Lemma 3.1.

Proof.

Proof of Theorem 1.3.

4 Construction of examples

Proof of Example 1.2.

Proof of Remark 1.4.

Acknowledgements

References

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