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
The same conclusion is obtained in that paper for more general
functions, imposing conditions that are satisfied for continuous
convex f with
A different approach to the decoupling of superposition operators
along sequences of functions rests on certain regularity
assumptions on f. For example, assume that
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
To keep the presentation simple and highlight the main idea, we
only treat the case
To explain our results we formalize the notion of decoupling.
Suppose that X and Y are Banach spaces. Consider a map
We say that
If u is a limit of
By [6], the map
then the induced superposition operator
We illustrate the distinction between BL-splitting and almost BL-splitting by the following examples.
If
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
In this context we now formulate our main theorem, in a slightly
more general setting than what we considered above.
A function
Denote by
Consider
and which is A-periodic in its first argument, for some
invertible matrix
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
Theorem 1.3 applies in particular to the functions
considered in Example 1.2 when
Of course, by Sobolev’s embedding theorem, a map
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
The uniform continuity of operators
We now discuss additional aspects and applications of the results
presented above. To this end we return to a simple setting on
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
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 [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
BL-splits along a weakly convergent sequence if the weak limit is
a function tending to 0 as
is uniformly continuous on bounded subsets of
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
2 Almost BL-splitting
In this section we prove a result on the almost BL-splitting of
superposition operators in
If
Consider
- (a)If
is bounded and converges in to a function u, then and almost BL-splits along with respect to u. - (b)If
and in , then almost BL-splits along with respect to u. - (c)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 haveand
For the proof, let
We prove the assertions one by one.
(a): From (1.4) and from the theory of
superposition operators [4] it follows that
The functions
Hence, for all
Consider a smooth cut off function
From the continuity of
Since
where C is independent of ε. Letting ε tend to 0 and using (2.2), we obtain
(b): The continuous embedding
and
(c): Since
3 Uniform continuity
Here we prove uniform continuity on bounded subsets of
For simplicity we will only prove the case
Let
and let
We first recall a functional consequence of Lions’ Vanishing Lemma [15, Lemma I.1.].
Suppose for a sequence
Note first that
If the claim were not true there would exist
Pick
for all n. We reach a contradiction since
The claim of the theorem now follows from
[15, Lemma I.1.] with
We start by proving the uniform continuity. Let
and that there is
Successively we will define infinitely many sequences
and
Here,
We need to say something about the extraction of subsequences.
In order to obtain
For
If
Pick
There are
By (3.5) and by
Theorem 2.1
(b) and (c)
there exists a sequence
and
Set
By the equivariance of
and, since by (3.11) the map
Equations (3.13) and (3.4) (for k) imply that
hence (3.4) for
hence (3.5) for
hence (3.6) for
Since (3.8) is true for k, together with
(3.16) we obtain (3.8) for
By the definition of
This proves (3.7) for
We now skip a finite number of elements of the sequences
constructed in this induction step and adapt
and
for all
Since
for all
We fix m with these properties, we write
and
for all
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
and hence
We claim that the
diagonal sequence
Note that by construction, for all
Hence we have the representation
First we show that
Fix
Then (3.19), (3.22), and the translation
invariance of the norm yield for
It is easy to see that the sequence
To finish the proof of (3.21), suppose for a
contradiction that
for every
as
with
is a subsequence of
We are now in the position to finish the proof of uniform
continuity of
and
By Lemma 3.1 and (3.21), we have
It only remains to prove BL-splitting for
as
4 Construction of examples
We first treat the case
for
for each
as
For the other example,
for each
and hence
as
The construction of these counterexamples is closely related to
Example 1.2. First consider the function
introduce
and
as
and the claim follows. Note that the example above has no
simple analogue in the case
Now we treat the function
Define
and
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
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.
References
- [1]↑
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004), no. 2, 423–443.
- [2]↑
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal. 234 (2006), no. 2, 277–320.
- [3]↑
N. Ackermann and T. Weth, Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math. 7 (2005), no. 3, 269–298.
- [4]↑
J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Math. 95, Cambridge University Press, Cambridge, 1990.
- [5]↑
Y. V. Bogdanskii, Laplacian with respect to a measure on a Hilbert space and an L 2 {L_{2}}-version of the Dirichlet problem for the Poisson equation, Ukrainian Math. J. 63 (2012), no. 9, 1336–1348.
- [6]↑
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490.
- [7]↑
J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, Singapore, 1999.
- [8]↑
G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Math. Soc. Lecture Note Ser. 293, Cambridge University Press, Cambridge, 2002.
- [9]↑
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Encyclopedia Math. Appl. 152, Cambridge University Press, Cambridge, 2014.
- [10]↑
Y. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations 29 (2007), no. 3, 397–419.
- [11]↑
Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251 (2007), no. 2, 546–572.
- [13]↑
W. Kryszewski and A. Szulkin, Infinite-dimensional homology and multibump solutions, J. Fixed Point Theory Appl. 5 (2009), no. 1, 1–35.
- [14]↑
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145.
- [15]↑
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283.
- [16]↑
P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987), no. 1, 33–97.
- [17]↑
E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math. 136 (1999), no. 3, 271–295.
- [18]↑
K. Tintarev and K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London, 2007.
- [19]↑
M. Willem, Minimax Theorems, Progr. Nonlinear Differential Equations Appl. 24, Birkhäuser, Boston, 1996.