Liftings, Young measures, and lower semicontinuity
This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs for under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional
to the space . Results in this spirit were first obtained in Fonseca & Müller [Arch. Ration. Mech. Anal. 123 (1993), 1–49], but our theorem is valid under more natural hypotheses than those currently present in the literature, requiring principally that be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising in the and variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow up procedure.
Finding an integral representation for the relaxation of the functional
to the space of functions of bounded variation, where has linear growth in the last variable, is of great importance in the Calculus of Variations. Defined by the formula
where, for the moment, we do not specify the notion of convergence “” with respect to which is computed, the relaxation is the greatest functional on which is both less than or equal to on and lower semicontinuous (with respect to ) over . From a theoretical point of view, identifying is a necessary step in the application of the Direct Method to minimisation problems with linear growth: indeed, if is coercive, minimising sequences for are merely bounded in the non-reflexive space and can only be expected to converge weakly* in . Since candidate minimisers are only guaranteed to exist in this larger space, there is a need to find a ‘faithful’ extension of to to which the Direct Method can be applied. If they exist, minimisers in of this extension of can then be seen as weak solutions to the original minimisation problem over .
From the perspective of applications, if then the Cauchy–Schwarz inequality implies that computing provides a (usually optimal) lower bound for the -limit of the sequence of singular perturbations
which arises in a vast number of phase transition problems from the physical sciences [8, 11, 18, 22, 25, 33, 37]. In this context, minimisers of can be seen as “physically reasonable” solutions to the highly non-unique problem of minimising the coarse-grain energy , which take into account the fact that transitions between phases should have an energetic cost.
The first general solution to the relaxation problem was provided by Fonseca & Müller in  (see also  for the -independent case). Motivated by problems in the theory of phase transitions, the authors showed that, if is quasiconvex in the final variable, satisfies
for some , and strong localisation hypotheses (see below) hold, then the relaxation of with respect to the strong -convergence in is given by
where is the recession function of defined by , and
is the surface energy density associated with . Here, , and we have used the usual decomposition
for the derivative of a function (see, for example ). Fonseca & Müller’s result, later improved in the subsequent papers [13, 19], makes use of the blow up method to obtain a lower bound for : if is such that is bounded then, upon passing to a subsequence, must converge weakly* in to some Radon measure . Next, one computes lower bounds for the Radon–Nikodym Derivatives , , via estimates of the form
To obtain the inequality “” in (3), it suffices to bound the right hand side of (4) from below by and to obtain analogous results for and . The authors of  achieve this by noting that the partial coercivity property combined with the rescaling in of each allows for to be replaced by a rescaled and truncated sequence , which is crucially weakly* and convergent in to a blow up limit .
Fonseca & Müller’s result confirms that, as in the case for problems posed over for , quasiconvexity is still the right qualitative condition to require for variational problems with linear growth. However, it has been an open question as to whether the -localisation hypotheses which have been used until now are truly necessary for (3) to hold. Up to an error which grows linearly in , these assumptions state that must be such that whenever is sufficiently close to and that uniformly in for sufficiently small.
It is known in the case of superlinear growth that for to be weakly lower semicontinuous over , need only be quasiconvex in the final variable, Carathéodory, and satisfy the growth bound . Similarly, in the special cases where is independent of or if (), so that only scalar-valued functions are considered, it is also known that (3) holds with Carathéodory, quasiconvex in the final variable with linear growth and such that exists [29, 15]. Reasoning by analogy, the implication is that (3) should hold under just the assumptions that be quasiconvex in the final variable and possesses sufficient growth and regularity to ensure that the right hand side of (3) is well-defined over .
To avoid the use of extraneous hypotheses, we must pass from to in (4) using only the behaviour of the sequence rather than any special properties of the integrand. In order to do this, we must improve our understanding of the behaviour of weakly* convergent sequences in , particularly under the blow up rescaling . These are much more poorly behaved than weakly convergent sequences in for thanks to interactions between and in the limit as , see Example 3.10. In the reflexive Sobolev case, powerful truncation techniques [21, 28] mean that these interactions can be neglected in the sequence under consideration, but no such tools are available in when .
The main contributions of this paper are the development of a new theory for understanding weakly* convergent sequences and blow up procedures in , together with the use of this theory to provide a new proof for the integral representation of the weak* relaxation of to under (nearly) optimal hypotheses. This representation is valid for Carathéodory integrands and does not require the -localisation properties of which have previously been used:
Let where and be such that
is a Carathéodory function whose recession function exists in the sense of Definition 2.8 and satisfies ;
satisfies a growth bound of the form
for some , , satisfying , and for all ;
is quasiconvex for every .
Then the sequential weak* relaxation of to is given by
The growth hypotheses of Theorem A are essentially the optimal conditions under which is guaranteed to be finite over all of , and this result consequently identifies the extension of to for the purposes of the Direct Method whenever such an extension is well-defined (in fact, our work allows for a more general but less simple lower bound for than that presented in (5) – see Definition 2.12 and Theorem 6.7). In addition, Theorem A can be seen as a -version of the optimal Sobolev lower semicontinuity theorems obtained by Acerbi & Fusco  and Marcellini .
We emphasise that our is the relaxation of to with respect to sequential weak* convergence, whereas the relaxation of interest from the perspective of some applications and the one which is the subject of the earlier works works [20, 7, 13, 19, 10] is , the relaxation of with respect to strong convergence in . In the absence of coercivity might be strictly less than , but it is always the case that in these applications is partially coercive in the sense that for some continuous and . In these circumstances it is therefore reasonable to expect that the problem of computing reduces to that of computing ‘locally’ in regions of where (and this is in fact the strategy followed in previous works), and so our work opens the possibility of new progress in this area. Indeed, in a forthcoming sequel to this paper we will use Theorem A to derive an integral representation for valid under improved hypotheses on the integrand .
Theorem A assumes that exists in a stronger sense than has been classically required in the literature (see Definition 2.8), where only the upper recession function is used. In fact, the other properties required of in  imply that their must exist in the sense of Definition 2.8 at every point of continuity for , that must be lower semicontinuous, and such that is continuous in for every .
Our proof of the lower semicontinuity ”” component of Theorem A is based on the idea of understanding joint limits for pairs under weak* convergence as objects in the graph space , rather than solely in . To do this, we develop a theory of liftings (ideas of this type were first introduced by Jung & Jerrard in ), which replaces functions by graph-like measures , where is the graph map of . Using a Reshetnyak-type perspective construction, the functional can be generalised to one defined on the space of liftings. The key point is that working in this more general setting means that we can think of sequences as converging weakly* to Radon measures in rather than merely in . This allows us to estimate more precisely from below by computing Radon–Nikodym derivatives at points with respect to the total variation of the (elementary) lifting rather than merely at points with respect to the derivative . In order to carry out these computations, we lay out a framework of generalized Young measures associated to liftings under weak* convergence that allows us to “freeze the -variable” for a wide class of functionals with linear growth by employing robust tools from Geometric Measure Theory. In particular we use a new type of Besicovitch Derivation Theorem which allows us to differentiate in with respect to graphical measures of the form , using very general families of sets (see the discussion which precedes Theorem 5.1).
The idea of understanding the joint weak limits of sequences of pairs by considering instead objects defined over the graph space has of course been explored before. This strategy is successfully followed in  to identify in the case and, for with coercive, convex, and isotropic in the final variable, currents are used to identify in [9, 10]. More generally, ‘graph-like’ objects have been widely used to better understand nonlinear functionals in the Calculus of Variations and Geometric Measure Theory in the context of currents (in particular, Cartesian currents) and varifolds [4, 17, 16, 3, 23, 24, 32]. Liftings seem to be situated at a sweet spot between the usual techniques of the Calculus of Variations and the higher abstractions of geometric analysis, admitting a simple yet surprisingly powerful calculus which appears to be well-suited for functionals of this type. We hope that this tool will prove to be useful in a variety of related problems.
The final step in the proof of Theorem A is to show that the lower bound for obtained via our theory of liftings is optimal. This is equivalent to constructing weak* approximate recovery sequences for and which are such that and
Perhaps surprisingly, Example 6.1 demonstrates that it is not always possible to construct genuine weak* recovery sequences which satisfy
even for continuous and convex in the final variable: in contrast to the -independent cases where or the scalar valued case where , it can occur in the absence of coercivity that admits a global minimiser for which no weakly* convergent minimising sequence exists. Nevertheless, we are able to construct approximate recovery sequences for using a novel ‘cut and paste’ technique based around the rectifiability of the measure combined with Young measure techniques.
This paper is organised as follows: after notation is established and preliminary results and concepts are introduced in Section 2, liftings are defined and their theory developed in Section 3. We prove a structure theorem, establish the convergence and compactness properties of liftings, and we show how can be extended to a functional defined on the space of liftings. In Section 4, we develop a theory of Young measures associated to liftings, including representation and compactness theorems. Section 5 introduces tangent Young measures and their associated Jensen inequalities, which together suffice to implement an optimal weak* blow up procedure and deduce the lower semicontinuity component of Theorem A. Finally, in Section 6 we construct approximate recovery sequences for , before combining these with the results Section 5 to state and prove Theorem 6.7, which is a slightly more general version of Theorem A.
The authors would like to thank Irene Fonseca, Jan Kristensen and Neshan Wickramasekera for several helpful discussions related to this paper. This work presented in this paper forms part of G.S.’s PhD thesis, undertaken at the Cambridge Centre for Analysis at the University of Cambridge, and supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1. F.R. gratefully acknowledges the support from an EPSRC Research Fellowship on “Singularities in Nonlinear PDEs” (EP/L018934/1).
Throughout this work, will always be assumed to be a bounded open domain with compact Lipschitz boundary in dimension , and , will denote the open unit ball in and its boundary (the unit sphere) respectively. The open ball of radius centred at is , although we will sometimes write if the dimension of the ambient space needs to be emphasised for clarity. The volume of the unit ball in will be denoted by , where is the usual -dimensional Lebesgue measure. We will write for the space of real valued matrices, and for the identity matrix living in . The map denotes the projection , and , represent the homotheties and . Tensor products and for vectors , , and real valued functions , , are defined componentwise by and respectively.
The closed subspaces of and consisting only of the functions satisfying are denoted by and respectively. We shall use the notation when the domain of integration might not be clear from context, as well as the abbreviation . We shall sometimes use subscripts for clarity when taking the gradient with respect to a partial set of variables: that is, if then and .
2.1. Measure theory
For a separable locally convex metric space , the space of vector-valued Radon measures on taking values in a normed vector space will be written as or just if . The cone of positive Radon measures on is , and the set of elements whose total variation is a probability measure, is . The notation will denote the usual weak* convergence of measures, and we recall that is said to converge to strictly if and in addition . Given a map from to another separable, locally convex metric space , the pushforward operator is defined by
If is continuous and proper, then is continuous when and are equipped with their respective weak* or strict topologies.
We omit the proof of the following simple lemma:
Let , satisfy and let be a continuous injective map. Then it holds that
Given a function which is positively one-homogeneous in the final variable (that is, for all and ) and a measure , we shall use the abbreviated notation
We note that, if is an injection then applying Lemma 2.1 to and lets us deduce
If is a measure on then we recall that the Disintegration of Measures Theorem (see Theorem 2.28 in ) allows us to decompose as the (generalised) product , where is the pushforward of onto and is a (-almost everywhere defined) parametrised measure. Here, is defined (uniquely) via
For , the -dimensional Hausdorff (outer) measure on is written as and, if is a Borel set satisfying , its restriction to defined by is a finite Radon measure. A set is said to be countably -rectifiable if there exists a sequence of Lipschitz functions () such that
and -rectifiable if in addition . We say that is a -rectifiable measure if there exists a countably -rectifiable set and a Borel function such that .
With assumed to be countably -rectifiable, we can define the Radon–Nikodym derivative for any with respect to , given for -almost every , by
The function is a Radon–Nikodym Derivative in the sense that is a -rectifiable measure and that we can decompose
in analogy with the usual Lebesgue–Radon–Nikodym decomposition.
A measure is said to be admit a (-dimensional) approximate tangent space at if there exists an (unoriented) -dimensional hyperplane and such that
The existence of approximate tangent spaces characterises the class of rectifiable measures in the sense that possesses a -dimensional approximate tangent space at -almost every if and only if is -rectifiable (see Theorem 2.83 in ).
The (column-wise) divergence of a measure , written as or , is an row vector-valued distribution on defined by duality via the formula
Any satisfying (column-wise) is non-atomic. That is, for every .
Let be arbitrary. Defining for , we therefore have that
Noting that and , we can let so that pointwise before using the Dominated Convergence Theorem to deduce
By varying through , we see that as required. ∎
Given a function , we recall the mutually singular decomposition of the derivative , where , is absolutely continuous with respect to , and is the countably -rectifiable jump set of . Each admits a precise representative which is defined -almost everywhere in . The jump interpolant associated to is then the function defined, up to a choice of orientation for the jump set of , for -almost every by
The need to fix a choice of orientation for in order to properly define is obviated by the fact that will only appear in expressions of the form
which are invariant of our choice of .
Given (the precise representative of) a function , the function associated to its graph is denoted by . If is a measure on satisfying both and (we will usually take ), its pushforward under then still makes sense as the Radon measure on .
A sequence is said to converge strictly to if in and strictly in as . We say that converges area-strictly to if in and in addition
as . It is the case that area-strict convergence implies strict convergence in and that strict convergence implies weak* convergence. That none of these notions of convergence coincide follows from considering the sequence given by for some fixed. This sequence converges weakly* to the function for any , strictly if and only if , but (since the function is strictly convex away from ) never area-strictly. Smooth functions are area-strictly (and hence strictly) dense in : indeed, if and is a family of radially symmetric mollifications of then it holds that area-strictly as .
If is such that is Lipschitz and compact, then the trace onto of a function is denoted by . The trace map is norm-bounded from to and is continuous with respect to strict convergence (see Theorem 3.88 in ). If are such that , then we shall sometimes simply say that “ on ”.
The following proposition, a proof for which can be found in the appendix of  (or Lemma B.1 of  in the case of a Lipschitz domain ), states that we can even require that smooth area-strictly convergent approximating sequences satisfy the trace equality :
For every , there exists a sequence with the property that
Moreover, if we can assume that and, if , then we can also require that .
Theorem 2.4 (Blowing up functions).
Let and write as the disjoint union
and as the mutually singular sum
where denotes the set of points at which is approximately differentiable, denotes the set of jump points of , denotes the set of points where is approximately continuous but not approximately differentiable, and satisfies . For and , define by
Then the following trichotomy relative to holds:
For -almost every ,
For -almost every ,
strictly in as .
For -almost every and for any sequence , the sequence contains a subsequence which converges weakly* in to a non-constant limit function of the form
where is non-constant and increasing. Moreover, if is a sequence converging weakly* in this fashion then, for any , there exists such that the sequence converges strictly in to a limit of the form described by (10).
In all three situations, we denote (or ) by . If the base (blow up) point needs to be specified explicitly to avoid ambiguity, then we shall write , and .
For points , the conclusion of Theorem 2.4 follows directly from the approximate continuity of at -almost every and the existence of the jump triple for -almost every . For points , the weak* precompactness of sequences follows from the fact that and combined with the weak* compactness of bounded sets in . The representation of is non-trivial and can only be obtained through the use of Alberti’s Rank One Theorem, see Theorem 3.95 in . It remains for us to show that, given a weakly* convergent sequence and , we can always find such that is strictly convergent in .
First note that we can assume that in since (see for instance Theorem 2.44 in ) for arbitrary it is always true that
Now let be such that and . This implies that and hence that strictly. Since
we therefore see that
as required. ∎
For , the function gives a ’vertically recentered’ description of the behaviour of near . It will be convenient to have a compact notation for also describing this behaviour when is not recentered.
For , define by
If the choice of base point needs to be emphasised for clarity, we shall write .
This definition is independent of the choice of orientation and, for -almost every , the rescaled function converges strictly to as .
The following proposition was first proved in :
Let be a domain with Lipschitz boundary and assume that . Then the embedding
is continuous when is equipped with the topology of strict convergence.
Theorem 2.7 (The chain rule in ).
Let and let be Lipschitz. It follows that and that