For example, since 1990 many organizations have become adapted to a commercial environment that includes electronic commerce (e-commerce).

Before 1990 the environment of organizations with products to sell was one in which buyers and sellers exchanged and transported goods from one place to another.

In the late 1990s, advancement of networked computer technology engendered an explosive growth in e- commerce.

E-commerce allowed the exchange of goods and services over the world wide web, increasing efficiency and precision in commercial transactions.

In North America business-to-consumer e-commerce transactions grew from $11.5 billion in 1998 to $44.5 billion in 2000.

Organizations that were set up for e-commerce transactions got the 100 business of organizations by aggressively improving delivery times.

Within a few years, airline tickets, hotel reservations, and all kinds of goods and services were available via the web.

This rapid change in the external environment selected organizations with technology processes best able to respond to customer demands.


Metacontingencies are relations between interlocking behavioral contingencies and their selecting environments (Glenn, 1989).

Together with behavioral contingencies, metacontingencies account for cultural selection and evolutionary change in organizations.

In organizations, metacontingencies have three components: interlocking behavioral contingencies, their aggregate product, and their receiving system.

The receiving system is the recipient of the aggregate product and thus functions as the selecting environment of the interlocking behavioral contingencies (cf.

Brethower, 2000).

Interlocking contingencies will cease recurring if there is no demand for their products.

Figure 4 illustrates the concept of metacontingency.


Analogous to operant reinforcement in individual behavior, the external environments of organizations deliver selecting consequences.

Customers "buy" (or don't buy) the organization's products, shareholders buy or sell their stocks, granting agencies award grants or don't, government regulators award passes or levy penalties, and so forth.

Most of these consequences are contingently related, however imperfectly, to the products of the interlocking behavioral contingencies.


Consider a restaurant as an organization.

The aggregate product of the restaurant's interlocking behavioral contingencies is the food served, and the receiving system is the Aggregate Product Interlocking Contingencies Behavioral Contingency Receiving System Demand 101 consumers.

The restaurant will survive only if its food and its physical features (ambience) meet the requirements of the selecting environment (people who eat there).

The food and the ambience may change as the restaurant's external environment (customer preferences or competition) changes.

The systems that contribute to the restaurant's product include purchasing, food preparation, service, financial management, and property maintenance.

Each process involves one or more metacontingencies.


Serving meals represents one set of interlocking behavioral contingencies involving several people's behavior: the wait staff's turning in the order, the chef's providing instructions, the cook's preparing food and placing it where the wait staff can pick it up.

The behavior of each individual is related to that of others in the interlocking contingencies.

The aggregate product of these interlocking contingencies is meals served.

If the meals are well adapted to demand, consumers are likely to continue patronizing the restaurant.


Other metacontingencies that have different aggregate products also affect consumer demand.

For example, the interlocking behavior of the wait staff affects timeliness and quality of service.

So multitudes of metacontingencies exist inside the boundaries of the restaurant.

The behavior of any individual, as well as the makeup of any of the interlocking behavioral contingencies may contribute to the fitting of the organization's products to the restaurant's environmental demands.


An organization as a whole can evolve, or change, as repetitions of its internal interrelated metacontingencies occur over time.

The aggregate products generated by the interlocking contingencies vary over time, and the environments in which they exist differentially select those variations.

Figure 5 is a diagram of a cultural lineage.

It shows three repetitions of the same metacontingency over time.

The participants in the metacontingency change across repetitions (illustrated by different shading of the human figures).


Consider a metacontingency in which interlocking behavioral contingencies produce a restaurant's most popular specialty.

Systematic variations in the product might result from slight differences in the interlocking contingencies that occur at lunch and at dinner.

As a result, lunch orders may decrease while dinner orders may increase or remain high.

If these variations in interlocking contingencies have similar effects on other meals, a discerning manager will attempt to analyze the differences in the interlocking behavioral contingencies and arrange contingencies differently for the lunch shift.


The major parts of an organizational ecosystem are its core systems.

In the ecology of an organization, the output of one system directly affects the functioning of other systems.

In organizations, core systems are essential parts, directly responsible for generating the aggregate product.

For example, food preparation is at the core of a restaurant's success.

If the food is bad, no matter how good the service might be, the restaurant likely will fail in the long run.

Production is a core system in a manufacturing company; merchandising is a core system in a retail company; sales is a core system in a marketing company (Malott, 1999).


The relations among the systems and their subsystems in an organization constitute a web of interlocking metacontingencies.

Whether the organization as a whole meets the requirements of the external environment depends almost entirely on the characteristics of these interlocking metacontingencies.

The greater the component complexity of any subsystem, the more interlocking metacontingencies are likely to exist.

Hierarchical complexity increases with the number of subsystem levels.


In this web of interlocking metacontingencies, any significant mismatch between product generation in a system (or subsystem) and the environmental requirements of a related system (or subsystem) is likely to be detrimental to both systems.

If an organization's systems result in products that match poorly with their external environments, either the environment or the systems must change to sustain the organization over extended time.

Because the environment of an organization's subsystems is managed internally, the environment of a subsystem can change so its products better support the organization.


Consider, for instance, a manufacturing company that produced plastic components for the automotive and telecommunication industries.

To fill a product niche in the health industry, the company started manufacturing plastic connectors used in infant heart transplants.

The customer had precise specifications for the product, which required a pollution-free manufacturing environment, imposing significant changes in the production system.

A special area was set up for pollution-free molding, unique safety gear was incorporated in the system, the dress code of the workers was altered, and new production specifications were added to the manufacturing information system.

Because a poor product could cost the life of an infant, other internal systems were adjusted.

For instance, legal agreements with the customers and shipping requirements were changed.


The interlocking metacontingencies in an organizational ecosystem ultimately determine the course of an organization's continuing evolution.

If, over time, a manufacturer's goods are not bought in sufficient quantity to sustain production and invest appropriately for the future, then the organization becomes increasingly less viable in a relatively stable environment.

A change in the external environment (e.g., disappearance of a competitor, or reduction in cost of raw materials) represents a change in metacontingencies that can avert extinction and make recovery possible (at least temporarily).


A fortuitous change in the external environment may result in an adequate match between an organization's systems and its selecting environment.

Such a "saved by the bell" outcome is not typical and organizations do not count on such lucky changes in their external selecting environments.

Rather, they focus on changing their internal environments.

The more complex those internal environments are, the harder it is to react quickly to changes in the external environment.

The safest course of action is to continually monitor the fit between the organization's products and the external environment, to identify current (and predictable future) requirements for continuing adaptedness, and then to plan and rearrange internal metacontingencies.


Because the systems in the organizational ecosystem continually affect one another, core systems in a web of organizational metacontingencies must co-evolve for the organization to thrive.

Co-evolution is the joint evolution of two systems that have a close ecological relationship.

In co-evolution, change in each system is matched by change in the other so that the two systems evolve in relation to one another.

Take, for instance, the use of contemporary cash registers in retail establishments.

Cash registers now calculate the total sale when a customer purchases several items; maintain a record of each sale and the department in which it was made; record whether the sale was by cash or credit; print the details of the sale on a sales slip, which serves as receipt for the customer; and keep track of the sales tax.

This technology could not be adequately implemented without altering several other subsystems, including inventory management.


If contingencies of competition exist between core systems, one core system will suffer at the expense of another.

By definition, all the core systems are essential to the organization's survival, so in designing metacontingencies in which related systems participate, care must be taken to insure co-evolution rather than competition between systems essential to the organization's capacity to meet the requirements of the selecting environment.

Unfortunately, most core systems compete with other systems for resources; and core systems often develop redundancies with other systems in the organization to accomplish their work.

For instance, information technology departments typically do not serve the organization's core systems because they are often overwhelmed with convoluted and fractionated technology infrastructure.

As a result, core systems, like production in a manufacturing company, often hire computer experts to facilitate the production process.

Training departments are similarly redundant in many cases because their personnel are too unfamiliar with critical aspects of the core systems to train employees adequately (Malott, 1999).


Metacontingencies are the units of analysis in organizational ecosystems, and their interlocking behavioral contingencies constitute the cultural entities that evolve via selection.

However, their constituent behavioral contingencies can be analyzed as units of analysis at the behavioral level.

Any intervention designed to better adapt an organization to its external environment requires changes in the interlocking metacontingencies.

And interventions in the interlocking metacontingencies require changes in behavioral contingencies for the individuals involved.


No organization could exist without operant behavior.

As we said earlier, an operant is behavior that has an effect on its environment.

It is acquired during the lifetime of individuals.

Its frequency, form, timing, accuracy or duration changes when behavioral contingencies change.

Although behavioral selection contingencies account for behavior of people in an organization, those contingencies occur in the context of the interlocking contingencies required by the system's external environment.

The complexity of organizations makes it hard to identify where behavioral change can best benefit an organization or where unrecognized changes in behavioral patterns may harm it.

However, the selection contingencies accounting for the behavior of individuals cannot be ignored because the entire ecosystem depends on them.


Not all behavior occurring within the boundaries of an organization is part of the systems that define and sustain the existence of that organization.

In fact, organizations change, sometimes in seemingly chaotic ways.

Take, for instance, an accounting department that produces reports that no one understands.

Although the reports have no function with respect to any other behavior in the organization, a supervisor may continue asking for them.

Behaviors like this can go undetected for indefinite periods, surviving because the receiving system (supervisor) maintains the interlocking contingencies that result in the product (the report).

Organizations can make significant improvements and 105 reduce costs by constantly analyzing the relevance of the products of individual behavior and of interlocking contingencies to overall organizational performance.


Behavior that seriously impedes the systems critical to the organization's survival is worse than wasteful.

An example is a milk processor whose employees fail to wash their hands before milking the cows or who add water to the milk in order to get larger volumes and higher pay.

The results of such behaviors result in milk contamination or low quality milk.

If these behaviors are widespread among employees, the organization can fail.

All of the aforementioned types of behavior are maintained by contingencies of behavioral selection, even those behaviors that have a harmful effect on the organization.

More milk results in more pay for the individual, regardless of the milk quality.

Only changes in behavioral contingencies can mitigate problems such as these.


Although not every behavior considered a "problem" is a threat to an organization, all organizational problems involve behavior.

In solving "behavior problems", we should consider their impact on the products to which they contribute.

Our first priority should be behavior problems that affect core systems, which in turn affect the performance of the organization.


Organizations are conglomerates of dynamic systems that constantly change.

The organizational boundaries as well as the boundaries of their internal systems are permeable.

Organizations are also complex in several different ways.

We identified three types of complexity: environmental (variations in the environment outside of the organization), component (number of organizational components and their relations), and hierarchical (number of levels and their relations).


What are the implications of complexity for managing organizational change? Although we can't eliminate complexity, we can manage it.

We can manage environmental complexity by aligning the internal systems to the environmental demands.

We can manage component complexity by analyzing the sets of interlocking contingencies and their products and eliminating redundancies and disconnects. And we can manage hierarchical complexity by attempting to simplify the levels of management, or reduce the disconnect between tiers.


Organizational change means alterations of metacontingencies and behavioral contingencies.

Metacontingencies are relations between the demand for aggregate products and the interlocking behavioral contingencies that produce them.

Behavioral contingencies are relations between environmental consequences and operant behavior of individuals.

The causes of organizational change are cultural and behavioral selection contingencies.


What are the implications of environmental selection for managing organizations?

First, because organizations evolve over time, any analysis of an organization is merely a snap shot of interrelated metacontingencies at a given time.

Repeated analyses allow us to understand the course of an organization's evolution.

Second, we alter metacontingencies 106 at all relevant levels of the organization, and implement contingency management for behavior critical to the organization's survival.


Organizations are, indeed, enormously complex entities, and Glenn and Malott (2004) have made an important contribution to the analysis of this complexity.

Each of these authors has also made significant individual contributions on related issues in a number of other works.

As we see it, the survival of behavior analysis as an interpretive enterprise critically depends on its adaptability to changing scholarly conditions.

Of particular note among these conditions at the present time is the burgeoning of interdisciplinary study.

The continuation of an intellectual enterprise in this climate may very well depend on the ease with which it is capable of becoming engaged in such study, and the speed with which it does so.

Glenn and Malott’s (2004) work shows an appreciation of this circumstance, and offers a path to our future in this regard.

The comments to follow are offered in support of their work.


Complexity


The analysis of a complex subject matter follows upon an initial construction of some set of units into which observations having some commonality may be grouped, such as to permit other scientific operations to be made with respect to this subject matter.

For example, the cell and the gene comprise units of this sort in different branches of biology, and it is upon these units that other abstracting and generalizing operations are performed in these sciences.

In the science of behavior, this unit is the operant, defined as a class of behaviors grouped by common controlling variables.

Accordingly, operations of prediction and control in the science of behavior pertain not to individual instances of behavior, but rather to the operants of which such instances are members.

Hence, to the extent that prediction and control of organizational change constitute the larger purposes of Glenn and Malott (2004) in depicting organizational complexity, units of this sort are a necessary feature of their formulation.


Toward this end, the authors have articulated a number of valuable constructs.

For example, organizations are characterized by three sorts of complexity, namely environmental, component, and hierarchical.

The first of these is not so much a characteristic of organizations per se as it is a description of the external circumstances under which they operate.

Environmental complexity is analogous to the controlling variables determining membership of individual responses in operant classes.

Different operants are distinguished by different controlling variables, however.

This does not appear to be the case for environmental complexity, as all organizations operate in a context of external circumstances of the sorts enumerated by the authors.

Hence, for this characteristic of organizations to serve analytical purposes, some such variables must be conceptualized as more pertinent than others for the characterization and differentiation of specific organizations.

An articulation of differences among organization types along these lines would thereby seem to constitute a useful refinement of this taxonomical element.


The categories of component and hierarchical complexity more specifically characterize organizations per se.

These categories are roughly analogous to the formal or topographical characteristics of behavior.

By way of illustration, a response may be comprised of multiple components, among which might be included muscular, perceptual, and glandular actions.

Further, a response might be conceptualized as a configuration of pre-current, interim, and final elements occurring in sequence.

Individual responses are identified by these means as are, presumably, individual organizations.


In behavior science, however, the topographical features of responses do not constitute grounds for their collection into operant classes.

Individual responses are collected into classes on the basis of their controlling variables.

This suggests that the categories of component and hierarchical complexity are descriptors of individual organizations, not criteria for the grouping of organizations into functional classes.

An articulation of differences among controlling variables is needed for the latter.


In summary, Glenn and Malott (2004) have identified two significant dimensions along which organizations may be differentiated, as well as the context of external circumstances in which all organizations operate.

Prediction and control of organizational performance or longevity cannot be achieved in the absence of criteria by which organizations may be grouped into classes, however.

In other sciences such groupings are made on the basis of specific classes of controlling variables.

It would seem that further work along these lines would be well worth pursuing.


Selection


Glenn and Malott (2004) do more than identify the context under which organizations operate.

They also articulate the role of this context in engendering organizational change.

Organizations are said to be subject to a process of cultural selection, which is roughly analogous to the process of natural selection in the biological domain.

The authors claim that natural selection causes differential preservation of genetic characteristics in the lineage of a species.

Similarly, cultural selection is said to account for differential preservation of aggregate product characteristics in the lineage of an organization.

This analysis is somewhat unclear.


Natural selection is not properly regarded as the cause of differential preservation of genetic characteristics.

Natural selection refers as much to this outcome as to the process by which the outcome is realized.

Similarly, cultural selection does not explain the differential preservation of aggregate product characteristics.

Rather, both natural selection and cultural selection are more properly understood as contingency relations, than as causes.

The same may be said of behavioral selection.

Reinforcement is a descriptive, not an explanatory, concept.


That said, the authors’ contention that meta-contingencies, together with behavioral contingencies, account for cultural selection and the evolution of organizations requires more clarification.

The same would be true were contingencies of survival said to account for natural selection and the evolution of organisms, or that behavioral contingencies account for behavioral selection and the evolution of repertoires.

As we see it, the contingencies, in each case, do not account for the process of selection and its outcome.

The contingencies express the relation between these processes and their outcomes.


While the distinction we are making in this regard may not seem to be an important one, we think the authors’ perspective on this matter contributes to a lack of clarity as to the units subject to the process of cultural selection.

To make this argument, let us return to the parallel processes of natural and behavioral selection, as articulated by the authors.

In the case of natural selection, genetic entities are selected by their environmental circumstances.

The relationships obtaining between these entities and their environmental circumstances are contingencies of survival.

In the case of behavioral selection, behaviors are selected by their consequences.

The relationships obtaining between these entities and their consequences are contingencies of reinforcement.

Accordingly, in the case of cultural selection, interlocking contingencies and their aggregate products, taken together, are selected by their receiving systems.

The relationships obtaining between these entities and their receiving systems are meta-contingencies.


The parallels drawn by the authors among selection processes at different levels lead the reader to assume that other similarities among these processes will be found and articulated.

This does not appear to be the case, however.

Instead, the authors go on to say that not only are interlocking contingencies and their aggregate products subject to cultural selection, but so also are the relations between these entities and their receiving systems.

That is to say, meta-contingencies are also held to be subject to selection at the cultural level.

The same is not assumed of contingencies of survival or contingencies of reinforcement.

These contingencies may change, of course, but they are not the entities selected.


The argument that meta-contingencies are subject to selection at the cultural level, therefore, is somewhat confusing.

Contingencies can’t be both the causes and the effects, or outcomes, of selection processes.

We don’t think a reasonable solution to this problem will be achieved by deciding into which of these categories contingencies ought to fall, as neither category seems suitable.

Contingencies neither account for the process of selection, nor are they outcomes of such processes.

Rather, as suggested above, contingencies express the relation between these two.


Measurement


For purposes of argument, let us assume that the units subject to cultural selection in the organizational domain are interlocking contingencies together with their aggregate products.

A unit of analysis is an abstraction, constructed by emphasizing the commonalities among some set of events at the expense of their unique features.

It is a class construction.

The value of such constructions is the opportunity they afford for the performance of other scientific operations, in the present case, operations of prediction and control.

These operations cannot be performed with respect to individual events.

They require classes of events whose members’ occurrences may be counted.

It is unclear whether the units subject to cultural selection, as articulated by the authors, meet this requirement.

In other words, it is unclear how individual, unique, instances of interlocking contingencies, together with their aggregate products, are collected into classes such that prediction and control of these classes of events may be achieved.


More to the point, it is unclear how the well-being of an organization is to be assessed.

The authors do not specify what is to be measured in making this assessment, nor do they specify the appropriate metric for this purpose.

Ultimately, the relative longevity of an organization among comparable entities captures its success in adapting to receiving system demands.

Longevity is not a useful measure of organizational well-being, however, in that it can be measured only once, namely when the organization ceases to be.

To impact the ultimate measure of organizational success, more immediate measures of adaptability to these demands must be identified.

For example, an organization’s profitability might be such a measure.

Redundancies, disconnects, and competition among sets of interlocking contingencies and their products impact this measure.

Inefficiencies and inadequacies of all sorts impact this measure.


The authors’ interpretation of complex events is not undermined by their failing to specify appropriate variables and metrics for an assessment of organizational well being.

Prediction and control of organizational well-being depends on it, though, and Glenn and Malott’s prescriptions for organizational change would be more complete were they to articulate these factors.


Organization


The organization of the section entitled, “Selection in Evolving Organizations” moves from the level of natural selection to cultural selection to behavioral selection.

This organization of topics produces an incomplete analysis of events at the level of individual behavior.

For example, at the level of individual behavior, the authors provide an example of an accounting supervisor continuing to request reports from an accounting department that no one understands, and which thereby “have no function with respect to any other behavior in the organization.

” The behavior of the accounting department in producing these reports is attributed to contingencies maintained by the supervisor.

The behavior of the supervisor also warrants analysis in this context, however, and in making this analysis it would become difficult to sustain the notion that these unintelligible reports “have no function with respect to any other behavior in the organization.” Obviously, they have some function with respect to the requesting behavior of the supervisor.


While the authors’ are more than capable of making an analysis of the supervisor’s behavior, to do so would entail some reiteration of the events involved in cultural selection just discussed.

A similar circumstance is seen in the example of milk processors producing more pay by diluting milk products with water.

The potential for greater pay by such practices does not seem to be an individual matter, but rather one predicated by cultural contingencies.

These examples of incomplete analyses of events at the level of individual behavior lead us to suggest that the topic of behavioral selection might be better placed before the consideration of cultural selection.


The organization of this section also gives rise to what appear to be implausible prescriptions for organizational change.

For example, the authors suggest that interlocking contingencies and their aggregate products are subject to deliberate, direct manipulation.

Changes in entities subject to natural selection occur only when environmental circumstances change, however.

Similarly, changes in entities subject to behavioral selection occur only when consequences change.

Hence, to suggest that entities subject to cultural selection are not wholly dependent on changes in receiving systems seems out of keeping with the authors’ more general formulation.

The organization of this section, we believe, is responsible for the truncation of analyses pertinent to practical applications in this field.

Hence, we would suggest a reorganization of these topics in future works.


Conclusion


To conclude, we wish to acknowledge that it is a whole lot easier to comment on analyses of complicated issues made by others than to make such analyses oneself.

The issues Glenn and Malott (2004) are dealing with are extraordinarily complex, and their efforts have brought considerable clarity and order to this complexity.

This is a very valuable contribution, one that is likely to foster the continued well-being of behavior analysis in a rapidly changing scholarly climate.


Hayes and Houmanfar (2004) pinpointed several of the conceptual challenges we faced in writing Complexity and Selection and provided us with an opportunity to enter a dialogue on some of those challenges.

They began by accurately and succinctly stating our purpose in delineating the classes that participate in functional relations at the organizational level of analysis.

If we can accurately describe the processes that account for organizational change, then we can use them to guide interventions designed to enhance organizational fitness.

We also believe that if we can use the model to intervene successfully in organizations where instances of the classes are fairly discernible, such as a factory, analysis and intervention in cultural phenomena of even greater complexity becomes feasible.


In discussing the three types of organizational complexity we described, Hayes and Houmanfar (2004) noted that environmental complexity addresses the complexity of the world outside the organization -- more specifically, those parts of the world that function with respect to an organization’s continuing activities.

We identified some of the external variables that affect some organizations.

Hayes and Houmanfar (2004) suggested that those variables pertain to all organizations, whereas different empirical events affect different operants in an operant analysis (for example, light onset reinforces flipping the switch whereas water flowing reinforces turning the faucet handle).


We think matters are not so clear cut.

In behavioral selection, certain types of events (e.g., praise) have the same function with respect to many operants.

Our examples of external environments affecting organizations were selected from many possibilities to illustrate some of the features of organizational environments in modern society.

If we were dealing with natural selection, we might name climate, predators, and vegetation as examples of environmental variables that often play a role in survival.

Similarly, we might mention food, drink and attention as variables that often have an environmental function.

For organizations, laws (for example) are often part of the external environment, but different laws are relevant to different organizations.

Laws pertaining to air pollution may be critical for a manufacturer but not for a software developer.

Similarly, food would rarely be acceptable as specification of a reinforcing consequence, although we get by with “praise” or “attention” in application.


Hayes and Houmanfar (2004) suggested that we refine our taxonomy of environmental events affecting organizational survival.

We agree that would be useful for practical purposes, which are among our interests in writing the article under discussion.

For analytical purposes, we are lacking ways to classify the environmental events functionally.

Whereas behavioral selection processes are refined as reinforcement, punishment and extinction, cultural selection processes have not been so refined.

If the concepts of behavioral selection processes can be further used to understand cultural selection, classification of environmental variables in functional terms will be necessary.


In response to our section on organizational complexity, Hayes and Houmanfar (2004) pointed out that component and hierarchical complexity are topographical descriptors.

Although the descriptors may apply to many organizations, they are not “criteria for the grouping of organizations into functional classes” (p. 107).

We agree.

However, the instances of the classes entering functional relations in metacontingencies are not organizations but interlocking behavioral contingencies that repeatedly produce an outcome.

The descriptors, component and hierarchical complexity, help us discern the relations among the classes (recurrences) of interlocking behavioral contingencies that reliably produce products needed elsewhere in an organization for the final product to be available to the selecting environment.


This brings us to Hayes and Houmanfar’s comments on selection.

Selectionist terminology is notoriously difficult.

Even in evolutionary biology, where the concept arose and has been worked over for more than 150 years, writers use the word in different ways.

It has been said that Darwin’s great achievement was to posit a mechanism that explained the evolution of species (Mayr, 1982).

That mechanism is natural selection.

Change over time in characteristics of organisms of a lineage is the effect of contingencies of natural selection.

Reduced to its most simplistic formulation, the elements of the mechanism are as follows:


a) Organisms of a lineage interact differentially with their environment.

b) The differential interactions cause differential preservation of some characteristics in the lineage.

c) The change over time in characteristics of a lineage of organisms is called evolution by natural selection.


Statement a specifies differential relations (contingencies) between features of organisms and features of the environment.

Statement b specifies the relation between a and its result; that result is differential preservation in the species (lineage) of some features over others.

Statement c describes the relation between a and b.


In the biological literature, the term selection is used for both a and b, engendering charges of circularity.

In behavior analysis, statement a is analogous to a reinforcement procedure; statement b is analogous to a reinforcement process (i.e., the change in behavior that results from response-consequence contingencies); and statement c describes the relation between the procedure and the process and is called learning.


Like selection, the term reinforcement is used for both a procedure and a process and engenders charges of circularity.

It is surely confusing to say that reinforcement (procedure of establishing a contingency) brings about a result (increased frequency) and also that the relation between the reinforcement procedure and the result is called reinforcement (process).

That confusion is endemic in the behavior analysis literature, as a similar confusion seems endemic in the literature of evolutionary biology.


Hayes and Houmanfar (2004) were concerned with our statement, “Together with behavioral contingencies, metacontingencies account for cultural selection and the evolution of organizations” (Glenn & Malott, 2004, p. 100) because it appeared circular.

It implied to them that we were saying cultural selection (metacontingencies) causes cultural selection and evolution. But in light of the explication above, the statement can be interpreted as follows: behavioral and cultural selection procedures (statement a) result in differential replication of some interlocking behavioral contingencies over others.

The procedure and its result constitute a cultural selection process (statement b).


We do not agree with Hayes and Houmanfar (2004), that “the contingencies express the relation between [selection] processes and their outcomes” (p. 108).

Instead, we believe a more accurate statement is “selection processes express the relation between contingencies and their outcomes.

” At the risk of trying the reader’s patience, selection as a procedure specifies the contingent relations between occurrences and their environments.

A selection process specifies the relation between a procedure and its resulting changes over time in lineages of occurrences.

Table 2 summarizes our perspective.


Hayes and Houmanfar (2004) elegantly stated the role of units of analysis in science and wondered whether the units of analysis in our selectionist view of organizations meet the class (lineage) requirement of recurrences.

They are concerned that organizations are the units of analysis and organizations do not recur.

We see organizations more like ecosystems.

They are loci where multiple lineages of interlocking behavioral contingencies undergo selection by demand for their products.

The units of interest are the recurring interlocking behavioral contingencies.


In conclusion, we appreciate the opportunity to clarify some points of confusion in the initial paper and thank Hayes and Houmanfar (2004) for the intellectual stimulation their comments provided.


This paper deals with the geometry of deformation of nonclassical continua mod- eled as media capable of (first-order) structured deformations introduced by Del Piero and Owen [1993; 1995].

1 The main objective of the theory of structured deformations is to describe how a continuous body with microstructure will deform under the applied forces.


In the original setting [Del Piero and Owen 1993; 1995], a structured deforma- tion is a triplet (K, g, G) of objects whose nature will now be roughly described.

The set K, the crack site, is a subset of vanishing Lebesgue measure of the ref- erence region , the map g : ∼ K → R 3 , the deformation map, is piecewise continuously differentiable and injective, and G is a piecewise continuous map A substantial step towards a concrete interpretation of the tensor G is offered by the approximation theorem [Del Piero and Owen 1993, Theorem 5.8].

That theorem shows that each structured deformation (K, g, G) is a limit of a suitable sequence of simple deformations (K k , g k , D a g k ) in the sense that K k → K, g k → g, and D a g k → G (2) with suitably defined convergences of the objects in (2).

I note that the nontrivial feature of the proof of the approximation theorem lies in proving the injectivity of g k .

Moreover, Del Piero and Owen [1993] prove the following the limit relation for the tensor M:


valid for any sequence (not just the one constructed in the proof of the approxi- mation theorem) (K k , g k , D a g k ) satisfying (2) and any x ∈ ∼ K, where B(x, ρ) is the open ball of center x and radius ρ, J (g k ) is the set of all points of (jump) discontinuity of g k , [g k ] is the jump of g k at the points of J (g k ), n k is the normal to J (g k ), and H 2 is the area measure.


To apply the relaxation techniques of the calculus of variations, Choksi and Fonseca [1997] later enlarged the space of structured deformations to contain all pairs (g, G) where g is in 3 SBV( , R m ) and G is in L 1 ( , M m×n ).

Here m and n are positive integers, the dimensions of the spaces R m and R n of dependent and independent variables, respectively.

Thus, in addition to weaker regularity, the authors relax the injectivity requirement and put the crack site K equal to ∅. (The cracks are described by the omnipresent discontinuities of g.) Choksi and Fonseca [1997, Theorem 2.12] prove the following version of the ap- proximation theorem, which is stated here in a slightly rephrased form as explained below:


Theorem 1.1.

Let ⊂ R n be a bounded open set, and let (g, G) ∈ L 1 ( , R m ) × L 1 ( , M m×n ).

Then there exists a sequence g k in SBV( , R m ) such that


Here D a g k is the absolutely continuous part of the generalized derivative of g k .

The statement of [Choksi and Fonseca 1997, Theorem 2.12] is narrower since (a) the authors assume, in accord with the overall framework of their paper, that g is in SBV( , R m ) and (b) since they replace the equality (4) 2 by the weak * convergence in the sense of measures (although they say that they will prove the equality).

Their proof also shows that g ∈ L 1 ( , R m ) suffices.


In connection with this generality, the question arises: what additional informa- tion beyond (4) can be imposed on the sequence g k if it is known that g belongs to the smaller space BV( , R m ) or even to SBV( , R m )?

An answer, one of the two goals of this note, given in the subsequent theorem, is proved for reference regions represented by admissible domains (which is a mild restriction on , satisfied, e.g., by all open sets with lipschitzian boundary).

4 Theorem 1.2 (approximation theorem).

If is an admissible domain in R n and (g, G) ∈ BV( , R m )×L 1 ( , M m×n ), then there exists a sequence g k ∈ SBV( , R m ) such that in addition to (4) the total variation M(Dg k ) of g k satisfies sup{M(Dg k ) : k = 1, . . . } < ∞; (5) hence, we have the following convergence (without passing to a subsequence):


3 I use the standard notations for function spaces throughout this introduction: thus, BV( , R n ) and SBV( , R n ) are spaces of R m -valued maps on of bounded variation and of special bounded variation, and L 1 ( , R m ) and L 1 ( , M m×n ) are spaces of (Lebesgue-)integrable R m - or M m×n - valued maps on .

M( , M m×n ) is the space of M m×n -valued measures on .

The reader is referred to Sections 2 and 3 below for detailed definitions.

Thus, the extra information stemming from the inclusion g ∈ BV( , R m ) is (5).

It is easy to see that, conversely if (g, G) ∈ L 1 ( , R m )× L 1 ( , M m×n ) is a pair sat- isfying (4) and (5) with g k ∈ SBV( , R m ), then necessarily g ∈ BV( , R m ); in this sense, (5) is optimal.

( Both directions are very intuitive.) The proof of the bound- edness in the approximation theorem is based on the observation in Lemma 5.1 below, but otherwise the construction of the sequence essentially follows that of Choksi and Fonseca.


for almost every point x of .


Here ess lim ρ→0 is the essential limit as ρ → 0, i.e., the limit neglecting an exceptional set of ρ's of vanishing Lebesgue measure.

6 Further, κ n is the volume of the unit ball in R n , J (g k ) is the set of all points of jump discontinuity of g k , [g k ] is the jump of g k at the points of J (g k ), n k is the normal to J (g k ), and H n−1 is the (n − 1)-dimensional Hausdorff measure.


The Appendix to the present paper also outlines a proof of a weaker version of the approximation theorem that does not use Alberti's theorem mentioned above.

In that version, the equality (4) 2 is replaced by the convergence (7) 2 .


Throughout, n is a positive integer, the dimension of the underlying space R n , and m is a positive integer, the dimension of the target space R m .

We denote by a · b the scalar product in both these spaces and by | · | the euclidean norm.

Further, M m×n is the set of all linear transformations from R n to R m .

The value of A ∈ M m×n on x ∈ R n is denoted by Ax.

We denote by A · B := tr( AB T ) the scalar product in M m×n , where A T ∈ M n×m is the transpose of A and tr denotes the trace.

We further denote by | A| = √ A · A the associated euclidean norm.

If f is a map with domain any set M and if N ⊂ M, then f |N denotes the restriction of f to N .


The interior, closure, and boundary of a set M ⊂ R n is denoted by int M, cl M, and bdry M. As in the introduction, B(x, ρ) denotes the open ball in R n of center x and radius ρ.

The symbol κ n denotes the volume of B(0, 1).


Throughout, let be an open subset of R n , later to be restricted by additional requirements.

Let Z be a finite-dimensional inner-product space.


We denote by L n the Lebesgue measure in R n [Federer 1969, §2.6.5], and if k is an integer, 0 ≤ k ≤ n, we denote by H k the k-dimensional Hausdorff measure in R n [ibid., §§ 2.10.2-2.10.60]; recall that H n = L n .

If A ⊂ R n is a Borel set, we denote by H k A the restriction of H k to A, which is the measure defined by (H k A)(B) = H k (A ∩ B) (9) for each Borel set B ⊂ R n .

If A ⊂ R n is a Borel set and f a Z -valued Borel map defined H k almost everywhere on A, integrable with respect to H k on A, then f H k A denotes the Z -valued measure on R n defined by ( f H k A)(B) = A∩B f dH k (10) for each Borel set B ⊂ R n .

The definitions (9) and (10) also apply to k = n, i.e., to L n ≡ H n , resulting in L n A and f L n A.


We denote by L 1 ( , Z ) the set of all (classes of equivalence of) Lebesgue- integrable maps on with values in Z ; we write | · | L 1 ( ,Z ) for the norm on L 1 ( , Z ), defined by | f | L 1 ( ,Z ) = | f | dL n for each f ∈ L 1 ( , Z ).

We denote by C ∞ 0 ( , Z ) the set of all of indefinitely differentiable Z -valued maps f on R n with compact support contained in .


We denote by M( , Z ) the set of all (finite) Z -valued measures on .

If µ ∈ M( , Z ), we denote by |µ| the total variation (measure) of µ, i.e., the smallest nonnegative measure on such that |µ(B)| ≤ |µ|(B) for each Borel subset B of .

We denote by M(µ) the mass of µ, defined by M(µ) = |µ|( ).

A standard result is that


We say that a measure µ ∈ M( , Z ) is supported by a Borel set A ⊂ if µ(B) = 0 for every Borel set B ⊂ such that A ∩ B = ∅. The reader is referred to [Am- brosio et al. 2000, Chapter 1] for further details of measures with values in finite- dimensional inner-product spaces.


If f is a Z -valued map defined L 1 almost everywhere in an interval (0, ) where > 0, we say that a ∈ Z is an essential limit of f at 0 and write a = ess lim ρ→0 f (ρ) (12) if there exists an L 1 null set N ⊂ (0, ) such that a = lim ρ→0 ρ∈(0, )∼N f (ρ), where the last limit is the ordinary limit relative to a subset of (0, ).

Note that, un- like the set N , the value a is uniquely determined, which justifies the notation (12).


We state some basic definitions and properties of the space BV of maps of bounded variation, of the space SBV of special maps of bounded variation, of sets of finite perimeter, and of admissible domains that will be needed in the sequel.

For more details, see [Ambrosio et al. 2000; Evans and Gariepy 1992; Ziemer 1989; Federer 1969].


Definition 3.1.

We denote by BV( , R m ) the set of all g ∈ L 1 ( , R m ) such that there exists a measure Dg ∈ M( , M m×n ) satisfying g ·div T dL n = − T · dDg (13) for each T ∈ C ∞ 0 ( , R m×n ).

Here div T is an R m -valued map on such that a · div T = tr(D(T T a)) for each a ∈ R m , where D(T T a) denotes the classical derivative of the map T T a. The elements of BV( , R m ) are called maps of bounded variation; the measure Dg is uniquely determined by g and is called the weak (or generalized) derivative of g. We denote by M(Dg) the mass of the measure Dg as defined in Section 2 and call M(Dg) the total variation of g. Equations (11) and (13) provide


The choice of T represented by a matrix function with only the (i, j) element different from 0, where i ∈ {1, . . . , m} and j ∈ {1, . . . , n}, reduces (13) to the usual index definition of BV as in, e.g., [Ambrosio et al. 2000, Equation (3.2)].

The set BV( , R m ) is a Banach space under the norm


Definition 3.2. Let g ∈ L 1 ( , R m ).

We say that g has an approximate limit at x ∈ if there exists a ∈ R m such that


The value a is uniquely determined and is called the approximate limit of g at x. The complement S(g) ⊂ in of the set of all x ∈ where the approximate limit of g exists is called the approximate discontinuity set of g.


Definition 3.3. Let g ∈ L 1 ( , R m ).

We say that x ∈ is an approximate jump point of g if there exist a, b ∈ R m , a = b, and n ∈ R n with |n| = 1 such that


The triplet (a, b, n), if it exists, is uniquely determined to within the interchange of a and b and a simultaneous change of the sign of n. In any case, the product [g] ⊗ n, (15) occurring frequently below, is uniquely determined, where [g] = a − b is the jump of g at x. We denote by J (g) the set of all approximate jump points of g and call any ±n the normal of J (g) at x.


The following result describes the relationship between the sets S(g) and J (g): Theorem 3.4.

If g ∈ BV( , R m ), then: (i) J (g) ⊂ S(g) and H n−1 (S(g) ∼ J (g)) = 0.


(ii) J (g) is countably (H n−1 , n − 1)-rectifiable in the sense that H n−1 almost all of J (g) can be covered by countably many class-1 surfaces C k , k = 1, . . . , of dimension n − 1 in R n .


The derivative of a map of bounded variation has the following well known structure.

The subsequent treatment uses especially the jump and the absolutely continuous parts of the derivative to be introduced now.


Theorem 3.5.

If g ∈ BV( , R m ), then:


(i) The derivative Dg has a unique decomposition Dg = D a g L n + D c g + D j g, where D a g, the absolutely continuous part of Dg, is a map in L 1 ( , M m×n ); D c g, the Cantor part of Dg, is a measure on singular with respect to L n and diffuse with respect to H n−1 , i.e., D c g is supported by a set of null Lebesgue measure in R n and D c g(B) = 0 for each Borel subset B of of finite H n−1 measure; and D j g, the jump part of Dg, is a measure absolutely continuous with respect to H n−1 .


(ii) The jump part D j g is supported by J (g), and in fact, D j g = [g] ⊗ n H n−1 J (g), where, for every point x of J (g), the value [g] ⊗ n is the product (15).


(iii) For L n almost every point x of , we have


Definition 3.6.

We denote by SBV( , R m ) the set of all g ∈ BV( , R m ) with D c g = 0.

The elements of SBV( , R m ) are called special maps of bounded variation.


SBV( , R m ) is a closed subspace of BV( , R m ) under the norm | · | BV( ,R m ) .

Theorem 3.7 [Alberti 1991].

If is bounded, then for any G ∈ L 1 ( , M m×n ), there exists a g ∈ SBV( , R m ) such that D a g = G; moreover, there exists a constant c ∈ R depending only on such that the map g as above can be chosen to satisfy


We conclude this section with basic information on sets of finite perimeter and on admissible domains.

Sets of finite perimeter fall in the framework of BV as will be explained below.

For a subset of the class of sets of finite perimeter called admissible domains (see below), we shall establish the approximation theorem.

The distinguishing feature of admissible domains is that maps from BV( , R m ) have well defined boundary values.


Definition 3.8.

A set E ⊂ R n is said to have a finite perimeter if 1 E ∈ BV(R n , R), where 1 E denotes the characteristic function of E. The perimeter of E is M(D1 E ).

The measure-theoretic boundary of E is the set S(1 E ) that differs from the reduced boundary bdry * (E) := J (1 E ) by a set of H n−1 measure 0.

Theorem 3.9.

If E is a set of finite perimeter, then for every x ∈ bdry * (E), the triplet (a, b, n) as in Definition 3.3 can be chosen to be (0, 1, n(x)); with this choice, n(x) is uniquely determined and is called the measure-theoretic normal to E at x. Equations (14) then imply the following well known formulas:


Thus, even 1 E ∈ SBV(R n , R).


Definition 3.10 [Ziemer 1989, Definition 5.10.1].

A bounded open set ⊂ R n is said to be an admissible domain if it has a finite perimeter and the following two conditions are satisfied:


(ii) There exists a constant M, and for each x ∈ bdry , there is a ball B(x, r ) with H n−1 (bdry * E ∩ bdry * ) ≤ MH n−1 ( ∩ bdry * E) whenever E ⊂ cl ∩ B(x, r ) is a set of finite perimeter.


Each open bounded set with lipschitzian boundary is an admissible domain [Ziemer 1989, Remark 5.10.2].

The following two theorems describe the main virtues of admissible domains: Theorem 3.11 (see [Ziemer 1989, Section 5.10]).

If is an admissible domain and g ∈ BV( , R m ), then there exist an H n−1 -measurable map g bdry on bdry such that g · div T dL n + T · dDg = bdry( ) T n · g bdry dH n−1 for every class-1 map T on with values in M m×n that has a continuous extension (again denoted by T ) to cl , where n is the measure-theoretic normal to .

There exists a c ∈ R depending only on such that


The map g bdry is determined to within a change on a set of H n−1 -measure 0 and is called the trace of g. One has


and there exists a c ∈ R depending only on such that


For the purpose of the approximation theorem and the limit relation (as stated in Section 1), we enlarge the set SBV( , R m ) × L 1 ( , M m×n ) of structured deforma- tions of Choksi and Fonseca [1997] to form the set BV( , R m ) × L 1 ( , M m×n ).

We furthermore interpret the elements g ∈ SBV( , R m ) as the macroscopic de- formations of the body with macroscopic crack site J (g).

We note that the space of structured deformations (K, g, G) of Del Piero and Owen [1993] as de- scribed in Section 1 with K = ∅ is a subset of SBV( , R m ) × L 1 ( , M m×n ) ⊂ BV( , R m ) × L 1 ( , M m×n ).

In a general, (g, G) ∈ BV( , R m ) × L 1 ( , M m×n ), the map g is the possibly discontinuous macroscopic displacement, of the body and G is a microscopic disarrangement as explained in the introduction and in accord with the original papers by Del Piero and Owen [1993; 1995].


The proof of the Approximation Theorem is based on the decomposition of R n into the disjoint union of sufficiently small cubes of equal edge length and with faces parallel to the natural coordinate planes in R n .

Various maps involved in the construction are then approximated by (generally) discontinuous maps constant on the cubes (as in the present section) or by discontinuous maps linear on the cubes (as in the Appendix below).


For each positive integer k, consider the decomposition of R n into the system of cubes


We only prove (21) since the other assertions of the lemma are immediate.

Let x ∈ P be arbitrary, and denote y(x) := x + n/k so that y(x) ∈ Q. Then


Consequently, integrating (22) over P, we obtain


Integrating over R n , we obtain


for each x ∈ R n , where p ∈ Z n /k is uniquely determined by the requirement that x ∈ C(k, p) and where


Then m k is piecewise constant, with all points of jump discontinuity contained in the union


Put l := g − h, which is an element of BV( , R m ).

Since is an admissible domain, the extension l 0 of l to R n equal to 0 outside satisfies l 0 ∈ BV(R n , R m ) by Theorem 3.12.

Let f k be a sequence of mollifications of l 0 on R n with the mollification parameter tending to 0 so that f k ∈ C ∞ 0 (R n , R m ),


Applying Proposition 5.2 with f replaced by f k , we find that for each k there exists an m k ∈ SBV(R n , R m ) such that


We put g k = m k | + h for k = 1, . . .

so that g k ∈ SBV( , R m ). Equations (29) and (30) imply m k | → l in L 1 ( , R m ) as k → ∞, and hence, we have (4) 1 .

Further, (31) and (27) imply (4) 2 .

Finally, (32), (28), and h ∈ SBV( , R m ) imply (5).

Assertion (6) then follows by an easy argument that is left to the reader.


Lemma 6.1. Let be bounded, let g be a map (not a class of equivalence) in BV( , R m ), let x ∈ , and let > 0 be such that B(x, ) ⊂ .

Then for L 1 almost every ρ ∈ (0, ), g| bdry B(x, ρ) is the trace of g|B(x, ρ) ∈ BV(B(x, ρ, R m ).


Proof.

By the Lebesgue differentiation theorem, there exists a Borel set E ⊂ with L n (E) = 0 such that for every y ∈ ∼ E we have


we see that for L 1 almost every ρ > 0 we have


For every such a ρ ∈ (0, ), we have (33) for H n−1 almost every y ∈ bdry B(x, ρ) and hence in particular also


This paper derives explicit formulas for disarrangement densities in structured deformations, focusing on submacroscopic separations, switches, and interpenetrations.

The authors employ relaxation techniques within a mathematical framework for structured deformations, confirming earlier results by Owen and Paroni 【1】.


The study provides alternative proofs and new explicit formulas for disarrangement measures, significant for applications in continuum mechanics.

The research also compares different relaxation schemes, showing that the same bulk disarrangement density arises from both the Choksi-Fonseca and Baía-Matias-Santos frameworks 【2】【3】.


The findings have implications for modeling the effects of submacroscopic defects in solids 【4】.


This paper deals with the relaxation of energies of media with structured deformations introduced by Del Piero and Owen [1][2].

Structured deformations provide a multiscale geometry that captures the contributions at the macrolevel of both smooth and nonsmooth geometrical changes (disarrangements) at submacroscopic levels.

The paper examines the special case of Choksi and Fonseca’s [3] energetics of structured deformations in which the unrelaxed energy does not contain the bulk contribution.

Thus, the energy is purely interfacial but of a general form.

New formulas for the relaxed bulk and interfacial energies are proved.

The bulk relaxed energy is shown to coincide with the subadditive envelope of the unrelaxed interfacial energy while the relaxed interfacial energy is the restriction of the envelope to rank-1 tensors.

Moreover, it is shown that the minimizing sequence required to define the bulk energy in the relaxation scheme of Choksi and Fonseca [3] can be realized in the more restrictive class required in the relaxation scheme of Baía, Matias and Santos [4], thus establishing the equality of relaxed energies of the two approaches for general purely interfacial energies.

The relaxations of the specific interfacial energies of Owen and Paroni [5] and Barroso, Matias, Morandotti and Owen [6] are simple consequences of our general results.


1. Introduction


This paper deals with the relaxation of nonclassical continua modeled as media with structured deformations introduced by Del Piero and Owen [1][2].

In their original setting, a structured deformation is a triplet (K, g, G) of objects of the following nature.

The set K ⊂ R3, the crack site, is a subset of vanishing Lebesgue measure of the reference region (cid:127), the map g : (cid:127) \ K → R3, the deformation map, is piecewise continuously differentiable and injective, and G is a piecewise continuous map from (cid:127) \ K to the set of invertible second-order tensors describing deformation without disarrangements.


Within this context, simple deformations are triples (K, g, ∇g) where g is a piecewise smooth injective map with jump discontinuities describing partial or full separation of pieces of the body.

In view of this, in the general case of a structured deformation (K, g, G), the tensor H = ∇g − G, the deformation due to disarrangements, measures the departure of (K, g, G) from the simple deformation (K, g, ∇g).


Choksi and Fonseca [3] introduced into the theory of structured deformations energy considerations and the ideas of relaxation.

For further studies in one- and multidimensional settings, see [7][8].

It is well-known that the existing techniques of relaxation of the calculus of variations and continuum mechanics are unable to cope with injectivity requirements.

Accordingly, Choksi and Fonseca neglect the injectivity requirement; in addition, they assume weaker regularity.

In their interpretation, structured deformations are pairs (g, G) where g : (cid:127) → Rn is a special Rn-valued map of bounded variation from the space SBV((cid:127)) and G : (cid:127) → Lin is an integrable Lin-valued map from the space Ł1((cid:127), Lin) [9].

Thus, SD((cid:127)) := SBV((cid:127)) × Ł1((cid:127), Lin) is the set of all structured deformations.

Structured deformations of the form (g, ∇g) with g ∈ SBV((cid:127)) are called simple deformations in this paper.


The relaxation starts from the energy


E(g) = \int_{(cid:127)} W (∇g) dV + \int_{J (g)} ψ([[g]], νg) dA


of a simple deformation g ∈ SBV((cid:127))

Here V and A are the Lebesgue measure and the (n − 1)-dimensional Hausdorff measure in Rn and ∇g is the absolutely continuous part of the derivative (= gradient) Dg of g, while the singular part Dsg := [[g]] ⊗ νg (cid:120) J (g) is a tensor-valued singular measure describing the discontinuities of g; that part is formed from the jump set J (g) ⊂ (cid:127) of g, the jump [[g]] of g on J (g) and the normal νg to J (g).

The reader is referred to (24), below, for a detailed description of these objects.

The material is characterized by the bulk energy density W : Lin → R and the interfacial (or cohesion) energy ψ : Dn → R, where we denote Dn = Rn × Sn−1.


The approximation theorem of Del Piero and Owen [1, Theorem 5.8] says that every structured deformation is a well-defined limit of simple deformations.

In the framework of Choksi and Fonseca [3] (see also [10]), this means that corresponding to each structured deformation (g, G) ∈ SD((cid:127)) there exists a sequence (gk, ∇gk) ∈ SD((cid:127)) (i.e., with gk in SBV((cid:127)))

such that


gk → g in Ł1((cid:127), Rn),
∇gk (cid:42)∗ G in M((cid:127), Lin),
sup{|∇gk|Ł1((cid:127),Rn) : k = 1, . . . } < ∞.


The relaxed energy of a structured deformation (g, G) ∈ SD((cid:127)) is defined by


I (g, G) = inf{lim inf_{k→∞} E(gk) : gk ∈ SBV((cid:127)) satisfies (2)}.


Thus, a sequence approaching the above infimum realizes the most economical way to build up the deformation (g, G) using approximations in SBV.

The relaxation theorem of Choksi and Fonseca [3, Theorems 2.6 and 2.17, Remark 3.3] says that, under some assumptions on W and ψ (a particular case of which are Assumptions 2.1, below), the relaxed energy admits the integral representation


I (g, G) = \int_{(cid:127)} H (∇g, G) dV + \int_{J (g)} h([[g]], νg) dA


where H and h are some functions determined explicitly in the cited theorems (Theorem 2.2 presents formulas for H and h for a particular case).


This paper deals with the relaxation of energy functions E for which the bulk contribution vanishes, i.e., with energy functions of the form


E(g) = \int_{J (g)} ψ([[g]], νg) dA


for each g ∈ SBV((cid:127)).

The main result, Theorem 2.3, below, gives explicit descriptions of the functions H and h from (4) and applies them to give simplified proofs of two particular cases Examples 2.5 and 2.6 given previously in [5][6].


2. The main result and examples


We make the following standing hypotheses about ψ.


2.1. Assumptions.


(i) The function ψ : Dn → R is continuous.


(ii) We have ψ(−a, −b) = ψ(a, b) and 0 ≤ ψ(a, b) ≤ C1|a| for every (a, b) ∈ Dn and some C1 > 0.


(iii) The function ψ( · , ν) is subadditive and positively homogeneous for each ν ∈ Sn−1.


To ease the statements of the results, we extend any function ζ : Dn → [0, ∞) to an identically denoted function ζ : Rn × Rn → [0, ∞) by homogeneity with respect to the second variable, i.e., by assuming that the extended function satisfies ζ (a, tb) = tζ (a, b) for any t ≥ 0 and (a, b) ∈ Rn × Rn.

This convention applies in particular to the functions ψ and h.


We need some notation to formulate the main results.

Let Q = (− 1/2, 1/2)^n, and for every M ∈ Lin, let wM : ∂ Q → Rn be given by wM (x) = M x for every x ∈ ∂ Q.


Furthermore, if (a, b) ∈ Dn, let Qb be any cube with unit edge, center at 0 ∈ Rn and two faces normal to b, and let za,b : Qb → Rn be the map defined by za,b(x) = 1/2 a(sgn(x · b) + 1), x ∈ Qb.


Finally, if u ∈ SBV((cid:127)), let us put (cid:57)(Dsu) := \int_{J (u)} ψ([[u]], νu) dA.


The following statement is a particular case W = 0 of the relaxation theorem of Choksi and Fonseca [3, Theorems 2.6 and 2.17, Remark 3.3].


2.2. Theorem.

The effective energies H and h are given by


H (A, B) = inf{(cid:57)(Dsu) : u ∈ SBV(Q), u = wA on ∂ Q, \int_{Q} ∇u dV = B}


for each A, B ∈ Lin and


h(a, b) = inf{(cid:57)(Dsu) : u ∈ SBV(Qb), u = za,b on ∂ Qb, ∇u = 0 on Qb}


for each (a, b) ∈ Dn.


The following theorem, the main result of this paper, shows that the functions H and h admit a much more explicit description in terms of a single function (cid:56).


2.3. Theorem.

The functions H and h in Theorem 2.2 are given by


H (A, B) = (cid:56)(A − B),
h(a, b) = (cid:56)(a ⊗ b)


for every A, B ∈ Lin and (a, b) ∈ Dn, where (cid:56) is a subadditive and positively homogeneous function on Lin defined by each of the following equivalent Assertions (i)–(iv); moreover, for dyadic arguments, we have an additional Assertion (v).


(i) (cid:56) is the biggest subadditive function on Lin satisfying (cid:56)(a ⊗ b) ≤ ψ(a, b) for every (a, b) ∈ Dn; i.e., (cid:56)(M) = sup{(cid:50)(M) : (cid:50) is subadditive on Lin and (cid:50)(a ⊗ b) ≤ ψ(a, b) for every (a, b) ∈ Dn}.


(ii) For every M ∈ Lin, (cid:56)(M) = inf{\sum_{i=1}^{m} ψ(ai , bi ) : (ai , bi ) ∈ Dn, i = 1, . . . , m, \sum_{i=1}^{m}

ai ⊗ bi = M}.


(iii) For every M ∈ Lin, (cid:56)(M) = inf{(cid:57)(Dsu) : u ∈ SBV(Q), u = wM on ∂ Q, ∇u = 0 on Q}.


(iv) For every M ∈ Lin, (cid:56)(M) = inf{(cid:57)(Dsu) : u ∈ SBV(Q), u = wM on ∂ Q, \int_{Q} ∇udV = 0}.


(v) For arguments of the form a ⊗ b, where (a, b) ∈ Dn, (cid:56)(a ⊗ b) = inf{(cid:57)(Dsu) : u ∈ SBV(Qb), u = za,b on ∂ Qb, ∇u = 0 on Qb}.


The proof of Theorem 2.3 is given in Sections 5 and 6, below.


2.4. Remarks.


(a) Since the pointwise supremum of any family of subadditive functions is subadditive (e.g., [11, Theorem 7.2.2]), (15) really defines a subadditive function.


(b) Among the above characterizations of (cid:56), the closely related novel forms (i) and (ii) must be considered as the most important.

The main advantage of (i) and (ii) is that they establish connections to the wealth of results of the convexity theory.

These will be employed to analyze the examples to be formulated below.


(c) In one dimension, one can orient the normals to jumps to be always the vector +1 (rather than −1) and hence the dependence of ψ on the second variable can be suppressed: ψ = ψ(a), a ∈ R. Assumption 2.1(iii) then says that ψ is subadditive and positively homogeneous.

Thus, the subadditive envelope (cid:56) of ψ is ψ itself, and all mentions of a subadditive envelope can be avoided.

This is not the case if Assumption 2.1(iii) is relaxed.

Indeed, working in one dimension, Del Piero [7][8] calculated the relaxation of the energy (1) with the interfacial energy ψ of a general form, avoiding Assumption 2.1(iii).

His main result contains the subadditive envelope of ψ also.

In light of the above discussion, this envelope plays a different but related role.

The relaxation of a purely interfacial energy of a more general form than that postulated in Assumptions 2.1 in arbitrary dimension will be treated in a future paper.


(d) The expressions in (iii)–(v) already occurred previously, albeit without noting that they are mutually equivalent and equivalent to (i) and (ii), except for some particular cases to be mentioned below.

The formula for H in (13a) with (cid:56) defined in (iv) and the formula for h in (13b) with (cid:56) defined in (v) are direct consequences of Choksi and Fonseca’s expressions in (11) and (12).

The formula for H with (cid:56) given by (iii) crops up in the relaxation schemes by Baía, Matias and Santos [4, (3.2)] and by Barroso, Matias, Morandotti and Owen [6, Theorem 3.2].

The relaxation schemes in the last two papers require among other things higher regularity of structured deformations and are not strictly comparable with that of Choksi and Fonseca described above.


(e) The infimum (iii) could be, in principle, bigger than (iv).

Nevertheless, the infima are generally the same.

This has been established previously in [6] for the special choices of ψ described in the following examples, which motivated the present study.


2.5. Example [5, Theorem 4, particular case L = I ].

If ψ| · |(a, b) = |a · b| and ψ±(a, b) = {a · b}± for every (a, b) ∈ Dn, where { · }+ and { · }− denote the positive and negative parts of a real number, then (cid:56)| · |(M) = |tr M|, (cid:56)±(M) = {tr M}± for every M ∈ Lin.

The effective energies H| · |, H±, h| · | and h± are determined through (cid:56)| · | and (cid:56)± by (13).


As shown in [5], {tr M}+ is a volume density of disarrangements due to submacroscopic separations, {tr M}− is a volume density of disarrangements due to submacroscopic switches and interpenetrations, and |tr M| is a volume density of all three of these nontangential disarrangements: separations, switches and interpenetrations.

The evaluation in [5] of H (equivalently, of (cid:56)) for (21) is rather complicated; a recent paper by Barroso, Matias, Morandotti and Owen [6] presents some simplification and the realization of the minimizing sequence in the narrower class (iv) in Theorem 2.3 mentioned earlier.

Our version of the derivation, which includes the minimizing sequence from (iv) via Theorem 2.3 also, is given in Section 7.


2.6. Example [6, (5.3)].

If ψ(a, b) = |a · p| for (a, b) ∈ Dn, where p ∈ Rn is a fixed vector, then (cid:56)(M) = |M T p| for any M ∈ Lin.


3. Notation and functions of bounded variation


We denote by Z the set of integers, by N the set of positive integers, by Sn−1 the unit sphere in Rn and by Lin the set of all linear transformations from Rn into itself, often identified with the set of n × n matrices with real elements.

We use the symbols “ · ” and “| · |” to denote the scalar product and the euclidean norm on Rn and on Lin.

The latter are defined by A · B := tr(ABT) and |A| = √(A · A) where AT ∈ Lin is the transpose of A and tr denotes the trace.


A real-valued function f defined on a vector space X is said to be subadditive if f (x + y) ≤ f (x) + f (y) for every x, y ∈ X and positively homogeneous if f (t x) = t f (x) for every t ≥ 0 and x ∈ X.


If (cid:127) is an open subset of Rn, we denote by Ł1((cid:127), Lin) the space of Lin-valued integrable maps on (cid:127).

We denote by M((cid:127), Lin) the set of all (finite) Lin-valued measures on (cid:127).

If µ ∈ M((cid:127), Lin), we denote by µ (cid:120) B the restriction of µ to a Borel set B ⊂ (cid:127).

If G, Gk ∈ Ł1((cid:127), Lin), k = 1, 2, .

. . , we say that Gk converges to G in the sense of measures, and write Gk (cid:42)∗ G in M((cid:127), Lin), if \int_{(cid:127)} Gk · H dV → \int_{(cid:127)} G · H dV for every continuous map H : Rn → Lin which vanishes outside (cid:127).


We state some basic definitions and properties of the space BV((cid:127)) = BV((cid:127), Rn) of maps of bounded variation and of the space SBV((cid:127)) = SBV((cid:127), Rn), special maps of bounded variation.

For more details, see [12][13][14][15].


We define the set BV((cid:127)) as the set of all u ∈ L 1((cid:127)) = L 1((cid:127), Rn) such that there exists a measure Du ∈ M((cid:127), Lin) satisfying \int_{(cid:127)} u · div T dV = −\int_{(cid:127)} T · dDu for each infinitely differentiable map T : Rn → Rn×n which vanishes outside some compact subset of (cid:127).

Here div T is an Rn-valued map on (cid:127) given by (div T )i = \sum_{j=1}^{n} Ti j, j , where the comma followed by an index j denotes the partial derivative with respect to j-th variable.

The measure Du is uniquely determined and called the weak (or generalized) derivative of u. We shall need the following form of the Gauss–Green theorem for BV: if (cid:127) is a domain with lipschitzian boundary and u ∈ BV((cid:127)), then there exists an integrable map u∂(cid:127) : ∂(cid:127) → Rn such that \int_{(cid:127)} dDu = \int_{∂((cid:127))} u∂(cid:127) ⊗ ν(cid:127) dA, where ν(cid:127) is the outer normal to ∂(cid:127).

The map u∂(cid:127) is determined within a change on a set of A measure 0 and is called the trace of u.


We define the set SBV((cid:127)) as the set of all u ∈ BV((cid:127)) for which Du has the form Du = ∇u V (cid:120) (cid:127) + [[u]] ⊗ νu (cid:120) J (u) A where ∇u, the absolutely continuous part of Du, is a map in Ł1((cid:127), Lin) and the term Dsu := [[u]] ⊗ νu (cid:120) J (u) A on the right-hand side of (24) is called the jump (or singular) part of Du.

The objects J (u) ⊂ (cid:127), [[u]] : J (u) → Rn and νu : J (u) → Sn−1 are called the jump set of u, the jump of u and the normal to J (u), respectively.

Here J (u) is the set of all x ∈ (cid:127) for which there exist νu(x) ∈ Sn−1 and u±(x) ∈ Rn such that we have the approximate limits u±(x) = ap lim_{y→x, y∈H ±(x,νu (x))} u(x), where H ±(x, νu(x)) = {y ∈ Rn : ±(y − x) · νu(x) > 0}. For a given x ∈ (cid:127), either the triplet (νu, u+, u−) = (νu(x), u+(x), u−(x)) does not exist or it is uniquely determined to within the change (νu, u+, u−) (cid:55)→ (−νu, u−, u+).

With one of these choices, one puts [[u]] = u+ − u− and notes that [[u]] ⊗ νu is unique.

Finally, we denote by (cid:104)r (cid:105) the integral part of r ∈ R. Clearly, r − 1 ≤ (cid:104)r (cid:105) ≤ r, 0 ≤ r − (cid:104)r (cid:105) ≤ 1.


Writing r = kt, where t ∈ R and k > 0, and dividing by k, we obtain 0 ≤ t − (cid:104)kt(cid:105)/k ≤ 1/k and hence (cid:104)kt(cid:105)/k → t as k → ∞ uniformly in t ∈ R.


4. Preliminary results


We put


C B (M) := {u ∈ SBV(Q) : u = wM on ∂ Q, ∇u = 0 on Q},


B (M) := {u ∈ SBV(Q) : u = wM on ∂ Q, \int_{Q} ∇u dV = 0}


for any M ∈ Lin.

We start with the following preliminary results.


4.1. Proposition.

If A, B ∈ Lin and u ∈ B (A) and v ∈ B (B), then u +v ∈ B (A + B) and (cid:57)(Dsu + Dsv) ≤ (cid:57)(Dsu) + (cid:57)(Dsv); if (J (u) ∩ J (v)) = 0, then we have the equality sign in (28).


Proof.

We have J (u + v) = Ku ∪ Kv ∪ L where L = J (u) ∩ J (v), Ku = J (u) \ K , Kv = J (v) \ K .

Next, we observe that on L we have νu(x) = ±νv(x) for A-almost every x ∈ L; since we have a freedom in the choice of the sign of νv, we assume νu(x) = νv(x) and denote µ = νu on L. Then [[u + v]] ⊗ νu+v = [[u]] ⊗ νu on Ku, [[v]] ⊗ νv on Kv, ([[u]] + [[v]]) ⊗ µ on L.


By the subadditivity of ψ, ψ([[u]] + [[v]], µ) ≤ ψ([[u]], µ) + ψ([[v]], µ) = ψ([[u]], νu) + ψ([[v]], νv) and hence (30) provides ψ([[u + v]], νu+v) = ψ([[u]], νu) on Ku, ψ([[v]], νv) on Kv, ≤ ψ([[u]], νu) + ψ([[v]], νv) on L.


Integrating over J (u + v) and using (29), we obtain (cid:57)(Dsu + Dsv) = \int_{J (u+v)}

ψ([[u

+ v]], νu+v) dA ≤ \int_{Ku} ψ([[u]], νu) dA + \int_{Kv} ψ([[v]], νv) dA + \int_{L} ψ([[u]], νu) dA + \int_{L} ψ([[v]], νv) dA = (cid:57)(Dsu) + (cid:57)(Dsv), which completes the proof of (28).


Remark.

If the interfacial energy density ψ has the special form ψ(a, b) = (cid:51)(a ⊗ b) where (cid:51) : Lin → [0, ∞) is a subadditive and positively homogeneous function, then (cid:57)(Dsu) is given by (cid:57)(Dsu) = (cid:51)(Dsu) where Dsu := [[u]] ⊗ νu (cid:120) J (u) is the singular part of the derivative Du of u and (cid:51)(Dsu) := \int_{J (u)} (cid:51)([[u]] ⊗ νu) dA is an instance of Reshetnyak’s [16] functional µ (cid:55)→ (cid:51)(µ) of a measure µ ∈ M(Q, Lin); see, e.g., [12, (2.29)].

The subadditivity and positive homogeneity of (cid:56) (asserted in Theorem 2.3) is then an instance of the general result [12, Proposition 2.37] asserting the same properties of the functional µ (cid:55)→ (cid:51)(µ).

Indeed, if Mi ∈ Lin and ui ∈ B (Mi ), i = 1, 2, then u1 + u2 ∈ B (M1 + M2) and therefore (cid:56)(M1 + M2) ≤ (cid:51)(Ds(u1 + u2)) = (cid:51)(Dsu1 + Dsu2) ≤ (cid:51)(Dsu1) + (cid:51)(Dsu2); taking the infimum over all u1 ∈ B (M1) and u2 ∈ B (M2) gives (cid:56)(M1 + M2) ≤ (cid:56)(M1) + (cid:56)(M2).

The positive homogeneity follows similarly.

We note that the interfacial energies in Examples 2.5 and 2.6 have the form (31), but this is not the case generally.


The following elementary result records some formulas to be employed below.


Remark. Let (cid:127) ⊂ Rn be an open bounded set with lipschitzian boundary.

A countable family (cid:127)α, α ∈ N, of pairwise disjoint subsets of (cid:127) with lipschitzian boundaries is said to be a partition of (cid:127) if one can write (cid:127) = \bigcup_{α=1}^{∞} (cid:127)α to within a set of null Lebesgue measure.

Let us agree to say that ϕ ∈ L 1((cid:127), R) is piecewise constant if there exists a partition (cid:127)α such that ϕ is constant on each (cid:127)α.

If να is the outer normal to (cid:127)α and if aα is the value of ϕ on (cid:127)α, then ϕ ∈ BV((cid:127), R) if and only if \sum_{(α,β)∈I} \int_{∂(cid:127)α∩∂(cid:127)β} |aα − aβ| dA < ∞, where I = {(α, β) ∈ N 2 : α < β, (∂(cid:127)α ∩ ∂(cid:127)β) > 0}.


If this is the case, we have the formulas J (ϕ) = \bigcup_{(α,β)∈I} (∂(cid:127)α ∩ ∂(cid:127)β), [[ϕ]]νϕ = (aα − aβ)νβ on ∂(cid:127)α ∩ ∂(cid:127)β for any (α, β) ∈ I, Dϕ = [[ϕ]]νϕ (cid:120) J (ϕ) A , to within changes on sets of null measure.

The total variation (mass) M(Dϕ) of Dϕ is equal to the sum in (32).


Proof.

Assume that (32) holds, and prove that ϕ ∈ BV((cid:127), R) and that the three formulas above hold.

We note that if (32) holds then µ := [[ϕ]]νϕ (cid:120) J (ϕ) is a (“finite”) measure in M((cid:127), Rn).

Let us prove that µ is the weak derivative of ϕ, which will also prove ϕ ∈ BV((cid:127), R).

Thus, we have to prove that \int_{(cid:127)} ϕ∇ f dV = −\int_{J (ϕ)} f [[ϕ]] dA for every class-infinity function f with support in (cid:127).

The application of the Gauss–Green theorem to each of the sets (cid:127)α provides \int_{(cid:127)α} ϕ∇ f dV ≡ aα \int_{(cid:127)α} ∇ f dV = aα \int_{∂(cid:127)α} f να dA. Summing these equations over all α and using that να = −νβ, one obtains (34) and hence we have ϕ ∈ BV((cid:127), R), (33) and all the remaining assertions of the remark.

The converse implication is proved by reversing the above arguments.


The function (cid:56)


The goal of this section is to prove that the functions defined in items (i)–(iv) of Theorem 2.3 coincide.

We denote these functions by (cid:56)1, (cid:56)2, (cid:56)3 and (cid:56)4, respectively, and prove that they are the same by establishing the cycle of relations (cid:56)1 ≥ (cid:56)2 ≥ (cid:56)3 ≥ (cid:56)4 = (cid:56)1.


Proposition. ( cid:56)1 ≥ (cid:56)2.


Proof.

It is easy to show that (cid:56)2 is a subadditive function.

Thus, the definition of (cid:56)1 gives the assertion.


The proof of the following lemma contains a construction of the central minimizing sequence uk ∈ C (M) for Theorem 2.3(iii).

This sequence will be defined as the superposition of (a slight modification of) the sequence of step deformations sk, k = 1, . . . , defined on Q by sk(x) = k−1a(cid:104)kx · b(cid:105), x ∈ Q. Clearly, ∇sk = 0, and in view of (27), sk(x) → a(x · b) on Q as k → ∞. Thus, sk satisfies the boundary condition sk = wa⊗b on ∂ Q in the asymptotic sense; however, the definition of C (a ⊗ b) requires the exact form of that boundary condition.

For this reason, we have to slightly modify sk near the boundary ∂ Q without violating the equation ∇sk = 0.


Lemma.

If M ∈ Lin and (ai , bi ) ∈ Dn, i = 1, . .

. , m, satisfy M = \sum_{i=1}^{m} ai ⊗ bi , then there exists a sequence uk ∈ C (M), k = 1, . . . , such that lim sup_{k→∞} (cid:57)(Dsuk) ≤ \sum_{i=1}^{m} ψ(ai , bi ).


We refer to Remark 5.3 for a mild condition on the sequence (ai , bi ) that guarantees that the lim sup in (36) strengthens to lim and the inequality sign to the equality sign.


Proof.

We shall first construct the sequence uk for the particular case when M = a ⊗ b is a dyad and then superimpose the sequences corresponding to the dyads ai ⊗bi , i = 1, . . . , m, to obtain the general case.

Thus, let (a, b) ∈ Dn and construct a sequence uk ∈ C (a ⊗ b), k = 1, . . . , such that lim_{k→∞} (cid:57)(Dsuk) = ψ(a, b).


Introduce the sets Ck = (1 − k−2)Q, Ll = (1 − (l + 1)−2)Q \ (1 − l−2)Q, k, l ∈ N, and observe that Q = Ck ∪ \bigcup_{l=k}^{∞} Ll with mutually disjoint summands for any k ∈ N. Here the product t S of a set S ⊂ Rn and a real number t is defined by t S = {t x : x ∈ S}. Equation (39) presents a decomposition of Q into the main set Ck, which is a large subset of Q for large k, while L k, L k+1, . . .

present infinitely many rectangular layers filling the gap Q \ Ck and becoming more and more refined towards the boundary of Q.


We use these sets Ck, L k, L k+1, . . .

to define a sequence of scalar functions ϕk : Q → R, k = 2, . . . , by ϕk(x) = (k − 1)−2(cid:104)(k − 1)2x · b(cid:105) if x ∈ Ck, l−2(cid:104)l2x · b(cid:105) if x ∈ Ll for some l ≥ k.


Let us use Remark 4.3 to prove that ϕk ∈ BV(Q, R).

Clearly, ϕk is a piecewise constant function in the sense of that remark.

Using (25a), one finds that x · b − 1 ≤ ϕk(x) ≤ x · b; hence, |ϕk| is bounded on Q and thus ϕk ∈ L 1(Q, R).

It remains to verify (32).


Let us show that in the present case (32) reads \int_{J (ϕk )} |[[ϕk]]| dA < ∞, where J (ϕk) = C ◦ k ∪ \bigcup_{l=k}^{∞} (L ◦ l ∪ L ∂ l ) is the jump set, with C ◦ k = {x ∈ Ck : k2x · b ∈ Z}, L ◦ l = {x ∈ Ll : l2x · b ∈ Z}, L ∂ l = (1 − l−2)∂ Q, and on J (ϕk) [[ϕk]]νϕk = (k − 1)−2b on C ◦ k, l−2b on L ◦ l where l ≥ k, ηlνk on L ∂ l where l ≥ k is the jump and normal to the jump set, with ηl(x) = l−2(cid:104)l2x · b(cid:105) − (l − 1)−2(cid:104}(l − 1)2x · b(cid:105) and with νk denoting the outer normal to the scaled cube (1−k−2)Q.


Equations (42) and (45) follow from the identities given in Remark 4.3.

One has to enumerate the regions of constancy of ϕk in an arbitrary way to obtain the system of sets (cid:127)α, α = 1, . . . , and use the formulas of that remark.

The details are left to the reader.

This establishes the equivalence of the inequalities (32) and (41).


To prove that (41) really holds, one finds from (45) that \int_{J (uk )} |[[ϕk]]| dA = (k − 1)−2 (C ◦ k A ) + \sum_{l=k}^{∞} l−2 (L ◦ l A ) + \sum_{l=k}^{∞} \int_{L ∂ l } |ηl| dA.


We estimate the terms (C ◦ k A ), (L ◦ l A ) and \int_{L ∂ l } |ηl| dA as follows.

First, prove that |(C ◦ k A ) − (k − 1)2 n(Ck)| ≤ 2n, |(L ◦ l A ) − l2 n(Ll)| ≤ 4n and hence (C ◦ k A ) ≤ 2n + (k − 1)2 n(Ck), (L ◦ l A ) ≤ 4n + l2 n(Ll).


Let us prove (48b); the proof of (48a) is similar.

Let ω : Ll → R be defined by ω(x) = l2x · b − (cid:104)l2x · b(cid:105), x ∈ Ll.

Then ω ∈ BV(Ll, R), Dω = l2b − b A (cid:120) L ◦ l , and hence the Gauss–Green theorem yields Dω(Ll) = l2 n(Ll)b − b (L ◦ l A ) = \int_{∂ Ll} ωνLl dA, from which |m n(Ll) − (L ◦ l A )| ≤ \int_{∂ Ll} |ω| dA.


We now observe that |ω| ≤ 1 on ∂ Ll and ∂ Ll = L ∂ l+1 ∪ L ∂ l .

Thus, \int_{∂ Ll} |ω| dA ≤ (L ∂ l+1 A ) + (L ∂ l A ) ≤ 4n since (L ∂ l+1 A ) ≤ 2n and (L ∂ l A ) ≤ 2n.

Thus, we have (48b).

Next prove that |ηl(x)| ≤ 2(l − 1)−2 on L ∂ l .

Indeed, writing |ηl(x)| = |(l−2(cid:104}l2x · b(cid:105) − x · b) − ((l − 1)−2(cid:104}(l − 1)2x · b(cid:105) − x · b)|, using the triangle inequality and the inequality (26) twice, with t = x · b and k = l2 and k = (l − 1)2, one obtains |ηl(x)| ≤ l−2 + (l − 1)−2 ≤ 2(l − 1)−2 and hence \int_{L ∂ l } |ηl| dA ≤ 2(l − 1)−2 (L ∂ l A ) ≤ 4n(l − 1)−2.


The estimates (49) and (51) and the formula (47) provide \int_{J (uk )} |[[ϕk]]| dA ≤ 2n(k − 1)−2 + n(Ck) + \sum_{l=k}^{∞} (4nl−2 + n(Ll)) + \sum_{l=k}^{∞} 4n(l − 1)−2 ≤ 1 + 2n(k − 1)−2 + 8n \sum_{l=k}^{∞} (l − 1)−2 < ∞, where we have used n(Ck) + \sum_{l=k}^{∞} n(Ll) = n(Q) = 1.


Thus, we have (41); hence, ϕk ∈ BV((cid:127), R) for every k and Dϕk = [[ϕk]]νϕk A (cid:120) J (ϕk) and ∇ϕk = 0.


Finally, note that the boundary trace ϕ∂ k of ϕk on ∂ Q satisfies ϕ∂ k (x) = x · b for every x ∈ ∂ Q.


While a rigorous proof of this can be given by using the essential limit of ϕk at x ∈ Q, we here only note that the definition of ϕk yields that lim_{j→∞} ϕk(x j ) = x · b for any x ∈ ∂ Q and any sequence x j ∈ Q converging to x. For this it suffices to note that in view of (39) one finds that x j must belong to some Ll for some l = l( j) ≥ k. The limit x j → x then implies that l(k) → ∞, and then the definition (40) and the formula (27) provide (54).


We define the sequence uk : Q → Rn, k = 2, . . . , by uk(x) = aϕk(x) for every x ∈ Q. By ϕk ∈ SBV(Q, R) and by (52) and (53), we have uk ∈ C (a ⊗ b).

Further, [[uk]] = [[ϕk]]a ⊗ νϕk ; consequently, by (45), ψ([[uk]], νuk ) = (k − 1)−2ψ(a, b) on C ◦ k, l−2ψ(a, b) on L ◦ l for any l ≥ k, ψ(ηla, νl) on L ∂ l for any l ≥ k and hence \int_{J (uk )} ψ([[uk]], νuk ) dA = (k − 1)−2ψ(a, b) (C ◦ k A ) + ρk, where ρk = \sum_{l=k}^{∞} l−2ψ(a, b) (L ◦ l A ) + \sum_{l=k}^{∞} \int_{L ∂ l } ψ(ηla, νl) dA.


Dividing (48a) by (k − 1)2, we obtain (k − 1)−2 (C ◦ k A ) → 1 as k → ∞. Using (6), we obtain that the nonnegative number ρk is bounded by (a constant multiple of) the quantity dk = \sum_{l=k}^{∞} l−2 (L ◦ l A ) + \sum_{l=k}^{∞} \int_{L ∂ l } |ηl| dA ≤ \sum_{l=k}^{∞} n(Ll) + 2n(k − 1)−2 + 4n \sum_{l=k}^{∞} (l − 1)−2 ≤ k−2 + 2n(k − 1)−2 + 4n \sum_{l=k}^{∞} (l − 1)−2 and hence ρk → 0.

Equations (55) and (56) then yield (37).


We now complete the proof in the general case.

By the preceding part of the proof, for each i ∈ {1, . .

. , m}, there exists a sequence ui k ∈ C (ai ⊗ bi , 0), k = 1, . . . , such that lim_{k→∞} (cid:57)(Dsui k ) = ψ(ai , bi ).

Define uk := \sum_{i=1}^{m} ui k for every k. By (28), (cid:57)(Dsuk) ≤ \sum_{i=1}^{m} (cid:57)(Dsui k ).

lim sup_{k→∞} (cid:57)(Dsuk) ≤ lim_{k→∞} \sum_{i=1}^{m} (cid:57)(Dsui k ) = \sum_{i=1}^{m} ψ(ai , bi ) by (57).


Hence, taking the infimum over all sequences ai and bi , one obtains from the definition of (cid:56)2 the inequality (cid:56)3(M) ≤ (cid:56)2(M).

The inequality (cid:56)3 ≥ (cid:56)4 is immediate.


5.5. Proposition. ( cid:56)4 = (cid:56)1.


Proof.

We seek to prove that (cid:56)4 is the biggest subadditive function satisfying (cid:56)4(a ⊗ b) ≤ ψ(a, b) for any (a, b) ∈ Dn.

To prove the subadditivity of (cid:56)4, let A, B ∈ Lin and u ∈ B (A) and v ∈ B (B).

Proposition 4.1 and (17) yield u + v ∈ B (A + B) and (cid:56)4(A + B) ≤ (cid:57)(Dsu + Dsv) ≤ (cid:57)(Dsu) + (cid:57)(Dsv).

Taking the infimum over all u and v then gives the subadditivity (cid:56)4(A + B) ≤ (cid:56)4(A) + (cid:56)4(B).


Next we note that the biggest subadditive function (cid:50) such that (cid:50)(a ⊗ b) ≤ ψ(a, b) for any (a, b) ∈ Dn is automatically positively homogeneous; thus, it suffices to prove the maximality of (cid:56)4 among all subadditive and positively homogeneous functions satisfying (60).

Thus, let (cid:50) be such a function and let M ∈ Lin and u ∈ B (M).

Then by (60) and by Jensen’s inequality for positively homogeneous subadditive functions, (cid:57)(Dsu) := \int_{J (u)} ψ([[u]], νu) dA ≥ \int_{J (u)} (cid:50)([[u]] ⊗ νu) dA ≥ (cid:50)(\int_{J (u)} [[u]] ⊗ νu dA) = (cid:50)(M).


Taking the infimum over all u ∈ B (M), we obtain (cid:56)4(M) ≥ (cid:50)(M).


This proves (cid:56)1 = (cid:56)2 = (cid:56)3 = (cid:56)4.

We define the function (cid:56) by (cid:56) = (cid:56)1.


Completion of the proof of Theorem 2.3


For this section, we put, for every (a, b) ∈ Dn, C(a, b) := {u ∈ SBV(Qb) : u = za,b on ∂ Qb, ∇u = 0 on Qb} and denote by (cid:56)5(a, b) the infimum in (19).

We then extend (cid:56)5 to Rn × Rn by homogeneity in the second variable.


Proposition.

We have H (A, B) = (cid:56)(A − B) for every A, B ∈ Lin.


Proof.

We employ Theorem 2.2 and the definition of (cid:56) in (18).

Invoking (11), we take any u ∈ SBV(Q) satisfying u = wA on ∂ Q, and \int_{Q} ∇u dV = B. Then v, given by v(x) = u(x) − Bx, x ∈ Q, satisfies v ∈ B (A − B) and (cid:57)(Dsu) = (cid:57)(Dsv).


Lemma.

We have (cid:56)5(a, b) ≤ (cid:56)(a ⊗ b) for every (a, b) ∈ Dn.


Proof. Let (a, b) ∈ Dn, and let (ai , bi ) ∈ Dn, i = 1, . . . , m, be a sequence satisfying a ⊗ b = \sum_{i=1}^{m} ai ⊗ bi .

Our goal is to construct a sequence uk ∈ C(a, b), k = 1, . . . , such that lim sup_{k→∞} \int_{J (uk )} ψ([[uk]], νuk ) dA ≤ \sum_{i=1}^{m} ψ(ai , bi ).


To define uk, let P = {x ∈ Rn : x · b = 0} be the plane through the origin perpendicular to b, let (cid:53) be the projection from Rn onto P, let F = P ∩ Qb, and put Bk = {x ∈ Rn : (cid:53)(x) ∈ (1 − k−1)F, 0 ≤ x · b < k−1} for any k ∈ N. Define uk : Qb → Rn by uk(x) = vk(x) if x ∈ Bk, za,b(x) else, x ∈ Qb, where vk(x) = \sum_{i=1}^{m} k−1ai (cid:104}k2x · bi (cid:105) for any x ∈ Rn and k ∈ N.


Employing Remark 4.2, we see that uk ∈ SBV(Qb); furthermore, clearly, uk = za,b on ∂ Qb and ∇uk = 0 on Qb; hence, uk ∈ C(a, b).


We proceed to prove (63).

We have J (uk) = Nk ∪ Mk ∪ L k ∪ Sk, where Nk = F \ (1 − k−1)F, Mk = {x ∈ ∂ B : 0 < x · b < k−1}, Sk = {x ∈ Rn : (cid:53)(x) ∈ (1 − k−1)F, x · b = k−1}, L k = \bigcup_{i=1}^{m} Li k where Li k = {x ∈ Bk : k2x · bi ∈ Z}.


The jump of uk and the normal to the jump set are [[uk]](x)νuk (x) = k−1 \sum_{i=1}^{m} ai ⊗ bi 1Li k (x) if x ∈ L k, a ⊗ b if x ∈ Nk, (a − vk(x)) ⊗ νk if x ∈ Mk, (a − vk(x)) ⊗ b if x ∈ Sk, x ∈ J (uk), where νk is the outer normal to Bk and 1Li k is the characteristic function of the set Li k. Hence, the subadditivity of ψ in the first variable yields \int_{L k} ψ([[uk]], νuk ) dA ≤ k−1 \sum_{i=1}^{m} ψ(ai , bi ) (Li k A ); consequently \int_{J (uk )} ψ([[uk]], νuk ) dA ≤ k−1 \sum_{i=1}^{m} ψ(ai , bi ) (Li k A ) + \int_{Mk} ψ(a − vk(x)), νk) dA + \int_{Sk} ψ(a − vk(x)), b) dA.


Let us now analyze the terms on the right-hand side of (67).

Using the considerations as in the proof of Lemma 5.2 (see (48) and (49)), one finds that k−1 (Li k A ) → 1 as k → ∞ for every i = 1, . . . , m. Thus, k−1 \sum_{i=1}^{m} ψ(ai , bi ) (Li k A ) → \sum_{i=1}^{m} ψ(ai , bi ).


Further, ψ(a, b) (Nk) → 0 since, obviously, (Nk) → 0.


Next note that, by (62) and (26), |ka(x · b) − vk(x)| = |ka(x · b) − \sum_{i=1}^{m} k−1ai (cid:104}k2x · bi (cid:105)| = |\sum_{i=1}^{m} ai (x · bi ) − k−2ai (cid:104}k2x · bi (cid:105)| ≤ \sum_{i=1}^{m} |ai ||(x · bi ) − k−2(cid:104}k2x · bi (cid:105)| ≤ k−1 \sum_{i=1}^{m} |ai |.


Then if x ∈ Mk, |a − vk(x)| ≤ |a − ka(x · b)| + |ka(x · b) − vk(x)| ≤ |a| + k|a||x · b| + k−1 \sum_{i=1}^{m} |ai | ≤ |a| + |a| + k−1 \sum_{i=1}^{m} |ai | since k|x · b| ≤ 1 on Mk.

Thus, |a − vk(x)| ≤ c < ∞ for any x ∈ Mk and any k = 1, . . . .

A combination with (6) and (Mk) → 0 then provides \int_{Mk} ψ(a − vk(x)), νk) dA → 0.


Similarly, if x ∈ Sk, then kx · b = 1 and hence |a − vk(x)| ≤ |ka(x · b) − vk(x)| ≤ k−1 \sum_{i=1}^{m} |ai | → 0.

Thus, (6) yields \int_{Sk} ψ(a − vk(x)), b) dA → 0 since (Sk) ≤ 1 for all k. Consequently, a combination of (67) with (68)–(71) provides (63) and hence the definition of (cid:56)5 gives (cid:56)5(a, b) ≤ \sum_{i=1}^{m} ψ(ai , bi ) for any sequence (ai , bi ) satisfying (62).

Taking the infimum of the right-hand side over all such sequences and using the definition of (cid:56)2 ≡ (cid:56), we obtain the assertion.


Lemma.

We have (cid:56)5(a, b) ≥ (cid:56)(a ⊗ b) for every (a, b) ∈ Dn.


Proof. Let u ∈ C(a, b).

Then, by Jensen’s inequality, \int_{J (u)} ψ([[u]], νu) dA ≥ \int_{J (u)} (cid:56)([[u]] ⊗ νu) dA ≥ (cid:56)(\int_{J (u)} [[u]] ⊗ νu dA) = (cid:56)(a ⊗ b) since the boundary condition u = za,b on ∂ Qb implies \int_{J (u)} [[u]] ⊗ νu dA = a ⊗ b. That is, we have \int_{J (u)} ψ([[u]], νu) dA ≥ (cid:56)(a ⊗ b) for every u ∈ C(a, b).

Taking the infimum, we obtain (cid:56)5(a, b) ≥ (cid:56)(a ⊗ b).


6.4. Proposition.

We have h(a, b) = (cid:56)(a ⊗ b) for every (a, b) ∈ Dn.


Proof.

This follows immediately from (12) and (19).


This completes the proof of Theorem 2.3.


Derivation of the examples


Derivation of Example 2.5 and (21).

We consider ψ| · |(a, b) = |a · b| first and prove (21a).

Clearly, the function (cid:50)(M) = |tr M| is a subadditive function satisfying (14) with ψ = ψ| · | and hence (16) gives (cid:56)| · |(M) ≥ |tr M| for any M ∈ Lin.

To prove the opposite inequality, we note that the definition (15) of (cid:56)| · | gives ψ| · |(a, b) = (cid:50)(a ⊗ b) ≤ (cid:56)| · |(a ⊗ b) ≤ ψ| · |(a, b) for every (a, b) ∈ Dn and hence (cid:56)| · |(a ⊗ b) = |a · b| and in particular (cid:56)| · |(a ⊗ b) = 0 if a · b = 0, which determines (cid:56)| · | on tensor products a ⊗ b. As a consequence, if N ∈ Lin can be written as N = \sum_{i=1}^{m} ai ⊗ bi where (ai , bi ) ∈ Rn × Rn, i = 1, . . . , m, where ai · bi = 0 for all i = 1, . . . , m, then (cid:56)| · |(N ) = 0 since 0 ≤ (cid:56)| · |(N ) ≤ \sum_{i=1}^{m} (cid:56)| · |(ai ⊗ bi ) ≤ \sum_{i=1}^{m} ψ(ai , bi ) = \sum_{i=1}^{m} |ai · bi | = 0.


To determine (cid:56)| · | on a general M ∈ Lin, we write M = A + W where A and W are the symmetric and skew parts of M. Let e1, . . . , en be an orthonormal basis of eigenvectors of A with the eigenvalues λi ; hence, A = \sum_{i=1}^{n} λi ei ⊗ ei .

Then M = B + N where B = (tr M)e1 ⊗ e1, N = W + \sum_{i=2}^{n} λi (ei ⊗ e1 − e1 ⊗ ei − (e1 + ei ) ⊗ (e1 − ei )).

Since W is a linear combination of the dyads ei ⊗ e j , 1 ≤ i (cid:54)= j ≤ n, one sees that N is of the form (72)–(73) and hence (cid:56)| · |(N ) = 0; consequently (cid:56)| · |(M) ≤ (cid:56)| · |(B) + (cid:56)| · |(N ) = (cid:56)| · |(B) = ψ((tr M)e1, e1) = |tr M|.


Equations 13 complete the proof of (21a).


To prove the two equations in (21b), we employ (21a) and (21b) as follows.

One has ψ±(a, b) = 1/2 (|a · b| ± a · b), and hence, if (ai , bi ) ∈ Dn and M ∈ Lin satisfy \sum_{i=1}^{m} ai ⊗ bi = M, then \sum_{i=1}^{m} ψ±(ai , bi ) = 1/2 (\sum_{i=1}^{m} ψ| · |(ai , bi ) ± tr M).


Taking the infimum as in (16) and using the above evaluation of (cid:56)| · | gives (cid:56)±(M) = 1/2 ((cid:56)| · |(M) ± tr M) = 1/2 (|tr M| ± tr M) = {tr M}±, which is (21b).


Derivation of Example 2.6 and (23).

The function (cid:50)(M) = |M T p| is a subadditive function satisfying (14), and we obtain in the same way as in the proof of Example 2.5 that (cid:56)(M) ≥ |M T p| for any M ∈ Lin and (cid:56)(a ⊗ b) = |a · p| and in particular (cid:56)(a ⊗ b) = 0 if a · p = 0.


To prove (cid:56)(M) ≤ |M T p|, we assume without loss of generality that | p| = 1 and let { p, e2, . . .

en} be any orthonormal basis.

In view of 1 = p ⊗ p + \sum_{i=2}^{n} ei ⊗ ei , M = 1M = p ⊗ M T p + \sum_{i=2}^{n} ei ⊗ M Tei ; normalizing the second members of the dyads, we obtain M = |M T p| p ⊗ sgn(M T p) + \sum_{i=2}^{n} |M Tei |ei ⊗ sgn(M Tei ).

The subadditivity of (cid:56) provides (cid:56)(M) ≤ (cid:56)(|M T p| p ⊗ sgn(M T p)) + \sum_{i=2}^{n} (cid:56)(|M Tei |ei ⊗ sgn(M Tei )) = |M T p| by (74).

Thus, (cid:56)(M) ≤ |M T p| and the proof of (23) is complete.


Acknowledgments


I thank David Owen for a detailed discussion on the previous version of this work.

His stimulating questions contributed to the final version of the results presented here.

Also, David Owen and the two anonymous reviewers pointed out to me that it is possible to replace my original coercivity assumption on ψ in Assumption 2.1(ii) by mere nonnegativity, which is necessary to cover Examples 2.5 and 2.6 and which is possible by [3, Remark 3.3].

I am also thankful for the support of the institutional research plan RVO 67985840.


An invited commentary on this article appears on page 1887, and the authors' response appears on page 1890.


Prenatal maternal smoking is an established risk factor for lower birth weight and reduced fetal growth, and it is associated with increased risk of major childhood diseases such as asthma, obesity, and cancers (1-3).

A complex chemical mixture of over 7,000 compounds make up cigarette smoke, including known reproductive toxicants such as carbon monoxide, arsenic, lead, cadmium, and nicotine.

Although prenatal maternal cigarette smoke is a well-recognized risk factor for reduced fetal growth, mechanisms are only partially understood and might include both direct effects on the fetus and morphological and func- tional changes to the placenta (4).


DNA methylation (DNAm), occurring mostly at cytosine- phosphate-guanine dinucleotides (CpG sites), can involve exten- sive erasure, rewriting, and reprogramming during embryogene- sis and cellular differentiation (5).

This reprogramming phase might pose a susceptibility window for early nutritional and environmental cues to program future health trajectories of newborns by altering cellular structure, lineage commitment, physiology, and metabolic functions, a hypothesis known as fetal programming (6).

For instance, prenatal smoking has been consistently associated with altered cord blood DNAm of several genes (7-10).

Previous studies have shown that cord blood DNAm variability associated with prenatal 1878 Am J Epidemiol.


Downloaded from https://academic.oup.com/aje/article/188/11/1878/5566169 by Stanford University user on 03 October 2025 smoking partially mediates the association of smoking and lower birth weight (11, 12).

While multiple studies have looked at maternal smoking and DNAm in cord blood, few have investigated placental DNAm variation, and even fewer have used an agnostic epigenome-wide approach (13-16).


Methylation of DNA is vastly cell-type specific, and early environmental influences on the epigenome are likely tissue- and cell-type specific (17).

Thus, investigating the role of target tissue in epigenetic studies is critical (18).

The placenta is the master regulator of the fetal environment, with active endocrine and metabolic functions, synthesizing hormones and regulating nutrient and waste exchange (19).

Nicotine, a parasympathomi- metic stimulant and primary constituent of tobacco smoke, acts as a vasoconstrictor reducing intervillous placental blood flow, increases hypoxia, and interferes with placental development (20, 21).

Additionally, many other reproductive toxicants present in cigarette smoke cross the placenta, exposing the fetus, and some accumulate within placental structures (22).

Consequently, the placenta might be particularly susceptible to prenatal tobacco smoke and a target tissue to mediate asso- ciations with fetal growth.

In this study, we hypothesized that DNAm variability of the placenta associated with prena- tal maternal smoking early in pregnancy would mediate the established association of prenatal smoking and lower infant birth weight.