How Animals may help understand men.

Concerning the Soul of Brutes (1672) that comparative anatomy does not allow us to draw any distinction between outward physical appearance of men and that of beasts. Should we conclude from this apparent contradiction that the difference between men and animals is invisible and cannot be understood from an anatomical point of view?

In this chapter, I will show that a close reading the work of Thomas Willis does not justify this conclusion. I begin with a brief overview of Anatomy of the Brain (1664), illustrating how Willis presents anatomy as a subject that can demonstrate the superiority of man to other creatures. Then I turn to the Two Discourses (1672) and examine how W illis emphasizes the limitations of comparative anatomy. This will lead me to the "paradoxical doctrine of the corporeal soul" that Willis presents to his readership as a way to reconcile the ancients and moderns and to explain what animals and men have in common. In conclusion, I will highlight the epistemological and moral implications of Willis's original answer to the classic question of the anthropological difference between men and beasts.
Dissecting the "Living Chapel of the Deity" <TX>In the Anatomy of the Brain (1664), Willis asks the following question: how is it possible to observe, describe and understand the human mind? Recalling his mission when he accepted the Sedleian Chair of Natural Philosophy in Oxford in 1660, 2 he stresses the difficulty of his task: "unlock[ing] the secret places of Man's Mind," "look[ing] into the living and breathing chapel of the deity" seem fraught with risk. 3 In his "Epistle dedicatory to the Archbishop of Canterbury" and in his "Preface to the Reader," Willis seems to follow a trend that had become traditional since the Renaissance: he writes that describing the fabric of the human is the ideal way to celebrate the "prerogative" of man over all other living creatures and to pay tribute to the Divine Architect. In reality, however, Willis breaks with this tradition, violating the prohibition expressed, for example, by André Du Laurens in his Histoire anatomique (1600): according to Du Laurens, the soul is above all natural forms, distinct from all material things, indivisible and an image of the deity; consequently, its essence cannot be made visible by the anatomist, who must limit himself to the physical objects that he can perceive with the senses. 4 As Willis acknowledges in the "Preface to the Reader," "to explicate the uses of the Brain," to inquire into its nature, and to describe the internal movement of Animal Spirits, "seems as difficult as to paint the Soul"; and in the "Epistle Dedicatory," he describes the effort to do so as the best "School-house of Atheism."

These difficulties and objections, however, do not prevent Willis from viewing the spherical shape of the brain and the disposition of its various organs as signs or proof of man's preeminent place in the Creation: <EXT>The figure or the skull in four footed beasts is narrow and prest down, but in man, the substance of whose brain is large, there is required a more capacious and almost spherical figure. For as God gave him an upright countenance to behold the Heavens; therefore his face is erect or lifted up: so the Brain it self is placed in a more eminent place, to wit, above the Cerebel and all the sensories. But in Brutes, and such whose faces are prone towards the Earth, and have a brain unfit for Speculation, the Cerebel, however serving to the more noted action and office of the Praecordia, is placed in the highest seat to which the mole of the brain is subjected. 5 <TX>If we paid attention exclusively to quotations such as this, we might wonder whether Willis's position was original at all. Animals here are "unfit for speculation," and their brains are designed merely to maintain lower vital functions. On account of this position, as Willis remarks in the "Epistle Dedicatory," he had "slain so many Victims, whole Hecatombs almost of all Animals in the Anatomical Court." If Willis wants to gain a "perfect knowledge of the Brain and its parts," he must dissect the brains of "all sorts of living creatures" and especially those of "four-footed beasts," like "dogs, calves, sheep." As he explains in the first chapter of Anatomy of the Brain, where he presents the method of comparative anatomy, this is because, first, "humane Heads or Bodies are not so readily to be had." Second, animal brains are simpler to study than "the immense bulk of an humane Head" and its "most intricate frame and various recesses." 6 Referring to Genesis, Willis justifies the analogy between men and four-footed beasts with reference to the point that they were created on the same day, just after "fishes and fowls" the day before: hence "there is in either twin species a like form of the brain," and the difference between human and animal brains is only one of "magnitude." 7 Although the designation "anatomia comparata" had been used previously by Francis Bacon in his Advancement of Learning (1605), Willis was one of the first anatomists "who understood by this expression the comparison of specific parts in various species of brutes, and in man" [START_REF] Isler | Doctor and Scientist[END_REF] (and not merely the investigation of many individuals of one species). [START_REF] Aurelius | Zootomia democritaea (Nuremberg: Lietris Endterianis, 1645). On the use of comparative anatomy at this time, see Charles Webster, "The College of Physicians: Salomon's House in Commonwealth England[END_REF] This comparative method allowed him not only to study "animals in general" (as Descartes did [START_REF]For the similarity of structures between men and animals, see René Descartes, Description du corps humain[END_REF] ), but also to observe commonalities and differences between the parts of various animals compared among themselves and with man. Comparative anatomy seems to be the best method to address the objections to Willis's project. The field of medicine, which entails the anatomical investigation of the complexity of the human body and of the human brain, can demonstrate the superiority of man to all other creatures. 11
The Limits of Comparative Anatomy <TX>Let us begin with a quotation from the end of Chapter 7 of Two Discourses ("The corporeal soul, or that of the Brutes, is Compared with the Rational Soul"), where Willis refers to his former treatise on the brain: <EXT>As we have shewn, by comparing the Corporeal Soul of the Brute, with the Rational (Soul) of Man, what vast difference there is between them, perhaps it might be to the purpose, to compare the Brains of either, and to observe their differences. But this Anatomy being elsewhere made (Cerebri Anatome), we have noted little or no difference, in the Head of either, as to the Figures and Exterior Conformations of the Parts, the Bulk only excepted; that from hence we concluded, the Soul Common to Man with the Brutes, to be only Corporeal, and immediately to use these Organs. [START_REF] Willis | Two Discourses Concerning the Soules of Brutes, Which is that of the Vital and Sentitive of Man. The First is Physiological, shewing the Parts, Powers, and Affections of the same. The other is Pathological, which unfolds the Diseases which affect it and its Primary Seat, to wit the Brain and Nervous Stock, and treat of their Cures[END_REF] <TX>Reading this comparison of the "corporeal soul" of brutes and the "rational soul" of men, we might be tempted to see proof of "inconsistencies" in Willis's thought. In Anatomy of the Brain, Willis had already underlined the similarity of human brains and those of quadrupeds, such as dogs, sheep, and monkeys. Comparative anatomy may help us to reveal the difference between the "fabrick of the brain in fowls and fishes" (inferior animals), on the one hand, and the fabric of the brain in "four footed beasts" (superior animals), on the other. 13 But the limitations of this approach appear when Willis tries to draw distinctions between members of the category of superior animals, namely, between "four footed beasts," such as dogs, foxes, and monkeys, and men: "We have already hinted, that the Brains of Men and of four footed Beasts, were alike in most things." 14 Here Willis clearly departs from Descartes's argument in Passions of the Soul. According to Willis, the pineal glandula cannot be considered as the physiological seat of the soul and of superior functions, which belong exclusively to men: "This Glandula is found in all perfect Creatures; for Man, all four footed Beasts, yea Fowl and Fishes are provided or endued with it." [START_REF] Descartes | Animals, which seem to be almost quite destitute of Imagination, Memory, and other superior Powers of the Soul, have this glandula or kernel large and fair enough[END_REF] Paradoxically, advances in the field of comparative anatomy during the seventeenth century had made it even more difficult to draw a visible (anatomical) boundary between animals and men. In 1641, in the first edition of his Observationum Medicarum, the physician Nicolas Tulp published the first physical description of the great ape or sylvan man. Although a quadruped, the ape nonetheless resembled the human species and was called "orang utang" or "man of the forest." [START_REF] Tulp | Apud Ludovicum Elzevirium, 1641), ch. 56[END_REF] Later, in 1690, Edward Tyson gave further thought to the physiology of the orang-utan, describing its brain and nervous system and showing that it is closer to man than to other animals. Dedicating the text to Lord Sommers, president of the Royal Society, Tyson presented the orang-utan as "the Nexus of the Animal and Rational." [START_REF] Tyson | To the Right Honourable John Lord Sommers, Baron of Evesham[END_REF] The anatomy and description of the great apes enabled Tyson to understand the gradual transition from plants to animals, and from animals to men. It may have helped him to approach "nearest to that kind of Beings which is next above us" and to "connect the Visible and Invisible World." [START_REF]Epistle Dedicatory[END_REF] The progress of the field of comparative anatomy led to a reconsideration of man's place in the "Great Chain of Being." [START_REF] Lovejoy | The Great Chain of Being:, A Study of the History of an Idea[END_REF] In consequence, it had to be acknowledged that medicine and anatomy could not specify the difference between animals and men. The Anatomy of the Brain had revealed "little or no difference" between the head of men and that of animals, Willis writes at the end of Chapter 7 of the Discourses Concerning the Soul of Brutes. Does this statement contradict the point made in the Anatomy of the Brain, namely, that the spherical shape of the human brain and the complexity of its structure were proofs of the superiority of man to animals? I would like to focus on another passage in Part 2 of Anatomy of the Brain (The Description and Use of the Nerves), in which Willis explains why the anatomy of the brain cannot provide clear criteria for identifying what it is to be human. In Chapter 26 of this work, Willis refers to two dissection experiments: first, the dissection of "the Carcass of a man that was a Fool from his birth" and, second, the dissection of a "monkey": <EXT>When of late we had dissected the Carcass of a man that was a Fool from his birth, we could find no defect or fault in the Brain, unless that its substance or bulk was very Small. . . . . Whilst we were writing these, we made an Anatomy of a Monkey, whose Brain differed little from that which is seen in a Dog or a Fox, unless that it was much more capacious in the proportion to the bulk of his Body, and the turnings and windings of it were larger. 20 <TX>Here we find the same affirmation as before. The difference between a man and a fool and between a man and a monkey is one of degree, not nature. The example of the fool in particular is significant, because it leads us to question the legitimacy of reason as the defining characteristic of a human being. Indeed, in the case of the fool, we have a human body that possesses the organs that should have enabled him to speak and reason. We therefore cannot point to any meaningful anatomical difference between the brain of a fool and the brain of a man endowed with reason. [START_REF] Willis | On this question[END_REF] The reader would conclude from Willis's account of these experiments that the anatomy of the brain cannot give us an explanation of what makes humans rational beings-which, however, does not exclude the possibility that we might understand what makes the difference from another point of view.

In fact, and as Willis reminds us in the passage preceding the text quoted above, "both the ancient Divines and Philosophers placed wisdom in the Heart." 22 As William Bynum notes, "Willis wrote in the aftermath of Harvey's Aristotelian emphasis on the primacy of the heart." 23 Although the brain is beginning to rival the heart as the centre and origin of vital functions, Willis still attributes morals and wisdom to the heart. This is perhaps an allusion to Francis Bacon, who quotes Proverbs 4:23 in Book 2 of Advancement of Learning: "ante omnia, fili, custodi cor tuum; nam inde procedunt actiones vitae" ("Keep thy heart with all diligence, for thereout come the actions of thy life"). [START_REF]On this quotation of Bacon and the relationship between the heart and the brain, see Milad Doueihi, Histoire perverse du coeur humain[END_REF] If we want to understand the difference between a "man of judgement" and a fool, or between a human being capable of "vertue and prudence" and a monkey, we must inquire into the "reciprocal affections of the Heart and Brain"; in other words, we must inquire into the dependence of our intellectual faculties on appetite, desire, affections, and passions (the dependence of the rational soul on the sensitive one). [START_REF] Willis | Hence both the ancient Divines and Philosophers placed wisdom in the Heart. Certainly the Works of Prudence and Vertue depend very much on the mutual commerce which happens[END_REF] It is true that the anatomy of nerves, and more specifically the description of the "infolding of the intercostal Nerve, which [Willis] call[s] the Internuncius of the Brain and Heart," provides us with some elements that help us to understand "to what injuries of changes or Diseases . . . the Brain and nervous Stock may be obnoxious." 26 But ultimately, we must turn to another kind of "speculation." The "anatomy" of the brain and nerves must be complemented by a "pathology or (the) Doctrine of the Passions of the Soul" or by what Willis also calls a "psycheology [sic] of the soul." or "discourse of the soul." 27 Yet turning from the brain, traditionally considered as the seat of thought, to the heart, that is, the seat of affections and passions, [START_REF] Willis | Galenic tradition, the brain is the centre of perception and the use of the cognitive faculties. However, followers of Aristotle[END_REF] does not help us to understand what makes man superior to fools and animals. Instead, it helps us to grasp what they have in common. This in turn may enable us to better understand two things: first, why the monkey "is too crafty and mimical above other Beasts, and can so aptly shew and imitate not only the gestures but the passions and some manners of a man"; 29 and, second, why we cannot find any visible or anatomical difference between a man of judgement and a fool. But for this to occur, Willis must abandon all attempts to describe the fabric of the human body as "a certain kingly Palace" with its "outward Courts and Porches." He must enter the "intimate Recesses and private Chambers" 30 of this palace. Here he will be able to observe the "various Metamorphoses or Changes" of the corporeal soul and to understand how these metamorphoses produce "strange species," [START_REF] Willis | The] whole soul suffers (sometimes) various Metamorphoses or Changes, and puts in strange species, as it "often happens in Melancholy diseases[END_REF] Pardies. Pardies regarded animals as endowed with a lower form of intelligence that is particular to the sensitive soul. 32 Willis, however, adopts a different standpoint in this debate. He presents the "paradoxical doctrine of the corporeal soul" in his "Preface to the Reader," both as a way to gain access to what remains invisible to the eye and as a way to redefine man as a "two soul'd animal" [START_REF] Willis | The Preface to the Reader[END_REF] or, as Willis writes later in Chapter 7 of Part 1, as an "amphibious animal . . . of a middle Nature and Order, between Angels and Brutes," which communicates with brutes "by the corporeal Soul" and with Angels by the "intelligent, immaterial and immortal soul." 34 It is striking that Willis presents this doctrine as a way to reconcile ancients and moderns.

Although he mentions the doctrines of the Pythagoreans, who believed in metempsychosis and thought that "the Souls of all living creatures [were] a certain part of the Universal Soul of the World," or the doctrine of the "Manichees" and "Origenists," who "granted the like Immortality to the Souls of the Brutes, yea of Plants," [START_REF] Willis | Soul of Brutes, 2. On these doctrines see Richard Sorabji, Animal Minds and Human Morals. The Origins of the Western Debate[END_REF] he does not consider them influential in his own time.

Instead, he links the hypothesis of the corporeity of the soul to the revival of Epicureanism. [START_REF] Wilson | Epicureanism at the Origins of Modernity[END_REF] Defining the soul as "plainly corporeal" or "made of a knitting together of subtil atoms," in agreement with Laërtius, is "as much to say . . . that the Animal is as it were the Loom in which the Yarn is the Body, and the Wood the Soul," in agreement with Gassendi. The Epicurean hypothesis, according to Willis, is the "basis" upon which "philosophers of this latter Age" (including Descartes) "have built all their doctrines of the soul." [START_REF] Willis | Soul of Brutes, 2. 38 Willis, Soul of Brutes[END_REF] We should remember that Willis is referring here to the soul "by which the Brutes as well as Men live, feel (and) move." 38 If we understand the "soul" as the principle responsible for the primary functions of life, sensation, and motion, and if we consider the fact that "for Descartes the primary functions of life, sensation, and motion are brought about by tiny, invisible particles such as those proposed by Epicurus . . . then it is reasonable to say that the soul, for Descartes, is corporeal." [START_REF] Wright | Locke, Willis, and the Epicurean Soul[END_REF] But we should not forget that Descartes made a clear distinction between the corporeal soul, "qui ne dépend que de la seule force des esprits animaux et de la configuration des parties, et que l'on pourroit appeler âme corporelle," and the soul as a "mind" ("esprit") or "thinking substance."

Characterizing the soul as the principle responsible for vital functions confounds (intentionally) what Descartes had asked Henry More to distinguish: "Il faut pourtant remarquer que je parle de la pensée, non de la vie, ou du sentiment; car je n'ôte la vie à aucun animal, ne la faisant consister que dans la chaleur du coeur. Je ne leur refuse pas même le sentiment autant qu'il dépend des organes du corps." 40 According to Willis, ancient and modern philosophers agree that "the soul of brutes" is "corporeal and divisible." [START_REF] George | Descartes also attributes a corporeal soul to animals in the same letter to Morus, but for him it is a purely mechanical principle[END_REF] They agree about the corporeity or materiality of the soul of brutes, but disagree about its passivity or activity. Whereas Pereira, Descartes, and Digby affirm that this soul is "deprived of all knowledge, sense, and appetite," that "Beasts [want] all Knowledg [sic] or Perception" and "that every action of the Brute consisted in it, as it were an artificial Motion of a Mechanical Engine," others, "also renowned Philosophers, both Ancient and Modern, professing themselves no less adverse to Atheism than the former, challenge in the behalf of the Beasts, not only the operations of an external and internal Sense, with Perception, Appetite, and spontaneous motions, but besides, grant to them a certain use of Judgement, Deliberation, and Ratiocination." 42 From this second group of philosophers, Willis quotes the first chapter of Nemesius's treatise On the Nature of Man, where we read that God did not suddenly create rational animals, but rather "first endowed the other animals also with certain natural forms of understanding, devices and resources for their preservation, so that they appear near to the rational animals." 43 Willis also mentions Gassendi's Physics and his "late Experimental Philosophy," from which the enumeration of "very many Instances (of) the Cunning and Wonderful Sagacity of brute Animals" leads him to attribute to them "a certain kind of reason." [START_REF] Willis | in his late Experimental Philosophy, when he had enumerated very many Instances, by which the Cunning and Wonderful Sagacity of brute Animals were declared. . . . From these we may easily understand, what dignity, and beyond the powers of any Machine, causing its Efficacy, he affirms to be in[END_REF] According to Willis, Descartes is right to think that "brutes are like Machines framed with a more simple furniture . . . determined for the doing still the same thing," whereas "in Man divers series of motions, and as it were complications of wheels within wheels, appear." 45 But his endeavours to "discriminate the Souls of Beasts from the humane" 46 cause him to neglect the fact that every machine requires a large amount of energy to function. If the soul of brutes is made of the subtlest atoms, these are active and dynamic atoms. Willis distinguishes two kinds of atoms: one has a fiery nature, arises in the blood, and circulates in the body (the vital part of the soul); while Willis compares the other kind to light to explain the movement of animal spirits: this second kind of atom circulates like rays of light within the body (the sensitive part of the soul). 47 Like Descartes, Willis uses the mechanistic model, but he uses it to describe the "self moving energy" ("energeia") that is at work in all "natural things" and is responsible for vital and involuntary functions, such as blood circulation, respiration, generation, and reproduction. [START_REF] Willis | Soul of Brutes, 33. On this broad sense of mechanism[END_REF] The "souls of Beasts" possess a "dignity" that is "beyond the powers of any Machine" and "caus[es] its Efficacy." [START_REF] Willis | Soul of Brutes, 4. 50 Willis[END_REF] Indeed, in Chapter 6, Willis cites various examples traditionally used to demonstrate the sensibility and intelligence of animals: "Many admirable Histories are reported concerning the subtilities and craft of the Fox, which he is wont to perform for the getting of his living. This creature, that he might allure the Hens within the compass of this Chain, with which he was tyed, lying all along, his legs stretched forth, feigns as if he were Dead, then they coming near him, he readily leaps upon them." 50 Willis, however, does not use these stories as grounds for attributing reason to foxes-at least reason as a capacity of reflection or abstraction separate from sensible things and experience. He uses the description of animals and these classic stories to demonstrate what animals and men have in common rather than to distinguish men from animals. 51
Conclusion <TX>In conclusion, I would like to discuss some implications of Willis's innovative position in the debate on the soul of brutes. First, it allows him to demonstrate that we share with brutes a kind of practical or empirical knowledge that is essential to the preservation of the human species. Second, it allows him to reevaluate the status of reason as a defining characteristic of man and to observe how men sometimes are induced to imitate animals or even persuade themselves that they are animals.

(1) The difficulty presented in Chapter 6 ("The Science of Brutes") is the following: how can we explain the fact that brutes are able to perceive, discern, and distinguish between objects, to desire and hate (appetite), and remember their master and pain or pleasure felt in the past? 52 To untangle this "knot," we should not grant the same "cognition" to brutes as to men: "For otherwise, if Cognition be granted to the Brutes, you must yield to them also Conscience, yea and deliberation and Election, and a Knowledge of universal things, and lastly an incorporeal and rational soul." 53 Instead, we should acknowledge that there is an activity inherent in matter, a "self-moving energy" that enables atoms to "cut forth for themselves Pores and Passage" in living bodies and is responsible for vital and involuntary functions, such as reproduction, conservation, respiration, circulation, and even instinct. [START_REF] Willis | Willis expresses his preference for the renewal of atomism and for a mechanical explanation in natural philosophy. See "Of the principles of natural things," in Of Fermentation, 2. But like many early modern philosophers, he is "attracted by the use of particles as carriers of sympathetic, dynamic . . . and other expansive qualities[END_REF] In Chapter 16 of Anatomy of the Brain, Willis attributes all involuntary natural functions, such as the pulse, respiration, and chylification, to the regular actions, "being made in the Brain without previous knowledge, . . . done by Instinct merely natural." Consequently, we may conclude, first, that the same principle explains why man lives and is able to preserve himself, why brute animals "being newly brought forth" are able to feed themselves, and why birds "build nests with wonderful Art" without needing "any shewing for example." 55 This principle also explains why brutes, as well as men, try to "preserve [themselves] as long as [they] can." 56 Second, this natural instinct may be "polished by frequent use and habit, and carried a little further," giving rise to a kind of "cunning" and "acquired knowledge." 57 Here, Willis describes how "the acts of the senses" and repeated experiences enable brutes to acquire "practical habits" [START_REF] Willis | For Secondly, besides the natural Instincts, living Brutes are wont to be taught by sensible species, to wit, to profit in the Knowledge of several things, and to acquire certain habits of practice[END_REF] and to develop a kind of practical ability to reason, which does not imply any abstract theories or conscious thought, but can be explained entirely by the powers of the "sensitive soul." Willis suggests as the sources of this practical reasoning first "nature" and "chance," and then "senses and experience." He concludes, "according to this sort of analyzing, the most intricate actions of brutes . . . may be explained, and reduced into competent notions of the sensitive soul." This kind of practical reasoning can explain the fox's "ruse," pretending to be dead in front of the hens, and the capacity of the "Ape" to imitate Man: "Hence it is that the Ape so plainly imitates Man." 59 Here we begin to perceive the answer to the problem raised in Chapter 26 of Anatomy of the Brain: if we want to understand why the monkey "can so aptly shew and imitate, not only the gestures, but the passions and some manners of a man," we need to inquire into the sensitive powers shared by brutes and men. This inquiry can help us to understand that cognition does not always imply conscience or abstract knowledge, but rather that there is a kind of cognition that enables us to choose the correct behavior or help us to "preserve ourselves" as long as we can.

However, the demonstration of this kind of practical knowledge, linked to instinct and to the preservation of the body, also has moral implications.

In the following chapter (Chapter 7: "The Corporeal Soul, or That of the Brutes, is Compared with the Rational Soul" 60 ), it seems that Willis intends to establish "the eminency of the Rational Soul above the Brutal or Corporeal" by "comparing" their "objects" and "acts or modes of knowing." First, Willis declares that the intellect "performs the government of the whole man." This may be shown from an epistemological point of view by recalling the superiority of the rational soul to the corporeal soul. The powers of the rational soul ("intellect, judgement, discourse, and other acts of reason") enable us to grasp any "ens" or immaterial object, whereas the corporeal soul's principal mode of knowing (fantasy or imagination) limits men and animals to sensible objects. We should also recall, from a moral and theological point of view, the "Precepts of Philosophers," "Moral Institutes" and the laws and precepts of "Sacred Religion," and how they have vindicated the supremacy of the rational soul to the sensitive soul.

Here, the rational soul remains the faculty that can "carry Man, not only beyond the Brutes, but .

. . above his natural State." [START_REF] Willis | Soul of Brutes, 43[END_REF] But we must not forget that man is "made, as it were, an Amphibious Animal, or of a middle Nature and Order," that he does not communicate merely with "Angels," but also with "Brutes." Willis must also show that this "Divine Politie . . . is not erected in Man without great contention." He must describe the "Dissentions of one Soul with another" and remember how the "corporeal soul, . . . being nearer to the Body, and so bearing a more intimate Kindness or Affinity towards the Flesh, is tied wholly to look to its Profit and Conservation." This leads him to conclude that "the Corporeal Soul does not so easily obey the Rational in all things, not in things to be desired, as in things to be known." This rebellion of the sensitive against the rational soul might prove to be morally reprehensible: it may induce man to "become like the Beast, or rather worse . . . , for as much as Reason becoming Brutal, leads to all manner of excess"; and, losing all "force and ardor of the mind," men become "little better than dull working beasts in fortitude and wisdom." 62 But it must also be understood, from a naturalist point of view, as a manifestation of a vital instinct of preservation: "From hence the Wars and Strivings between our two Appetites, or between the Flesh and Spirit, both Morally and Theologically inculcated to us, are also Physically understood." 63 (2) This leads me to the second point, the medical perspective on the human mind and on what man and animals have in common, which might help us to reconsider (against Descartes) the status of reason as a defining characteristic of human beings. Observing mental disorders is crucial to understanding how the rational soul depends on the powers and operations of the sensitive soul. "Ratiocination or Reasoning, howsoever strong" may diminish or even be "taken away" when the movement of the spirits within the brain is "hindred" and the faculty of fantasy is affected. 64 It is precisely this kind of "obstruction of the Spirits" that explains certain "eclipses" of our "faculties" in pathological states such as lethargy, stupidity, and foolishness: "hence it comes sometimes that the whole soul suffers various Metamorphoses or Changes, and puts in strange species, as often happens in Melancholy diseases, or to mad men." [START_REF] Willis | Melancholy and the Care of the Soul. Religion, Moral Philosophy and Madness in Early Modern England[END_REF] From his Oxford lectures (attended by John Locke) to the Pathology of the Brain and Nervous Stock (Pathologiae Cerebri et Nervosi Generis Specimen, London, 1667) and the second part of the Two Discourses (London, 1672), Willis devoted several pages to describing these mental conditions. Already in his Oxford Lectures, Willis mentions the "metamorphoses" of what he calls a corporeal soul or "animus sensitivus": "a contexture of animal spirits and . . . a sort of aetherial man made up of the most subtle atoms being coextensive to our body." [START_REF] Willis | Melancholy and the Therapeutic Language of Moral Philosophy in Seventeenth Century Thought[END_REF] Jackie Pigeaud has commented on what is at stake in this definition of the sensitive soul and how it gives Willis access to a doublet (or counterpart) of the physical man dissected by the anatomist: an interior man that is hidden from observation by the senses, 67 but rather must be described by a physician of the soul. As Jeremy Schmidt stresses, the "animus sensitivus was to become Willis's answer in De Anima brutorum to the Cartesian assertion that animals were, like machines, devoid of sense, feeling, and voluntary motion because they lacked an immaterial soul. Not only could a certain kind of matter in the form of the animus sensitivus think; according to Willis this material soul explained certain features of human thought as well." [START_REF] As | Melancholy and the Therapeutic Language of Moral Philosophy in Seventeenth Century Thought[END_REF] Moreover, observation of this "metamorphosis delirium" allows us to realize that the difference between man and animals may sometimes disappear. Indeed, men sometimes lose the ability to recognize other men or the ability to speak, behaving like brutes, as in the case of the "lethargist" Oxford gardener described in Chapter 3, Part 2 of Two Discourses Concerning the Soul of Brutes. 69 In cases of melancholy, and especially what Willis calls "occasional melancholy"-an emotional state of mind that no human being can ever wholly escape, and which can transport us from a joyful and merry state to one of temporary sadness, whereas the term "habitual" is used in physics to designate a severe form of mental pathology-some men might persuade themselves that they have lost their human shape and have transformed into a glass or a wild animal, such as a wolf in the case of lycanthropy. 70 These observations might be the starting point of a search for a different criterion that could help us to define what it is to be human: should we pay attention only to visible forms and anatomical structures? Yet we have already seen that comparative anatomy cannot answer the question of what it is to be human. Should we conclude that this distinguishing characteristic, being invisible (language, reason, the immortal soul), eludes the hands and sensible tools of the anatomist and must be addressed from a metaphysical point of view? Willis offers us a third option here. Animals are not "only certain Machines wonderful made by a Divine Workmanship," which could "without any Knowledg [sic], Sense, or Appetite, perform only Corporeal Motions." 71 We have much more in common with brutes than we might think. There is much to be learnt about our close relationship to animals from observing and describing the metamorphoses of the corporeal soul: a soul that we share with animals, that sometimes causes us to lose our ability to speak or recognize ourselves and others as human beings, a soul that sometimes leads us to imitate animal behavior and behave like brutes. Théophile Gelée (Rouen: Raphael du Petit Val, 1621), 1-2. 5 Willis, Anatomy of the Brain, 64. Willis refers here to Plato, Timaeus, 45a. See also Lo Presti's contribution in this volume. 6 Willis, Anatomy of the Brain, 56: "That the perfect knowledge of the Brain and its parts may be gained, it is necessary not only to dissect and look into mens [I assume no apostrophe is correct? yes] Heads, but all other kind of other Creatures heads: for besides, that the humane Heads or Bodies are not so readily to be had, that one may from day to day, behold the Brain and its frame, and carefully consider the situation of the parts, and search one after another their respects, habitudes and dependencies; besides also, the immense bulk of an humane Head is in itself an hindrance, whereby its most intricate frame and various recess[either put [sic] or fix?] and Appendices are the less accurately discerned and investigated: all which being reduced into an Epitomy, are plainly represented more commodiously in the dissection of Beasts." 7 Willis, Anatomy of the Brain, 56. About this argument and its use in the Aristotelian tradition, see Lo Presti above in this volume.

<CN>Chapter 10 <CT>How

 10 Animals May Help Us Understand Men <CST>Thomas Willis's Anatomy of the Brain (1664) and Two Discourses Concerning the Soules of Brutes (1672) <CA>Claire Crignon <TX>In a paper written in 1973 about the use of comparative anatomy in the medical thought of Thomas Willis (1621-1675), William Bynum notes some "inconsistencies" in Willis's work. 1 Although Willis describes the human brain as physical proof of human preeminence over animals in his Anatomy of the brain (1664), he declares in his Discourses

  Let us turn now to another important work by Thomas Willis, published in 1672: the De Anima Brutorum or Two Discourses Concerning the Soules of Brutes, a work that Willis announced in the Anatomy of the Brain as a necessary complement to his method of comparative anatomy.

  animals that seem to behave like men (or imitate them), and men who act sometimes foolishly or transform themselves into animals.
From Comparative Anatomy to the Psychology of the Soul <TX>The Two Discourses were published in 1672, at the same time as the Discours de la connoissance des Bestes written by the Jesuit physician and mathematician Ignace Gaston

<NHD>Chapter 10 :

 10 How Animals May Help Us Understand Men <UNN>For the wider context in which we can read and interpret Willis's thought, see in this volume Charles Wolfe, "Boundary Crossings. The Blurring of the Human/Animal Divide as Naturalization of the Soul in Early Modern Philosophy." <NTX> 1 William F. Bynum, "The Anatomical Method, Natural Theology, and the Functions of the Brain," Isis 64 (1973): 447. 2 Thomas Willis, "The Preface to the Reader," The Anatomy of the Brain, in Dr Willis's Practice of Physick, Being the whole Works of that Renowned and Famous Physician, trans. Samuel Pordage (London: T. Dring and J. Leigh, 1684), hereafter Anatomy of the Brain: "For the Province, which I hold in this Academy, requiring that I should comment on the Offices of the Senses, both external and also internal, and of the Faculties and Affections of the Soul, as also of the Organs and various provisions of all these . . . ." 3 Thomas Willis, "The Authors Epistle Dedicatory to his Grace Gilbert Archbishop of Canterbury," Anatomy of the Brain.

4

  André Du Laurens, Histoire Anatomique, in Toutes les OEuvres de M. André Du Laurens, trans.

Willis, Soul of Brutes, 188: "Fabulous antiquity scarce ever thought of so many metamorphoses of men, which some have not believed really of themselves; whilst some have believed themselves to be Dogs or Wolves, and have imitated their ways and kind by barking or howling; others imagining that their bodies were made of glass, were afraid to be touched left they should be broke to pieces."

Willis, Soul of Brutes, 3.


Samar Allouch
email: allouch@math.univ-paris13.fr

Vuk Milišić
email: milisic@math.univ-paris13.fr

Friction mediated by transient elastic linkages : extension to loads of bounded variation

 naturally extend to this new setting, while only partial results can be obtained following the comparison principle introduced in [12].

Introduction

Cell motility plays a central role in several important phenomenons in biology : cancer cell migration, neutrophils' extravasation, chemotaxis, etc. The present paper fits in the modelling framework presented in [START_REF] Oelz | Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments[END_REF][START_REF] Sfakianakis | Modelling cell-cell collision and adhesion with the filament based lamellipodium model[END_REF][START_REF] Manhart | An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals[END_REF]. The adhesive dynamics of actin filaments are at the heart of the project : they contribute to lamellipodium's stabilization and allow the cell to attach to the substrate or the surrounding tissue. This paper contributes to the better mathematical understanding of a minimal model introduced first in [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF], its aim is to extend results already obtained in [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF][START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF] to the case of stiffer external loads.

More precisely, we are interested in the motion of a single binding site, linked to a one-dimensional substrate and subjected to an external force f . As in [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF][START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF], the position of this binding site, denoted z ε , solves a Volterra integral equation

     1 ε ∞ 0 (z ε (t) -z ε (t -εa)) ρ ε (a, t) da = f (t), t ≥ 0, z ε (t) = z p (t), t < 0. (1)
The kernel ρ ε above solves a non-local age-structured problem :

         ε∂ t ρ ε + ∂ a ρ ε + ζ ε ρ ε = 0, t > 0, a > 0, ρ ε (a = 0, t) = β ε (t) 1 - ∞ 0 ρ ε (t, ã) dã , t > 0, ρ ε (a, t = 0) = ρ I,ε (a), a ≥ 0, (2)
where

β ε ∈ R + (resp. ζ ε ∈ R + )
is the kinetic on-rate (resp. off-rate) function. These possibly depend on the dimensionless parameter ε > 0. The past positions are stored in the Lipschitz function z p (t) ∈ R, prescribed for every t < 0.

Various mathematical issues related to this system have already been investigated [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF][START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF][START_REF] Milišić | Tear-off versus global existence for a structured model of adhesion mediated by transient elastic linkages[END_REF]. In [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF], the authors have introduced a specific Lyapunov functionnal in order to study the convergence of (2) when ε goes to 0. Indeed, due to the saturation effect in the non-local boundary condition in [START_REF] Anevski | Riemann-stieltjes integrals[END_REF], neither the Generalized Relative Entropy [START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Gabriel | Équations structurées en dynamique des populations[END_REF] nor more generic comparison principles [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF] do apply. Then, concerning [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF], under the assumptions that the force f is Lipschitz on R, and because the kernel ρ ε in (1) is non-negative, an extension of Gronwall's Lemma to integral equations, shows convergence of z ε towards z 0 the solution of (3), the limit equation associated to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. These two steps show that z ε -z 0 C 0 ([0,T ]) + ρ ε -ρ 0 C 0 (]0,T ];L 1 (R+)) → 0.

where z 0 it given by µ 1,0 (t)∂ t z 0 (t) = f (t), t > 0

z 0 (0) = z p (0) t = 0 (3)
where µ 1,0 (t) := ∞ 0 aρ 0 (a, t)da, and ρ 0 solves :

   ∂ a ρ 0 + ζ 0 (a, t)ρ 0 = 0, t > 0, a > 0, ρ 0 (a = 0, t) = β 0 (t) 1 - ∞ 0 ρ 0 (t, ã) dã , t > 0. (4)
In [START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF], the authors weakened some assumptions concerning the off-rate ζ ε , by assuming that ζ ε is not necessarily non-decreasing passed a certain age a 0 . Then, they introduce a new variable u ε related to z ε which transforms (1) into a transport problem with a non-local source term :

             ε∂ t u ε + ∂ a u ε = 1 µ 0,ε (t) ε∂ t f + ∞ 0 ζ ε (ã, t)u ε (ã, t)ρ ε (ã, t) dã , t > 0, a > 0, u ε (a = 0, t) = 0, t > 0, u ε (a, t = 0) = u I,ε (a) := z ε (0) -z p (-εa) ε , a ≥ 0, (5)
where µ 0,ε (t) := ∞ 0 ρ ε (ã, t) dã and according to [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF], it holds that z ε (0) = 1 µ 0,ε (0) ∞ 0 z p (-εa)ρ I,ε (a) da + εf (0) .

If f ∈ Lip(R), systems (1) and ( 5) are equivalent. Nevertheless, (5) admits a stability result that allows to show a weak-* convergence of u ε /(1 + a) towards u 0 /(1 + a) in L ∞ (R + × (0, T )), where u 0 is the solution of the limit problem

     ∂ a u 0 = 1 µ 0,0 ∞ 0 ζ 0 u 0 ρ 0 dã, t > 0, a > 0,
u 0 (a = 0, t) = 0, t > 0, [START_REF] Heida | Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space[END_REF] which in turn provides the strong convergence of z ε in C([0, T ]) towards z 0 solving (3).

In our analysis, however, when f ∈ BV((0, T )), the derivative of f is neither a function nor it is bounded, since it is a Radon measure. Therefore, we cannot apply directly results from [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF]. Instead, defining f δ to be a specific regularization of f [20, Section 5.3] provides a regular function u δ ε solving (5). To do this, we use the framework already established in [START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF]. Then we show that u δ ε satisfies certain a priori estimates that are uniform with respect to both δ and ε. These provide necessary compactness in order to pass to the limit with the regularization parameter δ and give existence and uniqueness of a weak solution u ε associated to [START_REF] Gabriel | Équations structurées en dynamique des populations[END_REF] with a load f ∈ BV((0, T )). The a priori estimates holding also in this weaker framework, we can consider convergence with respect to ε and prove consistency with the formal limit system. We show that, in the BV framework, the equivalence between ( 5) and (1) still holds. For the particular case when the kernel ρ ε is independent on time and on ε and under suitable hypotheses, we show error estimates to be compared with [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF], the comparison principle being applied to the integral of the error's modulus.

In order to clarify the interplay between parameters ε and δ, we make the following remarks : in the previous literature [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF][START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF], not only existence (and uniqueness) but also convergence results were strongly related to the Lipschitz regularity of the load f . This motivates the present work since it is not clear that the convergence occurs with respect to ε in this weaker framework. This explains also why we first regularize the problem with the parameter δ, make δ tend to zero and then consider the convergence with respect to ε.

The outline of the paper is as follows : in Section 2, collecting various results from the literature on BV-functions in one space dimension, we introduce the framework used in the rest of the paper. We make the link with the Riemann-Stieltjes integral, through a careful analysis of different definitions of BV-functions with respect to the boundary of the time domain (0, T ). In Section 3, we recall some results concerning (2) already established in [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF]. Then in Section 4, we establish uniform (with respect to ε) a priori estimates for the regularized system in u ε . After that, in Section 5, we show the weak convergence of u ε towards u 0 , the solution of the limit problem. This implies strong convergence of z ε in L ∞ (0, T ) as stated in Theorem 8. We establish, in Section ??, a specific comparison principle for Volterra equations when the density is constant in time and does not depend on ε.

Notations and main assumptions

We denote L p t L q a := L p ((0, T ); L q (R + )) for any real (p, q) ∈ [1, ∞] 2 and

X T := g ∈ L ∞ loc ((0, T ) × R + ) ; sup t∈(0,T ) g(t, a)w(a) L ∞ a < ∞ (8)
where w(a) := (1 + a) -1 . The space Lip(I) is the set of Lipschitz functions on the interval I.

Assumptions 1. For any T > 0 possibly infinite, we assume that : i) The past condition z p is L zp -Lipschitz on R -i.e. :

|z p (a 2 ) -z p (a 1 )| ≤ L zp |a 2 -a 1 |, ∀(a 2 , a 1 ) ∈ R -× R -.
ii) The function

β ε (t) is in L ∞ (0, T ) and ζ ε (a, t) is in L ∞ (R + × (0, T )). iii) For limit functions β 0 ∈ L ∞ t and ζ 0 ∈ L ∞ t L ∞ a it holds that ζ ε -ζ 0 L ∞ a,t → 0 and β ε -β 0 L ∞ t -→ 0
as ε → 0. iv) There are upper and lower bounds such that

0 < ζ min ≤ ζ ε (a, t) ≤ ζ max and β min ≤ β ε (a, t) ≤ β max , for all ε > 0, a ≥ 0 and t > 0. Assumptions 2. The initial condition ρ I,ε ∈ L ∞ a (R + ) satisfies i) positivity ρ I,ε (a) ≥ 0 , a.e. in R + ,
moreover, on has also that the total initial population satisfies 0

< R+ ρ I,ε (a)da < 1 ;
ii) boundedness of higher moments,

0 < R+ a p ρ I,ε (a)da < c p , for p = 1, 2,
where c p are positive constants depending only on p;

Next, we introduce definitions of functions with bounded variation in one dimension, as well as some related properties. Definition 1. Let f : (0, T ) → R be a Lebesgue measurable function. The pointwise variation (or Jordan variation) of f on (0, T ) is pvar(f, (0, T )) := sup P var(f, P )

where var(f,

P ) := n k=1 |f (t k ) -f (t k-1 )| and P = {0 < t 0 < • • • < t n < T } is a partition of (0, T ).
Moreover, we denote BPV((0, T )) := {f ∈ L((0, T )), s.t pvar(f, (0, T )) < +∞}, the space of measurable functions with pointwise bounded variation, see for example, [1, section 3.2], [9, chapter 2] and [7, section 2.2, 2.3]. The pointwise variation of f is clearly dependent on the value of f at each point of the domain, and it differs from one a.e.-representative of f to another. For this reason, for every measurable function f , one defines the essential pointwise variation : epvar(f, (0, T )) := inf {pvar(g, (0, T )) : f (t) = g(t) a.e. t ∈ (0, T )} [START_REF] Manhart | An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals[END_REF] In [START_REF] Leoni | A first course in Sobolev spaces[END_REF]Chapter 6], another functional space is defined :

Definition 2.
Given an open interval (0, T ) ⊂ R, the space of functions with bounded variation BV((0, T )) is defined as the space of all functions f ∈ L 1 ((0, T )) for which there exists a signed Radon measure µ f such that

(0,T ) f φ dt = - (0,T ) φ dµ f , for every φ ∈ C 1 c ((0, T )) (11)
for all φ ∈ C 1 c ((0, T )). The measure µ f is called the weak or distributional derivative of f . Remark 1.

i) We define the total variation of f ∈ L 1 ((0, T )) by

Df ((0, T )) = sup - (0,T ) f φ dt, φ ∈ C 1 c ((0, T )), |φ| ∞ ≤ 1 . (12)
Moreover, f ∈ BV((0, T )) if Df ((0, T )) < +∞. ii) Definitions ( 9) and ( 12) are not equivalent. For instance, the Dirichlet indicatrix function χ Q∩[0,1] is not of pointwise bounded variation in (0, 1) in the sense of Definition 1 but is well defined in the sense of Definition 2. The equivalence between the two definitions holds up to a.e. equality. Moreover every integrable function f : (0, T ) → R such that pvar(f, (0, T )) < +∞, is in BV((0, T )) and Df ((0, T )) ≤ pvar(f, (0, T )).

On the other hand, if f belongs to BV((0, T )), then f admits a right continuous representative f with bounded pointwise variation such that pvar( f , (0, T )) = Df ((0, T )).

Fore more details, see, e.g., [9, theorem 7.3] and [START_REF] Heida | Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space[END_REF]. iii) Under the norm

f BV := f L 1 + epvar(f, (0, T )) < ∞ BV((0, T )) is a Banach space.
Next, we provide existence of the left and right limits of functions with bounded variation [7, Proposition 2.2].

Lemma 1. Let f ∈ BV((0, T )), Then both the limits

f (0 + ) = lim s→0,s>0 f (s) and f (T -) = lim s→T,s<T f (s) exist.
Additionally, if f is integrable, the left and right limits are as follows:

Lemma 2. Suppose that f ∈ BV((0, T )), then

f (0 + ) = lim ρ→0 + 1 ρ ρ 0 f (t) dt, f (T -) = lim ρ→0 + 1 ρ T T -ρ f (t) dt.
Next, we present a result used in the proof of Proposition 3, which relates the pointwise variation to the Lebesgue measure:

λ(f, h, Ω) := {t∈Ω:t+h∈Ω} |f (t + h) -f (t)| dt, Lemma 3. If f is in BPV((0, T )), then λ(f, h, (0, T ))/|h| is bounded. Moreover, λ(f, h, (0, T )) ≤ |h| pvar(f, (0, T )).
For the proof we can see [START_REF] Leoni | A first course in Sobolev spaces[END_REF]Theorem 2.20].

Finally, in [START_REF] Heida | Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space[END_REF], the authors add a new notion of variation containing the boundary value in order to expand the total variation of f to [0, T ]. This variation is defined as varw(f ) := sup

φ∈C 1 c ([0,T ]) |φ|∞≤1 φ(T )f (T -) -φ(0)f (0 + ) - (0,T ) f φ dt (13)
Moreover, by summarizing the results of [7, Proposition 2.3, 2.6 and 2.7] all notions of variations coincide: epvar(f, (0, T )) = Df ((0, T )) = varw(f )

The previous result allows to extend Lemma 3 to BV((0, T )) functions :

Lemma 4. If f is in BV((0, T )), then λ(f, h, (0, T ))/|h| is bounded. Moreover, λ(f, h, (0, T )) ≤ |h| Df ((0, T ))
Proof. By taking the infimum over almost every equal measurable functions, one has

inf f = f a.e. λ( f , h, (0, T )) ≤ |h| inf f = f a.e.
pvar( f , (0, T )) = |h| epvar(f, (0, T )) = |h| Df ((0, T ))

Since the left hand side is a Lebesgue integral one has :

inf f = f a.e. λ( f , h, (0, T )) = λ(f, h, (0, T ))
which ends the proof.

Data regularization

Theorem 1. For every f ∈ BV((0, T )), there exists a sequence of smooth functions

(f δ ) δ in C ∞ ((0, T )) such that lim δ→0 (0,T ) |f δ -f |dt = 0 and lim δ→0 (0,T ) |f δ | dt = Df ((0, T )).
Although the proof is classical (see for instance [START_REF] Ziemer | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF]Theorem 5.3.3 p.225]), we need the explicit form of f δ in the rest of the paper. For this reason, we present in Section A.1 the proof of Theorem 1.

Lemma 5. Let f ∈ BV((0, T )) ∩ L ∞ ((0, T )). Then the regularization function f δ defined as (60) is bounded in (0, T ).
Next, we compare the left and right limits of f and its' approximation f δ on the boundary: Lemma 6. Let f ∈ BV((0, T )) and f δ defined as (60), then

f δ (0 + ) = f (0 + ) and f δ (T -) = f (T -).
First we need the following result : Proposition 1. Let f ∈ BV((0, T )). For every δ > 0, and t 0 ∈ {0, T },

lim τ →0 + 1 τ Iτ ∩(0,T ) |f δ -f | dt = 0, (14)
where

I τ = {t ∈ R : |t -t 0 | < τ }.
Proof. For a fixed t 0 ∈ {0, T } and t ∈ I τ ∈ (0, T ), we have

f δ (t) -f (t) = ∞ i=0 [χ δi * (φ i f ) -φ i f ]
by the definition of supp φ i (see ( 56)), we have 1/(j 0 +i+1) < τ < 1/(j 0 +i-1) then i > 1/τ -j 0 -1.

Since, R is an archimedean space then

∀τ > 0, ∃! i 0 := 1 τ -j 0 s.t i 0 ≤ 1 τ < i 0 + 1 ( 15
)
which implies by using (58) that

Iτ ∩(0,T ) |f δ -f | dt = ∞ i=i0 Iτ ∩(0,T ) [χ δi * (φ i f ) -φ i f ] dt ≤ ∞ i=i0 δ2 -i ≤ δ2 -i0 ∞ i=0 2 -i = 2 j0+1 δ 2 -1 τ then 1 τ Iτ ∩(0,T ) |f δ -f | dt ≤ C δ 2 -1 τ τ using again (15), we have 2 -1 τ τ = exp(-1 τ ln 2) τ ≤ 2 exp(-1 τ ln 2) τ Finally, we conclude that lim τ →0 + 1 τ Iτ ∩(0,T ) |f δ -f | dt = lim τ →0 + 2 exp(-1 τ ln 2) τ = 0.
Proof of Lemma 6. According to the Lemma 2,

lim τ →0 + 1 τ τ 0 |f δ (t) -f δ (0 + )| dt = 0 and lim τ →0 + 1 τ τ 0 |f (t) -f (0 + )| dt = 0.
Moreover, we have, thanks to Proposition 1

lim τ →0 + 1 τ τ 0 |f δ -f | dt = 0.
Thus, for all ε > 0, there exist δ > 0 such that 0 < τ < δ implies

f δ (0 + ) -f (0 + ) = 1 τ τ 0 f δ (0 + ) -f (0 + ) dt ≤ 1 τ τ 0 f δ (0 + ) -f δ (t) dt + 1 τ τ 0 |f δ (t) -f (t)| dt + 1 τ τ 0 f (t) -f (0 + ) dt ≤ 3ε
which proves the required result. Similarly, we can prove that

f δ (T -) = f (T -).
In the previous setting, the weak derivative of f ∈ BV((0, T )) defines a linear continuous form on C((0, T )). In the next section, we show how to extend this measure on functions in C([0, T ]).

Definition and basic properties of Stieltjes integral

The Riemann-Stieltjes integral (RS-integral) is a generalization of the Riemann integral. Let P a tagged partition of [0, T ] , defined as

P := {(ξ i , [t i-1 , t i ]) : 1 ≤ i ≤ n} (16)
where 0 = t 1 ≤ • • • ≤ t n = T , and on each interval [t i-1 , t i ] we choose a single value ξ i , for i ∈ {1, . . . , n}.

Definition 3. For any function f , g : [0, T ] → R and a partition P, we define the Riemann-Stieltjes sum by

S(f, dg, P, [0, T ]) := i f (ξ i )[g(t i ) -g(t i-1 )].
Moreover, the RS-integral of f with respect to g

(RS) [0,T ] f (t)dg(t)
exists and has a value I ∈ R, if, for every ε > 0, there exists δ > 0, such that the mesh size max i (t i -t i-1 ) < δ and for every

ξ i in [t i , t i+1 ], |S(f, dg, P, [0, T ]) -I| < ε. Lemma 7. Suppose that f is continuous on [0, T ] and g is of bounded pointwise variation on [0, T ], then [0,T ] f dg ≤ f ∞ pvar(g, [0, T ])
In the following Theorem we see that a Riemann-Stieltjes integral can be used to describe any bounded linear functional on C([0, T ]) (see [START_REF] Benedetto | Integration and modern analysis[END_REF]Theorem 7.1.1] and [8, Theorem 4.4-1] for more details)

Theorem 2. Let Γ f ∈ (C([0, T ])) , then there exist g ∈ BP V ([0, T ]) such that Γ f (ϕ) = [0,T ] ϕdg, ∀ϕ ∈ C([0, T ]).

Theorem 3. (Integration by parts). If one of the integrals

[0,T ]
f dg and

[0,T ]
gdf exists, then the other exists as well, and we have

[0,T ] f dg + [0,T ] gdf = [f g(t)] t=T t=0 . Moreover, If f ∈ C 1 ([0, T ]) and g ∈ BP V ([0, T ]), then df = f dt in the second term of the left hand side.
For the proof cf [14, Theorem 5.52] and [2, Lemma 2].

Lemma 8. Let f ∈ BV((0, T )). Then there exists g ∈ BPV([0, T ]) s.t

[f ϕ] t=T - t=0 + - (0,T ) f ϕ dx = [0,T ] ϕdg, ∀ϕ ∈ C 1 ([0, T ]) s.t. f (t) = g(t), a.e. t ∈ (0, T )
Proof. We regularize f ∈ BV((0, T )) by f δ ∈ C ∞ ((0, T )) as in Theorem 1, then we have :

s k t k f δ ϕdt + s k t k f δ ϕ dt = [f δ ϕ] t=s k t=t k =: L k , ∀ϕ ∈ C 1 ([0, T ])
where s k → T -and t k → 0 + . We define :

I k := s k t k f δ ϕdt, J k := s k t k f δ ϕ dt.
Thanks to Lebesgue's Theorem, one has that

lim k→∞ I k = I := T 0 f δ ϕdt, lim k→∞ J k = J := T 0 f δ ϕ dt
and thanks to Lemma 6 and the continuity of ϕ,

L k = L := f (T -)ϕ(T ) -f (0 + )ϕ(0), ∀k ∈ N
So that we have :

T 0 f δ ϕdt + T 0 f δ ϕ dt = [f ϕ] t=T - t=0 +
If we set

I f δ (ϕ) := T 0 f δ ϕdt,
it is a linear continuous form on C([0, T ]), since one has :

|I f δ (ϕ)| ≤ f δ L 1 ((0,T )) ϕ L ∞ ≤ { Df ((0, T )) + δ} ϕ L ∞ ((0,T )) (17)
where we used estimates from the proof of [START_REF] Ziemer | Weakly differentiable functions. Sobolev spaces and functions of bounded variation[END_REF]Theorem 5.3.3]. Since it is a continuous linear form on C([0, T ]), by Theorem 2 there exists h δ ∈ BPV([0, T ]) s.t.

I f δ (ϕ) = [0,T ] ϕdh δ , ∀ϕ ∈ C([0, T ])
in the Stieljes' sense. But by using the integration by parts from Theorem 3, we have that

[f ϕ] t=T - t=0 + - (0,T ) f δ ϕ dt = [0,T ] ϕdh δ = [h δ ϕ] t=T t=0 - [0,T ] h δ ϕ dt, ∀ϕ ∈ C 1 ([0, T ]) (18)
which implies that (0,T )

f δ ϕ dt = (0,T ) h δ ϕ dt, ∀ϕ ∈ D((0, T ))
and then we can apply [9, Lemma 7.4] and conclude that there exists c ∈ R, s.t.

f δ = h δ + c.
Then setting g δ := h δ + c provides a function s.t.

[

f δ ϕ] T 0 - T 0 f δ ϕ dt = T 0 ϕdg δ , ∀ϕ ∈ C 1 ([0, T ]) and s.t. f δ (t) = g δ (t), a.e. t ∈ (0, T ).
Thanks to [START_REF] Oelz | Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments[END_REF], I f δ is a linear continuous form on C([0, T ]) uniformly bounded with respect to δ. It can be identified via the Riesz representation theorem as a Radon measure µ δ on [0, T ]. Therefore, there exist µ ∈ M 1 ([0, T ]) and a sub-sequence µ δ k such that

µ δ k * µ in σ(M 1 ([0, T ]), C([0, T ]))
with respect to the weak-* topology. By Theorem 2, there exists h ∈ BPV([0, T ]) s.t.

µ(ϕ) = T 0 ϕdh
where the left side is a Radon measure and the right hand side is the Riemann-Stieltjes integral.

Because f δ tends to f in the L 1 (0, T ) topology, one has then that

[f ϕ] t=T - t=0 + - (0,T ) f ϕ dt = [0,T ] ϕdh, ∀ϕ ∈ C 1 ([0, T ])
then using again integration by parts from Theorem 3, one concludes that

f (t) = h(t) + c, a.e. t ∈ (0, T )
and setting g = h + c ends the proof.

Corollary 1. There exists a sub-sequence (f δ k ) k∈N , s.t.

I f δ k (ϕ) := T 0 ϕf δ k dt → T 0 ϕdg, ∀ϕ ∈ C([0, T ]) when k → ∞

Mathematical background for the linkages' density

We list here some of the results proved in [START_REF] Milišić | On the asymptotic regime of a model for friction mediated by transient elastic linkages[END_REF] used in the next sections of the paper.

Theorem 4. Let Assumptions 1 and 2 hold, then for every fixed ε > 0 there is a unique solution

ρ ε ∈ C 0 (R + ; L 1 (R + )) ∩ L ∞ (R 2 + ) of the problem (2). It satisfies (2) in the sense of characteristics, namely ρε(a, t) =      βε(t -εa) 1 -R + ρε(ã, t -εa) dã exp - a 0 ζε(ã, t -ε(a -ã) dã), ∀a < t ε ρI,ε(a -t ε ) exp -1 ε t 0 ζε t-t ε + a, t d t , ∀a ≥ t ε (19) 10
Moreover, it is a weak solution as well since it satisfies (cf [12, Lemma 2.1])

T 0 +∞ 0 ρ ε (a, t) (ε∂ t ϕ + ∂ a ϕ -ζ ε ϕ) da dt -ε +∞ 0 ρ ε (a, T )ϕ(a, T ) da + T 0 ρ ε (a = 0, t)ϕ(a = 0, t) dt + ε +∞ 0 ρ I,ε (a)ϕ(a, t = 0) da = 0 (20)
for every T > 0 and test function ϕ

∈ C ∞ (R 2 + ) ∩ L ∞ (R 2 +
). Now we define the moments of ρ ε which we denote by µ p,ε (t) := R+ a p ρ ε (a, t)da, with p = 1, 2.

Lemma 9. Let Assumptions 1 and 2 hold, then the unique solution

ρ ε ∈ C 0 (R + ; L 1 (R + )) ∩ L ∞ (R 2 + ) of (2) satisfies i) ρ ε (a, t) ≥ 0 for a.e (a, t) in R 2 + , ii) µ 0,min ≤ µ 0,ε (t) < 1, ∀t ∈ R + where µ 0,min < min µ 0,ε (0), βmin βmin+ζmax iii) µ p,min ≤ µ p,ε (t) ≤ k, where , µ p,min = min µ p,ε (0), µ p-1,min ζ max
The authors provide a Liapunov functional that reads :

H[u] := ∞ 0 u(a)da + ∞ 0 |u(a)|da ,
thanks to which they obtain the following convergence result for ρ ε :

Lemma 10. Let ζ min > 0 be the lower bound to ζ ε (a, t) according to Assumptions 1, and setting ρε := one has

H[(ρ ε -ρ 0 )(•, t)] ≤ H[ρ I,ε -ρ 0 (•, 0)] exp -ζ min t ε + 2 ζ min R ε L 1 a (R+) + |M ε | L ∞ t (R+) (21) with R ε := -ε∂ t ρ 0 -ρ 0 (ζ ε -ζ 0 ) and M ε := (β ε -β 0 )(1 - ∞ 0 ρ 0 da).
This ensures the convergence of ρ ε that reads :

Theorem 5. Let ρ ε the solution of the system (2) and let ρ 0 given by (4), then

ρ ε → ρ 0 in C 0 ((0, ∞); L 1 (R + )) as ε → 0 ,
where the convergence with respect to time is meant in the sense of uniform convergence on compact subintervals.

Existence, uniqueness and stability

Using the regularized function f δ introduced in Theorem 1, we consider an approximation of (1) : we denote by z δ ε := z δ ε (t) the function solving

   1 ε ∞ 0 z δ ε (t) -z δ ε (t -εa) ρ ε (a, t)da = f δ (t), t ≤ 0 z δ ε (t) = z p (t), t < 0 (22)
We also define u δ ε (a, t), an approximation of the elongation variable u ε , defined as the mild solution of

         ε∂ t u δ ε + ∂ a u δ ε = 1 µ 0,ε εf δ + ∞ 0 ζ ε u δ ε ρ ε da , t > 0, a > 0 u δ ε (a = 0, t) = 0, t > 0 u δ ε (a, t = 0) = u δ I,ε (a), a ≥ 0 (23)
where

u δ I,ε (a) := z δ ε (0 + ) -z p (-εa) ε (24)
and

z δ ε (0 + ) = 1 µ 0,ε (0) εf δ (0 + ) + ∞ 0 z p (-εa)ρ I,ε (a)da . ( 25
)
More precisely, u δ ε is a solution of system (23) in the sense of characteristics, namely

u δ ε (a, t) = a 0 h(t -εã) dã, if t > εa t/ε 0 h(t -εã) dã + u δ I,ε (a -t/ε), if t ≤ εa, (26)
where

h(t) := 1 µ 0,ε ε∂ t f δ + ∞ 0 ζ ε u δ ε ρ ε da .
By arguments similar to [12, Lemma 3], it is as well a weak solution of (23) i.e.

- T 0 ∞ 0 u δ ε (ε∂ t ϕ + ∂ a ϕ) da dt + ∞ 0 u δ ε (s, a)ϕ(s, a) da s=T s=0 = T 0 1 µ 0,ε εf δ + ∞ 0 ζ ε u δ ε ρ ε da ∞ 0 ϕ(t, ã) dã dt (27)
for any function

ϕ ∈ C ∞ c ([0, T ] × R + ). Although, problem (22)
can be defined for weaker data (typically L 1 ((0, T )) or the space of Radon measures M ((0, T ))), the elongation problem (23), requires to give a meaning to the time derivative of f , which is more restrictive. Nevertheless, as we are mainly interested in convergence results, f ∈ BV((0, T )) seems the weakest possible regularity to our knowledge. Theorem 6. Let Assumptions 1 hold, and let ρ ε be the unique solution of (2) then the system (23) has a unique solution u δ ε ∈ X T .

We are in the framework of [13, Theorem 6.1], but for sake of self-containtness, we recall in an abriged version the proof hereafter.

Proof. A Banach fixed point Theorem is used to prove this result. We define the mapping φ(v) = u such that by Duhamel's principle

u(a, t) = a 0 G(t -εã) dã, if t > εa t/ε 0 G(t -εã) dã + u δ I,ε (a -t/ε), if t ≤ εa. (28)
where

G(t) := 1 µ 0,ε εf δ + ∞ 0 ζ ε (a, t)v(a, t)ρ ε (a, t)da
As in [START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF], a simple computation shows that

u X T ≤ G L ∞ ((0,T )) T T + ε + u δ I,ε (•) 1 + a L ∞ (R+) Moreover, since ∂ t f δ ∈ L ∞ ((0, T )) G L ∞ ((0,T )) ≤ 1 µ 0,min ε f δ L ∞ ((0,T )) + ζ max (1 + k) µ 0,min v X T
where k is the upper bound of µ 1,ε proved in Lemma 9. Furthermore, by the same argument we can prove that φ is a contraction. Indeed, if

u i = φ(v i ) for i ∈ {1, 2} u 2 -u 1 X T ≤ C T T + ε v 2 -v 1 X T
for a constant C > 0. Then we can choose T < ε/C and we obtain the existence of a local solution in time of (23), by Banach-Picard's fixed point theorem. As the contraction time does not depend on the initial data, we shall extend the same result by continuation. This shows existence and uniqueness in X T for any T > 0.

Lemma 11. If Assumptions 1 holds, then the solution of system (23) satisfies the uniform a priori estimates

∞ 0 ρ ε (a, t)|u δ ε (a, t)| da ≤ t 0 |f δ | d t + R+ ρ I,ε |u δ I,ε | da ≤ C f BV((0,T )) , (1 + a)ρ I L 1 (R+) , z p L ∞ (R-) (29)
where C is independent on ε and on δ.

Proof. Again, we proceed as in [13, Lemma 5.1], multiplying (23) by sgn(u δ ε ), testing against ρ ε , and integrating with respect to a gives :

d dt R+ u δ ε ρ ε da + R+ u δ ε ζ ε ρ ε da ≤ ε |f δ | + R+ u δ ε ζ ε ρ ε da
the rigorous proof relies on arguments exposed in [12, Lemma 3.1] and is left to the reader. Finally, after integration with respect to time, we conclude that

∞ 0 ρ ε (a, t)|u δ ε (a, t)|da ≤ t 0 |f δ |dt + R+ ρ I,ε |u δ I,ε |da since |u δ I,ε (a)| ≤ z δ ε (0 + ) -z p (0) ε + z p (0) -z p (-εa) ε ≤ 1 µ 0,ε (0) f δ (0 + ) + 1 ε ∞ 0 (z p (-εa) -z p (0))ρ I,ε (a) da + z p (0) -z p (-εa) ε ≤ 1 µ 0,min z p L ∞ (R-) µ 1,ε (0) + f δ (0 + ) + z p L ∞ (R-) a ≤ max 1 µ 0,min z p L ∞ (R-) µ 1,ε (0) + f δ (0 + ), z p L ∞ (R-) (1 + a),
the result follows.

In order to establish the convergence of u δ ε in X T , for a fixed ε, we introduce an intermediate variable w defined as w(a, t) := u δ ε (a, t) -

f δ (t) µ 0,ε (t) . ( 30
)
It satisfies

                     ε∂ t w + ∂ a w = 1 µ 0,ε ε f δ ∂ t µ 0,ε µ 0,ε + ∞ 0 ζ ε u δ ε ρ ε da , t > 0, a > 0, w(a = 0, t) = -f δ (t) µ 0,ε (t) , t > 0, w(a, t = 0 + ) = u δ I,ε (a) - f δ (0 + ) µ 0,ε (0) , a ≥ 0, (31)
The following crucial result holds:

Lemma 12. For a fixed δ and ε, and under the Assumptions 1, the unknowns w and u δ ε , are uniformly bounded in X T with respect to δ and ε.

Proof. Using arguments from [12, Lemma 2.1], one can show that w defined as w(a, t) := w(0, t -εa)

+ a 0 G w (t -εã) dã, if t > εa w(a -t/ε, 0 + ) + t/ε 0 G w (t -εã) dã, if t ≤ εa. ( 32
)
is a weak solution of (31). In the latter definition G w (t

) := ε f δ ∂ t µ 0,ε µ 2 0,ε + 1 µ 0,ε ∞ 0 ζ ε ρ ε u δ ε da . A simple computation shows that w X T ≤ G w L ∞ ((0,T )) + w(0, .) L ∞ ((0,T )) + w(., 0) 1 + a L ∞ (R+)
.

It remains to estimate G w L ∞ (0,T ) . For every fixed ε, µ 0,ε is a Lipschitz continuous function. Indeed, µ 0,ε satisfies

ε∂ t µ 0,ε -β ε (1 -µ 0,ε ) + R+ ζ ε ρ ε da = 0
and then

ε∂ t µ 0,ε L ∞ t ≤ β ε L ∞ t + ζ ε L ∞ t,a ρ ε L ∞ t L 1
a , which shows, by using the result of Lemma 11 that G w , and also w, are uniformly bounded in L ∞ ((0, T )) with respect to δ and ε. Indeed,

G w L ∞ ((0,T )) ≤ ε∂ t µ 0,ε L ∞ ((0,T )) f δ L ∞ ((0,T )) µ 2 0,min + ζ max µ 0,min R+ ρ ε |u δ ε |da < +∞.
Finally, we have that

u δ ε X T ≤ w X T + f δ L ∞ ((0,T )) µ 0,min < +∞
which ends the proof.

Previous stability estimates allow to show :

Theorem 7. Under Assumption 1, one has for any fixed ε > 0,

u δ ε u ε weakly-* in X T
as δ → 0, where u ε solves the weak problem

- T 0 ∞ 0 u ε (ε∂ t + ∂ a )ϕ da dt + ε ∞ 0 u ε (a, s)ϕ(a, s) da s=T s=0 = ε T 0 ∞ 0 ϕ(ã, t)dã µ 0,ε dg + T 0 ∞ 0 ζ ε ρ ε u ε da µ 0,ε ∞ 0 ϕ(ã, t) dã (33) for any ϕ ∈ C 1 c ([0, T ] × R + ).
Proof. The uniform bound on u δ ε in X T , proved in Lemma 12, implies that

u δ ε 1 + a * u ε 1 + a in L ∞ ((0, T ) × R + )
in the weak-* sense and the limit function u ε belongs X T . For every ψ ∈ L ∞ ((0, T ) × R + ), we have

ζ ε (1 + a)ρ ε ψ ∈ L 1 ((0, T ) × R + ) and then T 0 ∞ 0 ζ ε ρ ε u δ ε ψ da dt → T 0 ∞ 0 ζ ε ρ ε u ε ψ da dt.
By Corollary 1, the first term of right-hand in (27) tends to

T 0 ∞ 0 ϕ(t, ã)dã/µ 0,ε dg as δ → 0, for any ϕ ∈ C 1 c ([0, T ] × R + ).
Regarding the second term of the right hand side in ( 27), one has that ζ ε ρ ε /µ 0,ε ∈ L 1 ((0, T ) × R + ) and this leads, thanks again to the weak-* convergence above to write :

t 0 1 µ 0,ε ∞ 0 ζ ε ρ ε u δ ε da d t -→ t 0 1 µ 0,ε ∞ 0 ζ ε ρ ε u ε da d t as δ → 0, (34)
which ends the proof.

The latter theorem allows to prove a convergence result when returning to the z ε variable :

Proposition 2. Under the same assumptions as above, it holds that

z δ ε → z ε strongly in L ∞ ((0, T )) as δ → 0,
where z ε satisfies

z ε (t) = z ε (0 + ) + t 0 ε µ 0,ε dg + t 0 1 µ 0,ε ∞ 0 ζ ε u ε ρ ε da d t (35)
which is also a solution of (1).

Before showing this result, we make some comments : if u δ ε is a solution of (23) then z δ ε defined as Proof. First, by using (34) in the proof of Theorem 7, and Lemma 6, we have that f δ (0 + ) = f (0 + ) and z δ ε given by (36) converge strongly in L ∞ ((0, T )) to z ε which verifies (35). Using [13, Lemma 4.2], if z δ ε is defined as (36) it solves (22). Multiplying ( 22) by a test function ϕ ∈ L 1 (0, T ) gives :

z δ ε (t) := z δ ε (0 + ) + t 0 1 µ 0,ε ( t) εf δ ( t) + ∞ 0 ζ ε u δ ε ρ ε da d t (36) solves (22). Conversely, if z δ ε solves (22) then u δ ε , given by u δ ε (a, t) =      z δ ε (t) -z δ ε (t -εa) ε , if t > εa z δ ε (t) -z p (t -εa) ε , if t ≤ εa (37)
1 ε T 0 +∞ 0 (z δ k ε (t) -z δ k ε (t -εa)) ρ ε (a, t) ϕ(t) da dt = T 0 f δ k (t) ϕ(t) dt (38)
As z δ k ε converges strongly in L ∞ (0, T ) to z ε , the difference z δ k ε (t)-z δ k ε (t-εa) converges almost every where for any fixed (a, t) in R + × (0, T ) towards z ε (t) -z ε (t -εa). Thanks to the L ∞ bounds on u δ k ε /(1 + a), and the bounds in L 1 (R + × (0, T )) on the first moment of ρ ε , there exists an integrable majorizing function g(a, t) on R + × (0, T ) s.t.

z δ k ε (t) -z δ k ε (t -εa) |ϕ(t)|ρ ε (a, t) ≤ g(a, t)
uniformly for every k. Thus one can apply the Lebesgue's Theorem in the right hand side of (38). Since f δ converges in L 1 (0, T ) the convergence occurs in (38) for every ϕ ∈ D(0, T ) and thus almost everywhere in (0, T ) and thus z ε solves (1).

Weak convergence when ε goes to zero

Next, we prove the weak convergence of u ε from which we deduce the strong convergence of z ε .

Theorem 8. Under the same assumptions as above, one has

u ε u 0 weakly-* in X T
as ε → 0, where u 0 satisfies ( 7) and ∞ 0 u 0 (a, t) ρ 0 (a, t) da = f (t) a.e t ∈ (0, T ).

Furthermore, it also holds that

z ε → z 0 strongly in L ∞ ((0, T )) as ε → 0.
Proof. The proof follows the same steps as in [START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF]Theorem 6.2]. First, by Lemma 12, u δ ε is uniformly bounded in X T with respect to δ and ε, and therefore u ε is uniformly bounded in X T with respect to ε, then u ε is weakly convergent to u 0 in X T . On the other hand, Theorem 3.2 and Lemma 3.

4 imply that (1 + a)ρ ε → (1 + a)ρ 0 strongly in L 1 ((0, T ) × R + ). These arguments justify that for every ψ ∈ L ∞ ((0, T ) × R + ) one has T 0 ∞ 0 ζ ε ρ ε u ε ψ da dt → T 0 ∞ 0 ζ 0 ρ 0 u 0 ψ da dt.
Indeed, one has

T 0 ∞ 0 {ζ ε ρ ε u ε -ζ 0 ρ 0 u 0 } ψ da dt = T 0 ∞ 0 (ζ ε -ζ 0 )ρ ε u ε ψ da dt + T 0 ∞ 0 ζ 0 (ρ ε -ρ 0 )u ε ψ da dt + T 0 ∞ 0 ζ 0 ρ 0 (u ε -u 0 ) ψ da dt
As ζ ε → ζ 0 by Assumptions 1, and thanks to the weak convergence of u ε , both terms on the righthand side tend to zero as ε → 0. Note that this implies the weak convergence of R+ ζ ε ρ ε u ε da in L 1 ((0, T )), since we can choose ψ ∈ L ∞ ((0, T )). Moreover, thanks to Lemma 7 we obtain that

[0,T ] +∞ 0 ϕ da µ 0,ε dg ≤ C ϕ L ∞ t,a µ 0,min pvar(g, [0, T ]) ≤ C f BV((0,T )) .
As in [13, Theorem 6.2], passing to the limit in the weak formulation (33) we obtain

- T 0 ∞ 0 u 0 ∂ a ψ da dt = T 0 ∞ 0 ζ 0 ρ 0 u 0 ψ µ 0,0 da dt which implies that u 0 satisfies      ∂ a u 0 = 1 µ 0,0 ∞ 0 ζ 0 (ã, t)u 0 (ã, t)ρ 0 (ã, t) dã, t > 0, a > 0, u 0 (a = 0, t) = 0, t > 0, (39)
Similarly, we have the weak convergence of

∞ 0 u ε (t, a) ρ ε (t, a) da towards ∞ 0 u 0 (t, a) ρ 0 (t, a) da in L 1 ((0, T )). Hence, one concludes that u 0 satisfies also ∞ 0 u 0 (t, a) ρ 0 (t, a) da = f (t), a.e. t ∈ (0, T ).
As the the right hand side of (39) does not depend on age, one has that u 0 = γ(t) a, where in order to satisfy the last compatibility condition implies that

γ(t) R+ aρ 0 (a, t)da = f (t), a.e. t ∈ (0, T ).
Thus u 0 (a, t) = f (t)/µ 1,0 (t)a for almost every t ∈ (0, T ) and every a ∈ R + . Using again the weak convergence of R+ ζ ε ρ ε u ε da in L 1 ((0, T )) combined with the strong convergence of µ 0,ε allows to pass to the limit in the third term of (35).

Moreover,

|z ε (0 + ) -z p (0)| = 1 µ 0,ε (0) ∞ 0 (z p (-εa) -z p (0)) ρ I,ε (a) da + εf (0 + ) ≤ εk z p L ∞ (R-) µ 0,min + ε|f (0 + )| µ 0,min -→ 0 as ε → 0
where k is the constant from Lemma 9.

All together this provides that z 0 solves :

z 0 (t) = z p (0) + t 0 f (τ ) µ 1,0 (τ ) R+ ζ 0 (a, τ )ρ 0 (a, τ )da µ 0,0 (τ ) dτ (40)
but because ρ 0 solves (4), one has that aρ 0 solves :

∂ a (aρ 0 ) -ρ 0 + aζ 0 (a, t)ρ 0 = 0,
which after integration in time shows that

R+ aζ 0 (a, t)ρ 0 (a, t)da µ 0,0 (t) = 1
and this shows in turn that (40) reduces to

z 0 (t) = z 0 (0) + t 0 f (τ ) µ 1,0 (τ ) dτ
which is the integrated version of (3).

A comparison principle

In this section, we give error estimates between z ε and z 0 , the solution of the problem

   1 ε ∞ 0 (z ε (t) -z ε (t -εa)) (a) da = f (t), t ≥ 0, z ε (t) = z p (t), t < 0, (41)
where is constant in time and satisfies

   ∂ a + ζ(a) = 0, a > 0, (0) = β 1 - ∞ 0 (ã) dã , (42)
where the data of ( 41) and (42) satisfy

Assumptions 3. i) f ∈ BV((0, T )), ii) z p ∈ Lip(R -), iii) β ∈ R * + , iv) ζ ∈ L ∞ (R + ) such that 0 < ζ min ≤ ζ(a) ≤ ζ max , a.e. a ∈ R + .
Setting ẑε (t) := z ε (t) -z 0 (t), it solves :

ẑε (t) = 1 µ 0 t ε 0 ẑε (t -εa) (a)da + hε (t)
where

hε (t) = ε µ 0 t/ε 0 z 0 (t) -z 0 (t -εa) ε -a∂ t z 0 (t) (a) da + ε µ 0 +∞ t/ε z 0 (t) -z 0 (0) ε -a∂ t z 0 (t) (a) da + 1 µ 0 +∞ t/ε (z p (t -εa) -z p (0)) (a) da (43)
where

µ 0 := +∞ 0 (a) da then |ẑ ε (t)| ≤ 1 µ 0 t ε 0 |ẑ ε (t -εa)| (a)da + | hε (t)| (44)
Then integrating in time and setting

Ẑε (t) := t 0 |ẑ ε (τ )| dτ
one has that :

Ẑε (t) = t 0 |ẑ ε (τ )| dτ ≤ 1 µ 0 t 0 τ ε 0 |ẑ ε (τ -εa)| (a) da dτ + t 0 | hε (τ )| dτ
then, we change the order of integration and the domain of integration becomes D := {(a, τ ) ∈ (0, t/ε) × (εa, t)}. We use the change of variable t = τ -εa in order to write :

t 0 τ ε 0 |ẑ ε (τ -εa)| (a) da dτ = t ε 0 t εa |ẑ ε (τ -εa)| dτ (a) da = t ε 0 t-εa 0 |ẑ ε ( t)| d t (a) da = t ε 0 Ẑε (t -εa) (a) da
So that finally Ẑε solves :

Ẑε (t) ≤ 1 µ 0 t ε 0 Ẑε (t -εa) (a) da + t 0 hε (τ ) dτ = t 0 Ẑε (ã)K ε (t -ã) dã + t 0 hε (τ ) dτ
where K ε (ã) := 1 εµ0 ã ε is the kernel of the integral operator. We use a comparison principle [6, the Generalised Gronwall Lemma 8.2 p. 257] and construct a majorizing function U ε of the form U ε (t) = ε(K 0 + K 1 t) where K 0 and K 1 are suitably chosen, such that U ε ≥ | Ẑε | and U ε ∼ ε. The following two lemmas are required in order to apply this comparison principle :

Lemma 13. The Volterra kernel K ε satisfies :

K ε L ∞ (0,T ) := ess sup t∈(0,T ) t 0 |K ε (ã)| dã < 1
Proof. To prove this result, we need to show that

0 ≤ t 0 |K ε (ã)| dã = t ε 0 (a) da +∞ 0 (a) da < 1. (45)
The kernel solves (42), thus it can be explicitly computed as

(a) = β 1 + βI exp - a 0 ζ(s)ds
one has the lower bound :

(a) ≥ β 1 + βI exp (-ζ max a) > 0, ∀a ∈ R + .
This in turn shows that

∞ t ε (a)da > 0
which is equivalent to the claim.

Lemma 14. Consider the expectation value of a given density with respect to the tail a > t/ ,

A 1 [ ](t) := +∞ 0 a (a + t ) da +∞ 0 (a + t ) da (46)
then under Assumptions 3, one has

A 1 [ ](t) ≤ ζ max ζ 2 min . Proof. Setting q(a, t) := (a + t ) ( t ) , (47)
it solves ∂ a q(a, t) + ζ(a + t/ ) q(a, t) = 0, q(0, t) = 1.

This problem admits an explicit solution of the form q(a, t) = exp -

a 0 ζ(ā + t/ ) dā = exp - a+t/ t/ ζ(â) dâ . (48)
Then, we shall rewrite (46) as : where C 1 depends on µ 2 ,µ 1 , ∂ t z 0 BV((0,T )) and on z p Lip(R-) but not on ε.

Proof. Recalling the definition of hε in (43), we split t 0 hε (τ ) dτ into three parts. First, we define

I 1 := ε µ 0 t 0 τ /ε 0 z 0 (τ ) -z 0 (τ -εa) ε -a∂ t z 0 (τ ) (a) da dτ
Since ∂ t z 0 ∈ BV((0, T )), then I 1 can be written in the form

I 1 = 1 µ 0 t 0 τ /ε 0 τ τ -εa ∂ t z 0 ( t) -∂ t z 0 (τ ) d t (a) da dτ
by switching the integration order between τ and a, and using the change of variable t = τ + h, we get that

I 1 ≤ 1 µ 0 t/ε 0 t εa τ τ -εa ∂ t z 0 ( t) -∂ t z 0 (τ ) d t dτ (a) da ≤ 1 µ 0 t/ε 0 t εa 0 -εa |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dh dτ (a) da ≤ 1 µ 0 t/ε 0 0 -εa t εa |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dτ dh (a) da (49)
and thus applying Lemma 3, one has the estimate of the inner integral of the latter right hand side :

t εa |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dτ ≤ |h| ∂ t z 0 BV , ∀h ∈ (-εa, 0)
which implies that

I 1 ≤ ∂ t z 0 BV µ 0 t/ε 0 0 -εa |h| dh (a) da ≤ ε 2 ∂ t z 0 BV 2µ 0 t/ε 0 a 2 (a) da.
Next, we set:

I 2 = ε µ 0 t 0 +∞ τ /ε z 0 (τ ) -z 0 (0) ε -a∂ t z 0 (τ ) (a) da dτ ≤ 1 µ 0 t 0 +∞ τ /ε τ 0 ∂ t z 0 ( t)d t -τ ∂ t z 0 (τ ) (a) da dτ + 1 µ 0 t 0 +∞ τ /ε |τ -εa| |∂ t z 0 (τ )| (a) da dτ =: I 2,1 + I 2,2
As in the estimates of I 1 , first, one switches the order of integration and then one integrates on

D := {(a, τ ) ∈ (0, t/ε) × (0, εa)} ∪ {(a, τ ) ∈ (t/ε, +∞) × (0, t)}, (50)
and one makes the change of variable t = τ + h in order to obtain

I 2,1 = 1 µ 0 t/ε 0 εa 0 0 -τ |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dh dτ (a) da + 1 µ 0 +∞ t/ε t 0 0 -τ |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dh dτ (a) da = 1 µ 0 t/ε 0 0 -εa εa -h |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dτ dh (a) da + 1 µ 0 +∞ t/ε 0 -t t -h |∂ t z 0 (τ + h) -∂ t z 0 (τ )| dτ dh (a)
da also, by using Lemma 4, we get that

I 2,1 ≤ ∂ t z 0 BV µ 0 t/ε 0 0 -εa |h| dh (a) da + ∂ t z 0 BV µ 0 +∞ t/ε 0 -t |h| dh (a) da ≤ ε 2 ∂ t z 0 BV 2µ 0 t/ε 0 a 2 (a) da + ∂ t z 0 BV 2µ 0 +∞ t/ε t 2 (a) da ≤ ε 2 ∂ t z 0 BV 2µ 0 t/ε 0 a 2 (a) da + ε 2 ∂ t z 0 BV 2µ 0 +∞ t/ε a 2 (a)da ≤ ε 2 µ 2 ∂ t z 0 BV µ 0 .
The second term I 2,2 is estimated in the same way as I 2,1 . We have

I 2,2 = 1 µ 0 t 0 +∞ τ /ε |τ -εa| |∂ t z 0 (τ )| (a) da dτ = 1 µ 0 t/ε 0 εa 0 + +∞ t/ε t 0 |τ -εa| |∂ t z 0 (τ )| dτ (a) da ≤ ε 2 ∂ t z 0 ∞ 2µ 0 t/ε 0 a 2 (a) da + ε 2 ∂ t z 0 ∞ 2µ 0 +∞ t/ε a 2 (a) da ≤ ε 2 µ 2 ∂ t z 0 ∞ µ 0 .
Finally, by similar computations, one has

1 µ 0 t 0 +∞ τ /ε |z p (τ -εa) -z p (0)| (a) da dτ ≤ ε 2 z p Lip(R-) µ 2 µ 0
Theorem 9. Under Assumptions 3, z ε tends to z 0 , the solution of (3), strongly in L 1 (0, T ) as ε goes to zero. Moreover, there exists a generic constant C depending only on the data of the problem but not on ε, such that :

z ε -z 0 L 1 (0,T ) ≤ εC.
Proof. We have proved in Lemma 13 that the Volterra kernel K ε is non-positive and bounded (with a bound strictly less than one) in the sense of [

Ẑε (ã)K ε (ã) dã ≤ h1,ε (t) + h2,ε (t)
We construct a function U ε which satisfies,

U ε (t) - t 0 U ε (ã)K ε (ã) dã ≥ h1,ε + h2,ε (51)
We split the integral operator applied to U ε in two parts

U ε (t)- t 0 U ε (ã)K ε (ã) dã = U ε (t) - 1 µ 0 t/ε 0 U ε (t -εa) (a) da = 1 µ 0 +∞ 0 (U ε (t) -U ε (t -εa)) (a) da :=H1,ε + 1 µ 0 +∞ t/ε U ε (t -εa) (a) da :=H2,ε
and we shall specify U ε such that H 1,ε ≥ Hε (t) and H 2,ε ≥ 0. To this end we set

U ε (t) := ε(K 0 + K 1 t), ∀t ∈ R (52)
with constants K 0 and K 1 to be specified. One has obviously that

H 1,ε (t) = ε 2 K 1 µ 1 µ 0 ≥ ε 2 C 1 ≥ Hε (t)
a.e. on R + , provided that K 1 is chosen as

K 1 > µ 0 µ 1 C 1
Using (52) and the change of variable ã = -t/ε + a, we obtain that

H 2,ε = +∞ 0 (K 0 -εK 1 ã) (ã + t/ε) dã = (K 0 -εK 1 A ε [ ](t)) R+ t ε + a da
We are in the hypotheses of Lemma 14 : A ε [ ](t), the expectation of a given density with respect to the tail a > t/ε is bounded by a positive constant

A max A ε [ ](t) := R+ a ( t ε + a) da R+ ( t ε + a) da ≤ A max .
Therefore it suffices to chose K 0 > εK 1 A max in order to obtain that H 2,ε ≥ 0. These computations show that U ε is a super-solution. Then the comparison principle implies that, for all 0

≤ t ≤ T , 0 ≤ Ẑε (t) = t 0 |z ε (s) -z 0 (s)| ds ≤ U ε (t) = ε(K 0 + K 1 t) → 0 as ε → 0, hence Ẑε → 0 in C([0, T ]), which ends the proof since z ε -z 0 L 1 (0,T ) ≤ Ẑε C([0,T ]) .

A simple example

We construct by hand solutions of problems ( 1) and (3) when the load f is explicitly defined as

f (t) :=      1/2 if 0 < t ≤ 1 3 , 1 if 1 3 < t ≤ 2 3 , 3/2 if 2 3 < t < 1, (53)
and the kernel is a simple exponential (see more precise statements below). So defined f is of course of bounded variation on (0, 1). The solution z ε solving (1) and its limit z 0 show different regularities (see Figure 1) : the adhesive approximation is rougher than the limit solution. This is an interesting feature of our approach. Assumptions 4. i) the load f is defined in (53), ii) the on and off rates are constants defined as :

ζ ε = ζ 0 = ζ, β ε = β 0 = β.
iii) the initial condition is at equilibrium :

ρ I,ε = ρ 0 = βζ β + ζ e -ζa .
Lemma 15. Under Assumptions 4, one has that µ

0,ε = µ 0,0 = β/(β + ζ), µ 1,ε = µ 1,0 = β/(ζ(β + ζ)
) and ρ 0 (a) = µ 0,0 ζe -ζa . Then the solution z ε of ( 1) is BP V ([0, 1]) and it is explicitly given by

z ε (t) = t 0 f µ 1,0 ds + ε f (t) µ 0,0 + 1 µ 0,0 +∞ 0 z p (-εa)ρ 0 da and hence, z ε (t) -z 0 (t) = ε f (t) µ 0,0 + 0 -∞ z p (s) exp ζs ε ds with z 0 (t) = z p (0) + t 0 f (s)ds/µ 1,0 is a continuous functions in [0, 1]
. Note that the last term is an ε order term according to Assumption 1, indeed it holds that

0 -∞ z p (s) exp ζs ε ds ≤ ε ζ z p W 1,∞ (R-) .
Proof. The specific choice of data and kernel allows to rephrase (1) as

z ε (t) - ζ ε t 0 z ε (s) exp - ζ(t -s) ε ds = ε f (t) µ 0,0 + ζ ε 0 -∞ z p (s) exp - ζ(t -s) ε ds.
Next, setting

q ε (t) = z ε (t) exp ζt ε , t ≥ 0.
Then we can rewrite (15) for all t ≥ 0 as

q ε (t) - ζ ε t 0 q ε (s)ds = ε exp ζt ε f (t) µ 0,0 + ζ ε 0 -∞ z p (s) exp ζs ε ds. (54)
By differentiating (54) in time and using (6), we prove that q ε solve the equation and therefore q ε is explicitly given by

         qε (t) - ζ ε q ε (t) =
q ε (t) = exp ζt ε ε f (t) µ 0,0 + ζ ε 0 -∞ z p (s) exp ζs ε ds + t 0 f (s) µ 1,0 ds
which gives the formula of z ε . Moreover, it is clear that z ε is of bounded variation since f is it and z 0 is an absolutely continuous function.

A Auxiliary proofs

In this appendix, the domain Ω is an open set of R n .

A.1 Proof of Theorem 1

As in the proof of [9, Theorem 14.9], the aim is to show that for every δ > 0, there exists a sequence {f δ } δ in C ∞ (Ω) such that Since the total variation Df (Ω) is bounded, lim j→+∞ Df (Ω \ {t, dist(t, ∂Ω) > 1/j, |t| < j}) = 0 then for fixed δ > 0, there exists a j 0 ∈ N such that for all j ≥ j 0 , Df (Ω \ {t, dist(t, ∂Ω) > 1/j, |t| < j}) ≤ δ.

For i ∈ N, we define the subdomain Ω i of Ω by Ω i := {t ∈ Ω, dist(t, ∂Ω) > 1/j 0 + i, |t| < j 0 + i} such that Ω i ⊂⊂ Ω i+1 and ∞ i=0 Ω i = Ω. Let W 0 = Ω 1 and W i = Ω i+1 \ Ωi-1 , where Ω -1 = Ω 0 := ∅, and let {φ i } be a partition of the unity subordinate to the covering

{W i } i∈N φ i ∈ C ∞ 0 (W i ), 0 ≤ φ i ≤ 1, ∞ i=0 φ i = 1.
For i ∈ N, the idea is to find δ i > 0 so small that supp χ δi * (φ i f ) ⊂ W i (56) which allows to write

χ δi * (φ i f ) -φ i f L 1 (Ω) = χ δi * (φ i f ) -φ i f L 1 (Wi) (57)
and then from the local approximation Theorem (see Theorem 2 p.125 in [START_REF] Evans | Measure theory and fine properties of functions[END_REF]),

χ δi * (φ i f ) -φ i f L 1 (Wi) < Cδ
and also by convenience, we can take C = 1 2 i . Hence we conclude that

Ω |χ δi * (φ i f ) -φ i f | dt < δ 2 -i (58) Ω |χ δi * (f φ i ) -f φ i | dt < δ 2 -i . ( 59
)
for a positive mollifiers χ δ defined as

χ δ (t) := 1 δ χ( t δ ), χ(t) = 1 if |t| < 1 0 if |t| > 2 Define f δ = ∞ i=0 χ δi * (φ i f ). ( 60
)
By the construction of {W i }, we have W i ∩ W i+1 = ∅ and W i ∩ W i+1 ∩ W i+2 = ∅ which give that for every t ∈ Ω, #{i ∈ N : χ δi * (φ i f )(t) = 0} ≤ 2 and since the finite sum of infinitely differentiable functions is infinitely differentiable, we conclude that f δ ∈ C ∞ (Ω) and Let ψ ∈ C ∞ c (Ω) be such that |ψ| ∞ ≤ 1. Since φ i f ∈ BV(Ω) for every i ∈ N, we have by Lemma 14.10 in [START_REF] Leoni | A first course in Sobolev spaces[END_REF] that

Ω |f δ -f |dt ≤ ∞ i=1 Ω |χ δi * (φ i f ) -φ i f | dt < ∞ i=1 δ 2 i ≤ δ,

  is a solution of (23). For more details see[START_REF] Milišić | On a structured model for load-dependent reaction kinetics of transient elastic linkages mediating nonlinear friction[END_REF] Lemma 4.1 and Lemma 4.2].

A 1 [

 1 t) da by using hypothesis iv) from Assumptions 3, one has exp(-ζ max a) ≤ q(a, t) ≤ exp(-ζ min a) , t)da ≤ +∞ 0 a exp(-ζ min a)da = 1 ζ 2 min which shows the final upper bound. Proposition 3. Under Assumptions 3, for 0 < t < T one has the estimates : Hε (t) := t 0 hε (τ ) dτ ≤ ε 2 C 1

Figure 1 :

 1 Figure 1: The solutions z ε and z 0 as a function of time, for a fixed ε = 10 -1 on the left y-axis. The load f on the right y-axis.

  t) + εf (t)) , t > 0, q ε (0 + ) = εf (0)/µ 0

Ω

  |f -f δ | dt < δ and |Df δ |(Ω) < Df (Ω) + δ.

∞

  i=0 f φ i . Thus f δ → f in L 1 (Ω) as δ → 0. Using the theorem of Lower semi-continuity of variation measure (see chap 5,[START_REF] Evans | Measure theory and fine properties of functions[END_REF]), we haveDf (Ω) ≤ lim inf δ→0 |Df δ |(Ω).(61)it remains to show that lim sup δ→0 |Df δ |(Ω) ≤ Df (Ω).

Ωχ

  δi * (φ i f ) ψ dt = Ω φ i f (χ δi * ψ) dt (62)let ψ δi = χ δi * ψ, and using the fact that support of ψ is compact and that the partition of unity is locally finite, we have thatΩ f δ ψ dt = +∞ i=1 Ω χ δi * (φ i f ) ψ dt = +∞ i=1 Ω φ i f ψ δi dt = +∞ i=1 Ω f ((φ i ψ δi ) -φ i ψ δi ) dt := I 1 + I 2 since supp(φ i ψ δi ) ⊂ W i and |φ i ψ δi | ∞ ≤ 1, it follows that I 1 = Ω f (φ i ψ δi ) dt + +∞ i=2 Ω f (φ i ψ δi ) dt ≤ Df (Ω) + +∞ i=2 Df (W i ) ≤ Df (Ω) + 3 Df (Ω \ Ω 1 )≤ Df (Ω) + 3δ since each t ∈ Ω belongs at most two of the sets U i . On the other hand, by Fubini's theorem we haveI 2 = -+∞ i=1 Ω χ δi * (f φ i )ψ dt = -+∞ i=1 Ω (χ δi * (f φ i ) -f φ i ) ψ dtby using the fact that +∞ i=1 φ i = 0. We now use (59) and the fact that |ψ| ∞ ≤ 1, to conclude that I 2 ≤ δ and then we obtain thatΩ f δ ψ dt ≤ Df (Ω) + 3δ.By taking the supremum and passing to the limit when δ → 0, we obtain that lim sup δ→0 Df δ (Ω) ≤ Df (Ω) (63)Finally, (61) and (63) together concludes the proof.

A.2 Proof of Lemma 5:

Let f ∈ BV(Ω) ∩ L ∞ (Ω). By using (60) and (56) we can write for all t in Ω,

which ends the proof.