A CHARACTERIZATION OF ENERGETIC AND BV SOLUTIONS TO ONE-DIMENSIONAL RATE-INDEPENDENT SYSTEMS

Abstract. The notion of BV solution to a rate-independent system was introduced in [8] to describe the vanishing viscosity limit (in the dissipation term) of doubly nonlinear evolution equations. Like energetic solutions [5] in the case of convex energies, BV solutions provide a careful description of rateindependent evolution driven by nonconvex energies, and in particular of the energetic behavior of the system at jumps. In this paper we study both notions in the one-dimensional setting and we obtain a full characterization of BV and energetic solutions for a broad family of energy functionals. In the case of monotone loadings we provide a simple and explicit characterization of such solutions, which allows for a direct comparison of the two concepts.


1.
Introduction. Over the last decade, the analysis of rate-independent systems has received notable attention (see e.g. [5] for a thorough survey of applications). The analytical theory of rate-independent evolutions encounters some mathematical challenges, which are apparent even in the simplest example of rate-independent evolution, viz. the doubly nonlinear differential inclusion ∂Ψ(u (t)) + DE t (u(t)) 0 in X * a.e. in (0, T ). (1) Here X * is the dual of a finite-dimensional linear space (and in the sequel we focus our analysis on the simplest case X = R), DE t is the (space) differential of a timedependent energy functional E ∈ C 1 (X × [0, T ]; R), and Ψ : X → [0, +∞) is a convex and nondegenerate dissipation potential: rate-independence requires that Ψ is positively homogeneous of degree 1. Notice that we do not allow Ψ to take the value +∞, which rules out, e.g., irreversible rate-independent evolution. It follows from the above conditions that the range of ∂Ψ equals K * := ∂Ψ(0), which is a proper convex subset of X. Hence, if E t (·) is not strictly convex, one cannot expect the existence of an absolutely continuous solution to (1). It turns out that the natural space for candidate solutions u of (1) is BV([0, T ]; X), and where Var Ψ is the pointwise total variation with respect to Ψ (see (14) for the definition). Let us emphasize that the energetic formulation neither involves the differential DE of the energy, nor derivatives of the function t → u(t), nor the gradient DΨ (which does not exists in 0). Thus, it is well suited to deal with nonsmooth energies and jumping solutions. Furthermore, as shown in [4,5] this formulation can be considered and analyzed in very general ambient spaces, even with no underlying linear structure. Because of these features, the energetic concept has been exploited in several applicative contexts, see [5,6], and the references therein.
In the case of nonconvex energies, the global stability condition (S) may lead the system to change instantaneously in a very drastic way, jumping into very far-apart energetic configurations. A different dynamical approach has been proposed in [8] (see also [2,7]), by considering rate-independent evolution as the limit of systems with smaller and smaller viscosity. One can thus address the viscous approximation of (1), viz.
∂Ψ ε (u (t)) + DE t (u(t)) 0 in X * a.e. in (0, T ), (2) where in the simplest case we have In fact, we focus on (3) just for simplicity, since much more general regularizations can be considered, see [8]. The main result of [8] is that any limit point as ε ↓ 0 of a family (u ε ) ε of solutions to (2) is a curve u ∈ BV([0, T ]; X) fulfilling the local stability condition −DE t (u(t)) ∈ K * for a.e. t ∈ (0, T ), (S loc ) and the energy balance Notice that (E Π,E ) features the (pseudo)-total variation Var Π,E , suitably defined from the vanishing viscosity contact potential Π(v, w) := Ψ(v) max 1, Ψ * (w) for (v, w) ∈ X × X * with Ψ * (w) := inf Ψ(v)≤1 w, v . We refer to Section 2 for all details on the definition of Var Π,E in terms of Π and E.
Still, let us emphasize that, in general, both the rate-independent and the viscous dissipation contribute to Π, and thus to the energy balance (E Π,E ) via the (pseudo)-total variation Var Π,E . In contrast, in the energy balance (E) for energetic solutions only the rate-independent dissipation is involved. In fact, (E Π,E ) reflects the main feature of BV solutions, viz. that rate-independent and viscous effects are encompassed in the description of the solution jump trajectories, in order to provide finer (in comparison to energetic solutions) information on the behavior of the system. This aspect was fully explored in [8], with a thorough description of the non-jumping and jumping regimes and the related energy balances, see also Proposition 2 later on.
Moreover, we point out that, contrary to (S), (S loc ) is a local stability condition. Therefore, one expects BV solutions to jump "later" and "less abruptly" than energetic solutions.
The latter crucial property is clearly revealed by the characterization of energetic and BV solutions, which we provide in the present paper. Like in the papers [12,13], which also provide explicit characterizations of rate-independent evolution, we focus on the one-dimensional case X = R. More precisely, , we consider energies E : R × [0, T ] → R and dissipation potentials Ψ : R → [0, +∞) of the form where W ∈ C 1 (R) is a (possibly nonconvex) energy, ∈ C 1 ([0, T ]) a given external loading, δ ± > 0, and u + , u − respectively denote the positive and negative parts of u. Then, (1) reduces to the rate-independent ODE Our main Theorems 3.1 and 5.1 characterize BV and energetic solutions to (6), and provide the starting point for the explicit representation formulae given by Theorems 4.3 and 6.3 when the loading function is monotone. In the case of a strictly increasing map and of an initial datumū satisfying a slightly stronger (local/global) stability condition, we have the following simple descriptions: • u is a BV solution to (6) if and only if it fulfills where J u is the jump set of u. In terms of the upper monotone envelope of W , defined by m u0 (u) := max u0≤v≤u W (v) (see §4.1), (7) can also be written as • u is an energetic solution to (6) if and only if it fulfills where ∂W (·; u 0 ) is the (convex analysis) subdifferential (see §6.1) of the function Introducing the convex envelope W * * (·; u 0 ) of W (·; u 0 ) (whose definition is given in (92)) and its derivative m u0 (u) := DW * * (u; u 0 ) for u > u 0 , (8) also yields the following equation for the energetic solution u: Therefore, BV and energetic solutions depend monotonically on increasing loadings. Furthermore, BV solutions are governed by the upper monotone envelope of W , whereas energetic solutions involve the derivative of the convexified energy (9). In both cases, the initial condition u 0 provides crucial information to construct the above monotone graphs. The case of a decreasing loading can be easily recovered from the increasing one thanks to the symmetry principle recalled in Proposition 3. The concatenation principle allows us to extend immediately our results to the case of piecewise monotone loadings. In Examples 1 and 3, we illustrate (7) and (8) in the simple, yet significant case of the double-well potential In this context, the input-output relation → u given by (10) for energetic solutions, corresponds to the so-called Maxwell rule, cf. the discussion in [14, §I.3]. The latter evolution mode prescribes that for all t ∈ [0, T ], the function u(t) only attains absolute minima of the function u → W (u)−( (t)−δ + )u. This corresponds to convexification of W , and causes the system to jump "early" into far-apart configurations. Instead, the evolution mode (7) follows the Delay rule, related to hysteresis behavior. The system accepts also relative minima of u → W (u) − ( (t) − δ + )u, and thus the function t → u(t) tends to jump "as late as possible". The plan of the paper is as follows: in Section 2, we recall some definitions and preliminary properties of BV functions, energetic and BV solutions in a finitedimensional setting. Section 3 is devoted to a refined characterization of BV solutions in the one-dimensional setting. The case of monotone loadings is carefully analyzed in Section 4, after some auxiliary results on upper and lower monotone envelopes of functions. One-dimensional energetic solutions are studied in Section 5; their explicit characterization when is monotone is carried out in the last Section 6.

2.
Preliminaries. In this section we recall some notation and properties related to functions in BV([0, T ]; X), with X a finite-dimensional vector space, and to energetic and BV solutions of a general rate-independent system. 2.1. BV functions. Hereafter, we shall consider functions of bounded variation u ∈ BV([a, b]; X) to be pointwise defined at every time t ∈ [a, b]. Notice that a function u ∈ BV([a, b]; X) admits left (resp. right) limits at every t ∈ (a, b] (resp. t ∈ [a, b)), viz. ∃ u l (t) = lim s↑t u(s) and ∃ u r (t) = lim s↓t u(s). We also adopt the convention u l (a) := u(a), u r (b) := u(b). The pointwise jump set J u of u is the at most countable set defined by We denote by u the distributional derivative of u (extended by u(a) in (−∞, a) and by u(b) in (b, +∞)) in D (R): it is a Radon vector measure with finite total variation |u | supported in [a, b]. In the one-dimensional case X = R, u admits the Hahn decomposition u = (u ) + − (u ) − as the difference of two positive and mutually singular measures, such that |u | = (u ) + + (u ) − . It is well known [1] that u can be decomposed into the sum of its diffuse part u co and its jump part u J : u = u co + u J , u J := u ess-J u , so that u co ({t}) = 0 for every t ∈ [a, b]. (12) 2.2. Energetic solutions to rate-independent systems. We consider a general rate-independent system (X, E, Ψ), where the dissipation potential for some W ∈ C 1 (X) and ∈ C 1 ([a, b]; X * ). We shall also use the notation ∂ t E t (u) := − (t), u for the partial time derivative of E, and we set for w ∈ X * . We recall the notion of energetic solution to the rate-independent system (X, E, Ψ), cf. [9,10,5].
and the energy balance where denotes the pointwise Ψ-total variation of u on the interval [a, t].
The following characterization of energetic solutions has been proved in [8,Prop. 2.2].  du co dµ (t) + DW (u(t)) (t) for µ-a.e. t ∈ (a, b), µ := L 1 + |u co |, (DN) and the following jump conditions at each point t ∈ J u : The jump conditions (J ener ) show that, in the case of an energetic solution u the jump set J u coincides with the essential jump set ess-J u .
2.3. BV solutions to rate-independent systems. As we mentioned in the introduction, we shall restrict to BV solutions arising in the vanishing viscosity limit of (2), with dissipation potentials Ψ ε of the form (3). In such a setting, the vanishing viscosity contact potential Π : X × X * → [0, +∞) associated with the family (Ψ ε ) ε is given by For a fixed t ∈ [a, b], the (possibly asymmetric) Finsler cost induced by Π and (the differential of) E at the time t is for everyū, u 1 ∈ X given by As already observed in [8], it is not difficult to check that the infimum in (16) is always attained by a Lipschitz curve ϑ ∈ Lip([r 0 , r 1 ]; X) such that Π(θ(r), −DE t (ϑ(r))) ≡ 1 for a.a. r ∈ (r 0 , r 1 ).
∆ Π,E (t; u l (t), u(t)) + ∆ Π,E (t; u(t), u r (t)) , (17) and the associated (pseudo-)total variation is where µ can be any nonnegative and diffuse reference measure, provided that u co is absolutely continuous w.r.t. µ (e.g. µ = L 1 + |u co |). In fact, since Ψ is 1homogeneous, the the value of the integral on the right-hand side of (18) is independent of µ (see e.g. [1]). Then, we are in the position to recall the definition of BV solution given in [8].
and the (Π, E)-energy balance In [8,Sec. 4] the following result has been proved.
As in the case of energetic solutions, (J BV ) yield that, for a BV solution u the jump set J u coincides with the essential jump set ess-J u .
2.4. Symmetry and concatenation principles for energetic and BV solutions. We state here two useful properties of energetic and BV solutions. Let us first introduce the modified energy and dissipation potentials is an energetic solution of the rate-independent system (X, E, Ψ) (resp. a BV solution of the rateindependent system (X, E, Π)) if and only if the curveũ(t) := −u(t) is an energetic solution of the rate-independent system (X,Ẽ,Ψ) (resp. of the rate-independent system (X,Ẽ,Π)).
The proof follows from easy calculations, observing that Another simple property concerns the behavior of energetic and BV solutions with respect to restriction and concatenation. The proof is trivial.

The restriction of an energetic
X is an energetic (resp. BV) solution in each one of the intervals [t j−1 , t j ], j = 1, · · · , M , then u is an energetic (resp. BV) solution in [a, b].

2.5.
The one-dimensional setting. From now on we consider the particular case X = R, which we also identify with X * . We will denote by v + , v − the positive and negative part of v ∈ R. Dissipation. A dissipation potential is a function of the form Hence, we have

RICCARDA ROSSI AND GIUSEPPE SAVARÉ
Energy functional. The energy is given by a function E : 3. BV solutions of rate-independent systems in R. In this section we will provide an equivalent characterization of BV solutions to the rate-independent system (R, E, Π), in the one-dimensional setting considered in §2.5.
if and only if the following properties hold: a) u satisfies the local stability condition (equivalent to (S loc )) and they coincide in (a, b): we denote their common value by W (u lr ). c) u satisfies the following precise formulation of the doubly nonlinear differential inclusion (DN) with obvious modifications at t = a, t = b as in point b). d) At each jump point t ∈ J u , u satisfies the jump conditions and for every ϑ such that min u l (t), u r (t) ≤ ϑ ≤ max u r (t), u l (t) . In particular, Proof. We split the argument in various steps. Claim 1: The local stability condition (S loc ) is equivalent to (S loc,R ). It is sufficient to recall that K * = [−δ − , δ + ]. Claim 2: the jump conditions (J BV ) are equivalent to (25) and (26).
Taking (15) and (21) into account, the Finsler cost ∆ Π,E (t; ·, ·) in fact reduces (up to a linear reparametrization) to Let us consider, e.g., the caseū ≤ u 1 and notice that, if ϑ ∈ AC([0, 1]; R) fulfils ϑ(0) =ū and ϑ(1) = u 1 , then, setting Therefore, the value of the integral in (27) surely diminishes if we just consider the restriction of ϑ to the interval [r 0 , r 1 ]. Hence we can assume that the range of a minimizing curve in (27) We can also suppose that the competing curves ϑ in (27) are nondecreasing. Indeed, if ϑ is absolutely continuous and connectsū to u 1 , we can consider the curveθ(r) := max s∈[0,r] ϑ(s). It is easy to check thatθ is nondecreasing and absolutely continuous, since for all 0 ≤ r 1 ≤ r 2 ≤ 1 It follows thatθ = ϑ a.e. on the coincidence set {θ = ϑ}, whereas one can easily check thatθ (r) = 0 whereθ(r) > ϑ(r) (viz., whereθ(r) = ϑ(r)). From the above considerations we obtain Therefore, it is not restrictive to assume that any minimizing curve ϑ is nondecreasing on [0, 1]. Then, with a change of variable, from (27) we deduce the identity Notice that and moreover, at a jump point t with u l (t) < u r (t) there holds Comparing (J BV ) with (28) and (31) we immediately see that the first condition of (J BV ) is equivalent tothe first of (26). If (25) also holds, then we easily get the other two conditions of (J BV ). Conversely, (J BV ) yields and this identity implies (25) by the positivity of the integrand in (28). The case u l (t) > u r (t) can be studied by a similar argument, see also Proposition 3. Claim 3, sufficiency: a function u satisfying a) -d) is a BV solution.
In view of Prop. 2 and Claim 2, we simply have to check that the differential inclusion (DN) holds: it follows from (23), (24), and (S loc,R ), since µ is diffuse and therefore u r (t) = u l (t) = u(t) for µ-a.e. t ∈ [a, b]. Claim 4: a) and d) imply b).
It is immediate, since the continuity of W and yields that the inequalities in (S loc,R ) hold also for u l and u r ; (26) provides the opposite inequalities. Claim 5, necessity: a BV solution u satisfies a) -d).
By the previous claims, it remains to check c). The identity in (23) is satisfied (u ) + -a.e. in [a, b] thanks to the differential inclusion (DN) and the jump/stability conditions (which yield (23) on the jump set). By Claim 4, we know that W (u lr ) is continuous, so that the identity in (23) holds on the support of (u ) + . The same argument applies to (24).
The previous general result has a simple consequence: a BV solution is locally constant in a neighborhood of a point where the stability condition (S loc,R ) holds with a strict inequality.
Proof. In view of (25) and (26), s is not a jump point of u, hence u is continuous at s and the set {t ∈ we have seen that J is open, and it is easy to check that it is also closed in (α, β), so that J = (α, β) and the thesis follows. 4. BV solutions of the rate-independent system (R, E, Π) with monotone loadings. As mentioned in the introduction, BV solutions of rate-independent systems in R, driven by monotone loadings, involve the notion of the upper and lower monotone (i.e. nondecreasing) envelopes of the graph of a given function (W in our setting). In this section we first focus on a few properties of these maps and their inverses, and then we exhibit the explicit formulae characterizing BV solutions when is increasing or decreasing.
We call mū(·) the upper monotone envelope of W in the interval (ū, +∞). The contact set is defined by The mapping mū(·) is monotone and surjective thanks to (22); it is single-valued on (ū, +∞) (where we identify the set mū(u) with its unique element with a slight abuse of notation). We can thus consider the inverse graph pū(·) : R ⇒ R of mū(·): it is defined by u ∈ pū( ) ⇔ ∈ mū(u) for u, ∈ R.
The identity W (pū r ( )) = is due to the continuity of W and the second property of (44). To prove the latter, we observe that, when u > pū r ( ) we have mū(u) > , and we know that there exists v ∈ [pū r ( ), u] such that W (v) > . Since u is arbitrary we get pū r ( ) ≥ inf{u ≥ū : W (u) > }. The converse inequality follows from (42).

4.2.
The lower monotone envelope of W . In a completely similar way we can introduce the maximal monotone map below the graph of W on the interval (−∞,ū], viz.

4.3.
Monotone loadings and BV solutions. We apply the notions introduced in §4.1 to characterize BV solutions when is monotone. First of all, we provide an explicit formula yielding BV solutions for an increasing loading . The case of a decreasing and of a piecewise monotone loading will easily follow by applying Propositions 3 and 4.
is a BV solution of the rate independent system (R, E, Π) of §2.5. In particular, (51) yields and (52) is equivalent to (51) when is strictly increasing.
Applying Proposition 3 and the discussion of §4.2 we have: is a BV solution of the rate independent system (R, E, Π). In particular, (55) yields and (56) is equivalent to (55) when is strictly decreasing.
The next result states that, under a slightly stronger condition on the initial data, any BV solution driven by an increasing loading admits the representation (51).
Then u can be represented as in Theorem 4.3, i.e. it satisfies and Proof. We apply Theorem 3.1 and we split the argument in a few steps; as usual we use the short-hand notationū := u(a).
If a ∈ Σ we set α := a and Σ a := ∅. If a ∈ Σ we denote by Σ a the connected component of Σ containing a and we set α = max Σ a . If α > a then contrary to (58)), so that u r (a) = u(a) =ū. Since is nondecreasing, with a similar argument and still invoking (58) we conclude that u(t) = u r (t) ≡ū and (t) ≡ (a) in [a, α]. If α = b the claim is proved.
The first statement follows from Claim 1 and Lemma 3.2. To prove the second property in (63), we argue by contradiction and we suppose that a point s ∈ (β, b] exists such that if W (u(s)) + δ + > (s). Since is nondecreasing, Claim 1 and Lemma 3.2 show that u(t) ≡ u(s) for every t ∈ [α, s], so that s ≤ β.
The first identity in (63) then follows by a continuity argument and the stability condition (S loc,R ). Inequality (64) ensues from (63) and the first characterization in (44). Claim 4: For every t ∈ [a, b] we have u r (t) ≤ pū r ( (t) − δ + ). If u r (t) =ū there is nothing to prove. Otherwise, let t ≥ β and take z ∈ (ū, u r (t)). Since u is nondecreasing, there exists s ∈ [β, t] such that u l (s) ≤ z ≤ u r (s), so that by (63) and (26) since is nondecreasing. Being z < u r (t) arbitrary, the claim follows from the second of (44). Conclusion: Relation (60) follows from Claim 2, (64), and Claim 4. At every continuity point t for u we have u l (t) = u r (t) = u(t), so that (59) is due to (60) and (63).
A straightforward consequence of Proposition 3 and the discussion of §4.2 concerns the characterization of BV solutions in the case of a decreasing load: it can be deduced from the analysis of the increasing case.
Corollary 2 (Nonincreasing loadings). Letū ∈ R, let ∈ C 1 ([a, b]) be a nonincreasing loading satisfying (a) ≤ W (ū) + δ + , and let us suppose that W > W (ū) in a right neighborhood ofū if (a) = W (ū) + δ + . (65) and therefore We conclude this section with some examples illustrating the previous results. In particular, Example 2 shows that the characterization (60) may not hold if the loading does not comply with (58).
Example 1 (BV solutions driven by a double-well energy). The double-well potential energy clearly fulfills condition (22). Note that W (u) = u 3 − u, and Let us set * := W (u 1 ) = 2 √ 3 9 , and, for later convenience, We also introduce the intervals There are points t 1 ∈ (0, 1/2) : (t 1 ) = * + δ + ; All the BV solutions are then given by Example 2 (Bifurcation of BV solutions driven by critical loadings). We consider the same potential energy (68) as in Example 1, but we suppose now that (1/2) = * + δ + (in this case (65) is not satisfied at a = 1/2). In addition to the solution considered before, we have another family of solutions, among which e.g.
5. Energetic solutions of rate-independent systems in R. In this section we provide a general characterization of energetic solutions to the rate-independent system (R, E, Ψ) considered in §2. 5.
In order to express the global stability and jump conditions for energetic solutions, let us introduce the following one-sided global slopes where the subscripts i r and s l stands for inf-right and sup-left respectively. They satisfy and it not difficult to check that they are continuous. Indeed, it is sufficient to introduce the continuous function V : and observe, e.g. for W i,r , that W i,r (u) = min{V (u, z) : z ≥ u} and for u in a bounded set the minimum is attained in a compact set thanks to (22). Taking (72) into account, we observe that the global stability condition (S) can be reformulated as the system of inequalities for all t ∈ [a, b] The continuity property of the one-sided slopes also yields for all t ∈ [a, b] We can state the main characterization theorem concerning energetic solutions, which the reader may compare with Thm. 3.1 for BV solutions. W (u r (t)) = W i,r (u r (t)) = (t) − δ + for every t ∈ S + := supp (u ) + , W (u r (t)) = W s,l (u r (t)) = (t) + δ − for every t ∈ S − := supp (u ) − .
c) At each point t ∈ J u , u fulfills the jump conditions: and In particular, u is locally constant in the open set Since any jump point belongs either to the support of (u ) + , or of (u ) − , combining (78), (76), and (79) and (78), (77), and (80) we also get at every jump point We now develop the proof of Thm. 5.1.
Proof. It is easy to check that, if u ∈ BV([a, b]; R) satisfies conditions a) -c) then u is an energetic solution to (R, E, Ψ): we omit the simple details. We discuss here the converse implication. Hence, let u be an energetic solution of (R, E, Ψ). We have already shown point a); let us first consider point c). The first property of (78) easily follows by summing the identities of the jump conditions (J ener ), thus obtaining Ψ(u r (t) − u l (t)) = Ψ(u r (t) − u(t)) + Ψ(u(t) − u l (t)).
In particular, this implies that min(u l (t), u r (t)) ≤ u(t) ≤ max(u r (t), u l (t) , so that u(t) is a convex combination of u l (t) and u r (t) with a uniquely determined coefficient θ ∈ [0, 1]. Conditions (J ener ) then yield the corresponding property for W (u(t)).
Let us now consider, e.g., the case u l (t) < u r (t) and prove (79). From the first of the jump conditions (J ener ) and the definition of the right global slope W i,r (·), we find Combining this inequality with (75) we conclude that the identities in (79) hold. If now z ∈ [u l (t), u r (t)) we obtain ≥ W i,r (u l (t))(z − u l (t)) − W i,r (u l (t))(u r (t) − u l (t)) = W i,r (u l (t))(z − u r (t)), where the second inequality ensues from the definition (71) of W i,r . Dividing by z − u r (t) we then have that W i,r (z) ≤ W i,r (u l (t)). The proof of (80) is completely analogous.
Concerning b), we notice that (DN) yields so that (76) holds by continuity and by (74) in supp u + \ J u . On the other hand, for every t ∈ J u ∩ supp u + we have u l (t) < u r (t) so that, dividing inequality (84) by z − u r (t) and passing to the limit as z ↑ u r (t) we obtain and applying (74) we conclude the proof of (76). The identities in (77) follow by the same argument. It remains to check (81): by (79) and (80) any t ∈ I is a continuity point for u; the continuity properties of W i,r (·) and W s,l (·) then show that a neighborhood of t is also contained in I, so that I is open and disjoint from J u . Relations (76) and (77) then yield that u = 0 in the sense of distributions in I, so that u is locally constant.
6. Energetic solutions of the rate-independent system (R, E, Ψ) with monotone loadings.
6.1. Convex envelopes and their subdifferentials. This section is devoted to some preliminary convex analysis results, which turn out to be useful for the characterization of energetic solutions driven by monotone loadings.
Let W : R → (−∞, +∞] be a function with proper non-empty domain D(W) := {u ∈ R : W(u) < ∞}. For our purposes, we will assume that D(W) is a closed interval, W is of class C 1 in D(W) and it is bounded from below.
(86) The (convex analysis) subdifferential of W is the multivalued map ∂W : R ⇒ R defined by Clearly, ∂W(u) is empty if u ∈ D(W). In the present one-dimensional setting, we have a simple characterization of the subdifferential in terms of the one-sided slopes defined in (71): If u ∈ int D(W) , then ∂W(u) either coincides with {W (u)} or it is empty. By (88), the former case can also be characterized by if this is the case, the common value on the right-hand side of (89) is the unique element of ∂W(u).
The Fenchel-Moreau conjugate of W is the function and a further iteration of the conjugation yields Indeed, W * * coincides with the l.s.c. convex envelope of W, i.e. the maximal convex and l.s.c. function less than W. It can also be defined by We introduce the contact set where the latter set is the convex hull of (∂W) −1 ( ). We apply the above notions to the functions W(·) := W (·;ū),ū ∈ R, defined by We consider their conjugates W * (·;ū), and the subdifferentials There is a last interesting property which relates the one-sided slopes W s,l (·), W i,r (·), their upper monotone envelopes, and the convex envelope of W(·) = W (·;ū). (103) Then, there holds ∂W * * (u;ū) = rū(u) for every u ∈ R.
(104) In particular, and Proof. Let us start by proving (104): it is sufficient to consider the case u >ū.
Theorem 6.2. Letū ∈ R and ∈ C 1 ([a, b]) be a nondecreasing loading such that is an energetic solution of (R, E, Ψ). In particular, (109) yields Notice that, if is strictly increasing, then any selection u(t) of pū c ( (t) − δ + ) is also strictly increasing, so that the second condition of (109) is automatically satisfied.
Proof. Notice that (108) and (109) yield Indeed, (111) holds also at t = a in view of (88) applied to W(u) := W (u;ū), since in this case W s,l (ū) = −∞ and W i,r (ū) = W i,r (ū), cf. (99). A further application of (88) yields the global stability condition (73). Since the subdifferential has a closed graph, we also have Let us set α := inf{t > a : u(t) > u(a)}. Since u(a) satisfies the global stability condition by (108), the function u is clearly a constant energetic solution in [a, α]. Thus, Proposition 4 shows that it is not restrictive to assume that α = a.
In this case, u r (t) > u(a) for every t > a, so that u r (t) belongs to the interior of the domain of W (·;ū); it follows from (109) that W (u r (t)) = W i,r (u r (t)) = (t) − δ + in (a, b], and also at t = a by passing to the limit in the equation as t ↓ a. The formulation (76) of (DN) is thus satisfied in [a, b].
Let us check now the point c) of Theorem 5.1 concerning the jump conditions. If t ∈ J u , then in view of (112) pū c ( (t) − δ + ) contains two distinct points u l (t) < u r (t), so that the graph of W * * (·;ū) is linear on [u l (t), u r (t)]. Since u l (t), u(t), u r (t) belong to the contact set Cū, in view of (95) the jump conditions (78) and (79) are also satisfied, and therefore u is an energetic solution.
Then σ ∈ J u . We argue by contradiction and assume that σ ∈ J u . In view of (82a), necessarily u l (σ) > u r (σ), and (77) shows that u r is nondecreasing in [σ , σ). Let R := {ρ ∈ [a, σ] : u r (t) ≡ u l (σ), (t) ≡ (σ) for all t ∈ [ρ, σ]}. R is clearly closed and contains σ. Let us show that R is open in [a, σ]: it is sufficient to prove that for every ρ ∈ R∩(a, σ], the set R contains a left neighborhood of ρ (R obviously contains also a right neighborhood of ρ if ρ < σ).
When ρ > σ we can easily check that we can apply Claim 2. Indeed, u r (t) ≤ u r (ρ) in a left neighborhood of ρ since u r is nondecreasing in [σ , σ). Further, ρ cannot be a jump point for u thanks to Claim 1, which prevents a jump point with u l (ρ) < u r (ρ), and the fact that (ρ) − W (u r (ρ)) > −δ − , which prevents a jump point with u l (ρ) > u r (ρ). By the way, this shows that R ⊃ [σ , σ].
(123) If a ∈ Σ we denote by Σ a the connected component of Σ containing a, and we set α = sup Σ a . If α > a, then W (u r (t)) = (t) + δ − for every t ∈ [a, α], so that by (76) u is nonincreasing in [a, α]. Assumption (114a) and the jump conditions (82b) imply that a ∈ J u and u r (a) = u(a). Since also is nondecreasing we conclude by (114b) and (77) that u(t) ≡ u(a) and (t) ≡ (a) in [a, α]; moreover, by the same argument, α ∈ J u so that α ∈ Σ. When a ∈ Σ we simply set α := a and Σ a = ∅. The Claim then follows if we show that Σ \ Σ a is empty. This is trivial if α = b. If α < b we suppose Σ \ Σ a = ∅ and we argue by contradiction.
To prove the second identity in (126) for u r (t), we argue by contradiction and we suppose that a point s ∈ (β, b] exists such that if W i,r (u r (s)) + δ + > (s). Then, by (82a) s is not a jump point, and therefore in view of point b) of Thm. 5.1, u