a characterization of energetic and solutions to one ... · key words and phrases. abstract...

DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2013.6.167DYNAMICAL SYSTEMS SERIES SVolume 6, Number 1, February 2013 pp. 167–191

A CHARACTERIZATION OF ENERGETIC AND BV SOLUTIONS

TO ONE-DIMENSIONAL RATE-INDEPENDENT SYSTEMS

Riccarda Rossi

Dipartimento di MatematicaUniversita di Brescia

Via Valotti 9, I–25133 Brescia, Italy

Giuseppe Savar

´

e

Dipartimento di Matematica “F. Casorati”Universita di Pavia

Via Ferrata 1, I– 27100 Pavia, Italy

Abstract. The notion of BV solution to a rate-independent system was intro-duced in [8] to describe the vanishing viscosity limit (in the dissipation term)of doubly nonlinear evolution equations. Like energetic solutions [5] in thecase of convex energies, BV solutions provide a careful description of rate-independent evolution driven by nonconvex energies, and in particular of theenergetic behavior of the system at jumps.

In this paper we study both notions in the one-dimensional setting and weobtain a full characterization of BV and energetic solutions for a broad familyof energy functionals. In the case of monotone loadings we provide a simple andexplicit characterization of such solutions, which allows for a direct comparisonof the two concepts.

1. Introduction. Over the last decade, the analysis of rate-independent systemshas received notable attention (see e.g. [5] for a thorough survey of applications).The analytical theory of rate-independent evolutions encounters some mathematicalchallenges, which are apparent even in the simplest example of rate-independentevolution, viz. the doubly nonlinear di↵erential inclusion

@ (u0(t)) + DEt

(u(t)) 3 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 focusour analysis on the simplest case X = R), DE

t

is the (space) di↵erential of a time-dependent energy functional E 2 C1(X ⇥ [0, T ];R), and : X ! [0,+1) is aconvex and nondegenerate dissipation potential: rate-independence requires that is positively homogeneous of degree 1. Notice that we do not allow to take thevalue +1, which rules out, e.g., irreversible rate-independent evolution.

It follows from the above conditions that the range of @ equals K

⇤ := @ (0),which is a proper convex subset of X. Hence, if E

t

(·) is not strictly convex, onecannot expect the existence of an absolutely continuous solution to (1). It turnsout that the natural space for candidate solutions u of (1) is BV([0, T ];X), and

2010 Mathematics Subject Classification. Primary: 34C55, 47J20, 49J40; Secondary: 74N30.Key words and phrases. Abstract evolution equations, rate-independent systems, energetic and

BV solutions, hysteresis e↵ects.The authors have been partially supported by a MIUR-PRIN 2008 grant for the project “Op-

timal mass transportation, geometric and functional inequalities and applications”.

167

168 RICCARDA ROSSI AND GIUSEPPE SAVARE

this fact has motivated the development of various weak formulations of (1), whichshould also take into account the behavior of u at jump points.

First and foremost, we recall the notion of (global) energetic solution, proposedby A. Mielke and coauthors, cf. [9, 10, 5]. For the simplified rate-independentevolution (1), an energetic solution is a curve u 2 BV([0, T ];X) satisfying twoconditions for all t 2 [0, T ]: the global stability

Et

(u(t)) Et

(z) + (z�u(t)) for every z 2 X, (S)

and the energy balance

Et

(u(t)) + Var (u; [0, t]) = E0(u(0)) +

Z

t

0@

t

Es

(u(s)) ds , (E)

where Var is the pointwise total variation with respect to (see (14) for thedefinition). Let us emphasize that the energetic formulation neither involves thedi↵erential DE of the energy, nor derivatives of the function t 7! u(t), nor thegradient D (which does not exists in 0). Thus, it is well suited to deal withnonsmooth energies and jumping solutions. Furthermore, as shown in [4, 5] thisformulation can be considered and analyzed in very general ambient spaces, evenwith no underlying linear structure. Because of these features, the energetic concepthas been exploited in several applicative contexts, see [5, 6], and the referencestherein.

In the case of nonconvex energies, the global stability condition (S) may lead thesystem to change instantaneously in a very drastic way, jumping into very far-apartenergetic configurations. A di↵erent dynamical approach has been proposed in [8](see also [2, 7]), by considering rate-independent evolution as the limit of systemswith smaller and smaller viscosity. One can thus address the viscous approximationof (1), viz.

@ "

(u0(t)) + DEt

(u(t)) 3 0 in X

⇤ a.e. in (0, T ), (2)

where in the simplest case we have

"

(v) := (v) +"

2 2(v). (3)

In fact, we focus on (3) just for simplicity, since much more general regularizationscan be considered, see [8]. The main result of [8] is that any limit point as " # 0of a family (u

"

)"

of solutions to (2) is a curve u 2 BV([0, T ];X) fulfilling the local

stability condition

�DEt

(u(t)) 2 K

⇤ for a.e. t 2 (0, T ), (Sloc)


Var⇧,E(u; [0, t]) + Et

(u(t)) = E0(u(0)) +

Z

t

0@

t

Es

(u(s)) ds for all t 2 [0, T ]. (E⇧,E)

Notice that (E⇧,E) features the (pseudo)-total variation Var⇧,E, suitably definedfrom the vanishing viscosity contact potential

⇧(v, w) := (v)max�

1, ⇤(w)�

for (v, w) 2 X ⇥X

⇤ (4)

with ⇤(w) := inf (v)1hw, vi. We refer to Section 2 for all details on the definitionof Var⇧,E in terms of ⇧ and E.

Still, let us emphasize that, in general, both the rate-independent and the vis-cous dissipation contribute to ⇧, and thus to the energy balance (E⇧,E) via the(pseudo)-total variation Var⇧,E. In contrast, in the energy balance (E) for energeticsolutions only the rate-independent dissipation is involved. In fact, (E⇧,E) reflects

CHARACTERIZATION OF ENERGETIC AND BV SOLUTIONS 169

the main feature of BV solutions, viz. that rate-independent and viscous e↵ectsare encompassed in the description of the solution jump trajectories, in order toprovide finer (in comparison to energetic solutions) information on the behaviorof the system. This aspect was fully explored in [8], with a thorough descriptionof the non-jumping and jumping regimes and the related energy balances, see alsoProposition 2 later on.

Moreover, we point out that, contrary to (S), (Sloc) is a local stability condi-tion. Therefore, one expects BV solutions to jump “later” and “less abruptly” thanenergetic solutions.

The latter crucial property is clearly revealed by the characterization of energeticand 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 energiesE : R⇥ [0, T ] ! R and dissipation potentials : R ! [0,+1) of the form

Et

(u) := W (u)� `(t)u, (u) = �+ u

+ + �� u

� for all (u, t) 2 R⇥ [0, T ], (5)

where W 2 C1(R) is a (possibly nonconvex) energy, ` 2 C1([0, T ]) a given externalloading, �± > 0, and u

+, u� respectively denote the positive and negative parts ofu. Then, (1) reduces to the rate-independent ODE

@ (u0(t)) +W

0(u(t)) 3 `(t) a.e. in (0, T ). (6)

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 byTheorems 4.3 and 6.3 when the loading function ` is monotone. In the case of astrictly increasing map ` and of an initial datum u 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

u is nondecreasing on [0, T ], W

0(u(t)) = `(t)� �+ for all t 2 [0, T ] \ Ju

, (7)

where Ju

is the jump set of u. In terms of the upper monotone envelope of W

0,defined by mu0(u) := max

u0vu

W

0(v) (see §4.1), (7) can also be written as

mu0(u(t)) = `(t)� �+, u0 := u(0), `(0)� �+ � W

0(u0).

• u is an energetic solution to (6) if and only if it fulfills

u is nondecreasing on [0, T ], @W (u(t);u0) 3 `(t)� �+ for all t 2 [0, T ], (8)

where @W (·;u0) is the (convex analysis) subdi↵erential (see §6.1) of the func-tion

W (u;u0) :=

(

W (u) if u � u0,

+1 if u < u0.(9)

Introducing the convex envelope W

⇤⇤(·;u0) of W (·;u0) (whose definition isgiven in (92)) and its derivative mu0(u) := DW

⇤⇤(u;u0) for u > u0, (8) alsoyields the following equation for the energetic solution u:

mu0(u(t)) = `(t)� �+ for all t 2 [0, T ]. (10)

Therefore, BV and energetic solutions depend monotonically on increasing loadings.Furthermore, BV solutions are governed by the upper monotone envelope of W 0,whereas energetic solutions involve the derivative of the convexified energy (9). Inboth cases, the initial condition u0 provides crucial information to construct theabove monotone graphs. The case of a decreasing loading ` can be easily recovered


from the increasing one thanks to the symmetry principle recalled in Proposition3. The concatenation principle allows us to extend immediately our results to thecase of piecewise monotone loadings.

In Examples 1 and 3, we illustrate (7) and (8) in the simple, yet significant caseof the double-well potential

W (u) =1

4(u2

�1)2, u 2 R.

In this context, the input-output relation ` ! u given by (10) for energetic so-lutions, corresponds to the so-called Maxwell rule, cf. the discussion in [14, §I.3].The latter evolution mode prescribes that for all t 2 [0, T ], the function u(t) onlyattains absolute minima of the map u 7! W (u)� (`(t)� �+)u. This corresponds toconvexification of W , and causes the system to jump “early” into far-apart configu-rations. Instead, the evolution mode (7) follows the Delay rule, related to hysteresis

behavior. The system accepts also relative minima of u 7! W (u)� (`(t)� �+)u, andthus the function t 7! u(t) tends to jump “as late as possible”.

The plan of the paper is as follows: in Section 2, we recall some definitionsand preliminary properties of BV functions, energetic and BV solutions in a finite-dimensional setting. Section 3 is devoted to a refined characterization of BV so-lutions in the one-dimensional setting. The case of monotone loadings is carefullyanalyzed in Section 4, after some auxiliary results on upper and lower monotone en-velopes 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 relatedto functions in BV([0, T ];X), with X a finite-dimensional vector space, and toenergetic and BV solutions of a general rate-independent system.

2.1. BV functions. Hereafter, we shall consider functions of bounded variationu 2 BV([a, b];X) to be pointwise defined at every time t 2 [a, b]. Notice thata function u 2 BV([a, b];X) admits left (resp. right) limits at every t 2 (a, b](resp. t 2 [a, b)), viz. 9ul(t) = lim

s"t u(s) and 9ur(t) = lims#t u(s). We also adopt

the convention ul(a) := u(a), ur(b) := u(b). The pointwise jump set Ju

of u is theat most countable set defined by

Ju

: =�

t 2 [a, b] : ul(t) 6= u(t) or u(t) 6= ur(t)

� ess-Ju

:=�

t 2 (a, b) : ul(t) 6= ur(t)

.

(11)

We denote by u

0 the distributional derivative of u (extended by u(a) in (�1, a)and by u(b) in (b,+1)) in D0(R): it is a Radon vector measure with finite totalvariation |u

0| supported in [a, b]. In the one-dimensional case X = R, u0 admits

the Hahn decomposition u

0 = (u0)+ � (u0)� as the di↵erence of two positive andmutually singular measures, such that |u0

| = (u0)+ + (u0)�.It is well known [1] that u

0 can be decomposed into the sum of its di↵use partu

0co and its jump part u0

J:

u

0 = u

0co+u

0J, u

0J := u

0 ess-Ju

, so that u0co({t}) = 0 for every t 2 [a, b]. (12)

2.2. Energetic solutions to rate-independent systems. We consider a generalrate-independent system (X,E, ), where the dissipation potential

: X ! [0,+1) is 1-positively homogeneous, convex, with (v) > 0 if v 6= 0,


andE 2 C1(X ⇥ [a, b]), E

t

(u) := W (u)� h`(t), ui,

for some W 2 C1(X) and ` 2 C1([a, b];X⇤). We shall also use the notation@

t

Et

(u) := �h`

0(t), ui for the partial time derivative of E, and we set

K

⇤ := @ (0) = {w 2 X

⇤ : ⇤(w) 1} ⇢ X

⇤, where ⇤(w) := sup

(v)1hw, vi (13)

for w 2 X

⇤. We recall the notion of energetic solution to the rate-independentsystem (X,E, ), cf. [9, 10, 5].

Definition 2.1 (Energetic solution). A curve u 2 BV([a, b];X) is an energeticsolution of the rate-independent system (X,E, ) if for all t 2 [a, b] it satisfies theglobal stability

8 z 2 X : Et

(u(t)) Et

(z) + (z � u(t)) (S)


Et

(u(t)) + Var (u; [a, t]) = Ea

(u(a)) +

Z

t

a

@

t

Es

(u(s)) ds, (E)

where

Var (u; [a, t]) := supn

M

X

m=1

�

u(tm

)� u(tm�1)

�

:

a = t0 < t1 < · · · < t

M�1 < t

M

= t

o

(14)

denotes the pointwise -total variation of u on the interval [a, t].

The following characterization of energetic solutions has been proved in [8, Prop.2.2].

Proposition 1 (Di↵erential characterization of energetic solutions). A curve u 2

BV([a, b];X) is an energetic solution of the rate-independent system (X,E, ) if andonly if it satisfies the global stability condition (S), the doubly nonlinear di↵erentialinclusion in the BV sense

@ ⇣du0

co

dµ(t)⌘

+DW (u(t)) 3 `(t) for µ-a.e. t 2 (a, b), µ := L 1 + |u

0co|, (DN)

and the following jump conditions at each point t 2 Ju

:

Et

(ur(t))� Et

(ul(t)) = � (ur(t)�ul(t)),

Et

(u(t))� Et

(ul(t)) = � (u(t)�ul(t)),

Et

(ur(t))� Et

(u(t)) = � (ur(t)�u(t)).

(Jener)

The jump conditions (Jener) show that, in the case of an energetic solution u thejump set J

u

coincides with the essential jump set ess-Ju

.

2.3. BV solutions to rate-independent systems. As we mentioned in the in-troduction, we shall restrict to BV solutions arising in the vanishing viscosity limitof (2), with dissipation potentials

"

of the form (3). In such a setting, the van-

ishing viscosity contact potential ⇧ : X ⇥X

⇤! [0,+1) associated with the family

( "

)"

is given by

⇧(v, w) := (v) max(1, ⇤(w)) =

(

(v) if w 2 K

⇤,

(v) ⇤(w) if w 62 K

⇤.

(15)


For a fixed t 2 [a, b], the (possibly asymmetric) Finsler cost induced by ⇧ and (thedi↵erential of) E at the time t is for every u, u1 2 X given by

�⇧,E(t; u, u1) := infn

Z

r1

r0

⇧(#(r),�DEt

(#(r))) dr :

# 2 AC([r0, r1];X), #(r0) = u, #(r1) = u1

o

.

(16)

As already observed in [8], it is not di�cult to check that the infimum in (16) is al-ways attained by a Lipschitz curve # 2 Lip([r0, r1];X) such that⇧(#(r),�DE

t

(#(r))) ⌘ 1 for a.a. r 2 (r0, r1).For every u 2 BV([a, b];X) and every subinterval [↵,�] ⇢ [a, b], the jump varia-

tion of u induced by (⇧,E) on [↵,�] is

Jmp⇧,E(u; [↵,�]) := �⇧,E(↵;u(↵), ur(↵)) +�⇧,E(�;ul(�), u(�))

+X

t2Ju\(↵,�)

�

�⇧,E(t;ul(t), u(t)) +�⇧,E(t;u(t), ur(t))�

,

(17)

and the associated (pseudo-)total variation is

Var⇧,E(u; [↵,�]) :=

Z

�

↵

✓

du0co

dµ

◆

dµ+ Jmp⇧,E(u; [↵,�]), (18)

where µ can be any nonnegative and di↵use reference measure, provided that u

0co

is absolutely continuous w.r.t. µ (e.g. µ = L 1 + |u

0co|). In fact, since is 1-

homogeneous, the value of the integral on the right-hand side of (18) is independentof µ (see e.g. [1]).

Then, we are in the position to recall the definition of BV solution given in [8].

Definition 2.2. A curve u 2 BV([a, b];X) is a BV solution of the rate-independentsystem (R,E,⇧) if it satisfies the local stability condition

`(t)�DW (u(t)) 2 K

⇤ for a.e. t 2 [a, b] \ Ju

, (Sloc)

and the (⇧,E)-energy balance

Var⇧,E(u; [a, t])+Et

(u(t)) = E0(u(0))+

Z

t

a

@

t

Es

(u(s)) ds for all t 2 [a, b]. (E⇧,E)

In [8, Sec. 4] the following result has been proved.

Proposition 2 (Di↵erential characterization of BV solutions). A curve

u 2 BV([a, b];X) is a BV solution of the rate-independent system (R,E,⇧) if and

only if it satisfies the doubly nonlinear di↵erential inclusion (DN), and it fulfills at

each point t 2 Ju

the jump conditions:

Et

(ur(t))� Et

(ul(t)) = ��⇧,E(t;ul(t), ur(t)),

Et

(u(t))� Et

(ul(t)) = ��⇧,E(t;ul(t), u(t)),

Et

(ur(t))� Et

(u(t)) = ��⇧,E(t;u(t), ur(t)).

(JBV)

As in the case of energetic solutions, (JBV) yield that, for a BV solution u thejump set J

u

coincides with the essential jump set ess-Ju

.


2.4. Symmetry and concatenation principles for energetic and BV solu-tions. We state here two useful properties of energetic and BV solutions. Let usfirst introduce the modified energy and dissipation potentials

Et

(u) := Et

(�u), (u) := (�u), ⇧(u,w) := ⇧(�u,�w). (19)

Proposition 3 (Symmetry principle). A curve u 2 BV([a, b];X) is an energetic

solution of the rate-independent system (X,E, ) (resp. a BV solution of the rate-

independent system (X,E,⇧)) if and only if the curve u(t) := �u(t) is an energetic

solution of the rate-independent system (X, E, ) (resp. of the rate-independent sys-

tem (X, E, ⇧)).

The proof follows from easy calculations, observing that

Et

(u(t)) = Et

(u(t)), u

0(t) = �u

0(t),

(u0(t)) = (u0(t)), ⇤(w) = ⇤(�w), K

⇤ = �K

⇤

Another simple property concerns the behavior of energetic and BV solutions withrespect to restriction and concatenation. The proof is trivial.

Proposition 4 (Restriction and concatenation principle).

1. The restriction of an energetic (resp. BV) solution in [a, b] to an interval

[↵,�] ⇢ [a, b] is an energetic (resp. BV) solution in [↵,�].2. If a = t0 < t1 < · · · < t

M�1 < T

m

= b is a subdivision of [a, b] and u : [a, b] !X is an energetic (resp. BV) solution in each one of the intervals [t

j�1, tj ],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 caseX = R, which we also identify with X

⇤. We will denote by v

+, v

� the positive andnegative part of v 2 R.Dissipation. A dissipation potential is a function of the form

(v) := �+ v

+ + �� v

�, v 2 R, for some �± > 0. (20)

Hence, we have

@ (v) =

8

>

<

>

:

�+ if v > 0,

[��, �+] if v = 0,

�� if v < 0

for all v 2 R,

K

⇤ = [��, �+], ⇤(w) =1

�+w

+ +1

��w

� for all w 2 R.

Energy functional. The energy is given by a function E : R ⇥ [a, b] ! R of theform

Et

(u) := W (u)� `(t)u, (21)

with ` 2 C1([a, b]) and W : R ! R such that

W 2 C1(R), limx!�1

W

0(x) = �1, limx!+1

W

0(x) = +1. (22)

3. BV solutions of rate-independent systems in R. In this section we will pro-vide an equivalent characterization of BV solutions to the rate-independent system(R,E,⇧), in the one-dimensional setting considered in §2.5.

Theorem 3.1. A function u 2 BV([a, b];R) is a BVsolution of the rate-independent

system (R,E,⇧) of §2.5 if and only if the following properties hold:


a) u satisfies the local stability condition (equivalent to (Sloc))

� �� `(t)�W

0(u(t)) �+ for every t 2 [a, b] \ Ju

. (Sloc,R)

b) The function W

0(ul) is continuous in [a, b), the function W

0(ur) is continuous

in (a, b] and they coincide in (a, b): we denote their common value by W

0(ulr).c) u satisfies the following precise formulation of the doubly nonlinear di↵erential

inclusion (DN)

W

0(ulr(t)) = `(t)� �+ for every t 2 supp�

(u0)+�

\ (a, b), (23)

W

0(ulr(t)) = `(t) + �� for every t 2 supp�

(u0)��

\ (a, b), (24)

with obvious modifications at t = a, t = b as in point b).

d) At each jump point t 2 Ju

, u satisfies the jump conditions

min�

ul(t), ur(t)�

u(t) max�

ur(t), ul(t)�

, (25)

and

`(t)�W

0(#) � �+ if ul(t) < ur(t), `(t)�W

0(#) �� if ul(t) > ur(t), (26)

for every # such that min�

ul(t), ur(t)�

# max�

ur(t), ul(t)�

. In particular,

Ju

= ess-Ju

.

Proof. We split the argument in various steps.Claim 1: The local stability condition (Sloc) is equivalent to (Sloc,R).

It is su�cient to recall that K⇤ = [��, �+].Claim 2: the jump conditions (JBV) are equivalent to (25) and (26).

Taking (15) and (21) into account, the Finsler cost �⇧,E(t; ·, ·) in fact reduces(up to a linear reparametrization) to

�⇧,E(t; u, u1) = minn

Z 1

0max

⇣

1, ⇤�

`(t)�W

0(#(r))�

⌘

(#0(r)) dr :

# 2 AC([0, 1];R), #(0) = u, #(1) = u1

o

,

(27)

Let us consider, e.g., the case u u1 and notice that, if # 2 AC([0, 1];R) fulfills#(0) = u and #(1) = u1, then, setting

r0 := sup{r 2 [0, 1] : #(t) u}, r1 := inf{r 2 [r0, 1] : #(r) � u1},

there holds

#(r0) = u, #(r1) = u1, #(r) 2 (u, u1) for all r 2 (r0, r1).

Therefore, the value of the integral in (27) surely diminishes if we just consider therestriction of # to the interval [r0, r1]. Hence we can assume that the range of aminimizing curve in (27) is contained in [u, u1].

We can also suppose that the competing curves # in (27) are nondecreasing.Indeed, if # is absolutely continuous and connects u to u1, we can consider thecurve #(r) := max

s2[0,r] #(s). It is easy to check that # is nondecreasing andabsolutely continuous, since for all 0 r1 r2 1

#(r2)� #(r1) sups1,s22[r1,r2]

|#(s2)� #(s1)|

Z

r2

r1

|#

0(s)| ds.


It follows that #0 = #

0 a.e. on the coincidence set {# = #}, whereas one can easilycheck that #0(r) = 0 where #(r) > #(r) (viz., where #(r) 6= #(r)). From the aboveconsiderations we obtain

Z 1

0max

�

1, ⇤(`(t)�W

0(#(r)))�

(#0(r)) dr

Z 1

0max

�

1, ⇤(`(t)�W

0(#(r)))�

(#0(r)) dr.

Therefore, it is not restrictive to assume that any minimizing curve # is nondecreas-ing on [0, 1]. Then, with a change of variable, from (27) we deduce the identity

�⇧,E(t; u, u1) = �+

Z

u1

u

max�

1, ⇤(`(t)�W

0(r))�

dr. (28)

Notice that

�+ max�

1, ⇤(`(t)�W

0(r))�

� max�

�+, `(t)�W

0(r)�

� `(t)�W

0(r), (29)

and

�+ max�

1, ⇤(`(t)�W

0(r))�

= `(t)�W

0(r) , `(t)�W

0(r) � �+; (30)

moreover, at a jump point t with ul(t) < ur(t) there holds

Et

(ur(t))� Et

(ul(t)) = �

Z

ur(t)

ul(t)

�

`(t)�W

0(r)�

dr. (31)

Comparing (JBV) with (28) and (31) we immediately see that the first condition of(JBV) is equivalent to the first of (26). If (25) also holds, then we easily get theother two conditions of (JBV). Conversely, (JBV) yields

�⇧,E(t;ul(t), ur(t)) = �⇧,E(t;ul(t), u(t)) +�⇧,E(t;u(t), ur(t)),

and this identity implies (25) by the positivity of the integrand in (28). The caseul(t) > ur(t) can be studied by a similar argument, see also Proposition 3.Claim 3, su�ciency: 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 di↵erentialinclusion (DN) holds: it follows from (23), (24), and (Sloc,R), since µ is di↵use andtherefore ur(t) = ul(t) = u(t) for µ-a.e. t 2 (a, b).Claim 4: a) and d) imply b).

It is immediate, since the continuity of W 0 and ` yields that the inequalities in(Sloc,R) hold also for ul and ur; (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(u0)+-a.e. in (a, b) thanks to the di↵erential inclusion (DN) and the jump/stabilityconditions (which yield (23) on the jump set). By Claim 4, we know that W

0(ulr)is continuous, so that the identity in (23) holds on the support of (u0)+. The sameargument applies to (24).

The previous general result has a simple consequence: a BV solution is locallyconstant in a neighborhood of a point where the stability condition (Sloc,R) holdswith a strict inequality.


Lemma 3.2. Let u 2 BV([a, b];R) be a BV solution of the rate-independent system

(R,E,⇧) of §2.5, and let us suppose that �� < `(s) � W

0(u(s)) < �+ at some

s 2 [a, b]. Then, setting

↵ := max�

t 2 [a, s] : ⇤(`(t)�W

0(u(s))) = 1

,

� := min�

t 2 [s, b] : ⇤(`(t)�W

0(u(s))) = 1

we have ↵ < s < � and u(t) ⌘ u(s) for every t 2 [↵,�].

Proof. In view of (25) and (26), s is not a jump point of u, hence u is continuousat s and the set {t 2 [a, b] : ⇤(`(t) �W

0(u(t))) < 1} contains a (relatively) openneighborhood I of s; since u = ur = ul in I, point c) of Theorem 3.1 shows thatu(t) ⌘ u(s) in I. Consider now the set J := {r 2 (↵,�) : u(r) = u(s)

: we haveseen that J is open, and it is easy to check that it is also closed in (↵,�), so thatJ = (↵,�) and the thesis follows.

4. BV solutions of the rate-independent system (R,E,⇧) with monotoneloadings. As mentioned in the introduction, BV solutions of rate-independent sys-tems 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 0 inour setting). In this section we first focus on a few properties of these maps andtheir inverses, and then we exhibit the explicit formulae characterizing BV solutionswhen ` is increasing or decreasing.

4.1. The upper monotone envelope of W 0.

Definition 4.1 (Upper monotone envelope). For every u 2 R, we set ¯ := W

0(u),and we define the maximal monotone map mu(·) : R ◆ R

mu(u) := maxuvu

W

0(v) if u > u, mu(u) := (�1,W

0(u)],

mu(u) = ; if u < u.

(32)

We call mu(·) the upper monotone envelope of W 0 in the interval (u,+1). Thecontact set is defined by

C

u := {u} [

�

u > u : W 0(u) = mu(u)

. (33)

The mapping mu(·) is monotone and surjective thanks to (22); it is single-valuedon (u,+1) (where we identify the set mu(u) with its unique element with a slightabuse of notation). We can thus consider the inverse graph pu(·) : R ◆ R of mu(·):it is defined by

u 2 pu(`) , ` 2 mu(u) for u, ` 2 R. (34)

Clearly, pu(·) is a maximal monotone graph in R, and it is uniquely characterized bya left-continuous monotone function p

u

l (·) and a right-continuous monotone functionp

u

r (·) such that

pu(`) = [pul (`), pu

r (`)], i.e. mu(u) 3 ` , p

u

l (`) u p

u

r (`). (35)

We also consider a further selection in the graph of pu(·):

pu

c (`) :=�

u 2 pu(`) : W 0(u) = `

=�

u 2 C

u : mu(u) 3 `

= pu(`) \ C

u

. (36)

By introducing the set

A

u :=�

f : (u,+1) ! R : f is nondecreasing and fulfills f � W

0 , (37)


we have

mu(·)|(u,+1)

2 A

u

, W

0(u) mu(u) f(u) for all f 2 A

u

, u 2 (u,+1), (38)

so that mu(·) is the minimal nondecreasing map above the graph of W 0 in (u,+1).It immediately follows from (38) that

mu(u) = inf{f(u) : f 2 A

u

} for all u > u. (39)

The following result collects some simple properties of pul (·) and p

u

r (·).

Lemma 4.2. Assume (22). Then

1. the functions p

u

l (·) and p

u

r (·) are nondecreasing in R and we have

¯= W

0(u) `1 < `2 ) p

u

r (`1) < p

u

l (`2). (40)

2. For every ` 2 R there holds

lim�"`

p

u

l (�) = p

u

l (`) and lim�#`

p

u

r (�) = p

u

r (`). (41)

3. For every ` �

¯= W

0(u) there holds

W

0(u) ` if u 2 [u, pur (`)]. (42)

4. For every ` �

¯= W

0(u) there holds

W

0(pul (`)) = W

0(pur (`)) = W

0(pu

c (`)) = `. (43)

5. For every ` 2 R there holds

p

u

l (`) = min�

u � u : W

0(u) � `

, p

u

r (`) = inf�

u � u : W

0(u) > `

. (44)

Proof. First of all, (40) and (41) are general properties of the inverse of a maximalmonotone map.

As for (42), it is an immediate consequence of the inequality W

0 mu(·) in

[u,1).Let us check (43) and (44). The identity W

0(pul (`)) = ` and the first propertyof (44) ensue from (32): it is su�cient to notice that W

0(u) mu(u) < ` ifu u < p

u

l (`) and mu(u) = ` if u = p

u

l (`).The identityW

0(pur (`)) = ` is due to the continuity ofW 0 and the second propertyof (44). To prove the latter, we observe that, when u > p

u

r (`) we have mu(u) > `,and we know that there exists v 2 [pur (`), u] such that W

0(v) > `. Since u isarbitrary we get p

u

r (`) � inf{u � u : W 0(u) > `}. The converse inequality followsfrom (42).

4.2. The lower monotone envelope of W

0. In a completely similar way wecan introduce the maximal monotone map below the graph of W 0 on the interval(�1, u], viz.

nu(u) := infuvu

W

0(v) if u < u, nu(u) := [W 0(u),+1),

nu(u) = ; if u > u,

(45)

which satisfies

nu(u) = sup�

f(u) : f 2 B

u

for u < u, (46)

where

B

u :=�

f : (�1, u) ! R : f is nondecreasing and fulfills f W

0 .


As before, the inverse graph qu(·) := (nu(·))�1 : R ◆ (�1, u] can be representedas qu(u) = [qul (u), q

u

r (u)], where

q

u

l (`) = sup�

u u : W

0(u) < `

, q

u

r (`) = max�

u u : W

0(u) `

, (47)

and we set

qu

c (`) :=�

u 2 qu(`) : W 0(u) = `

. (48)

If we consider the symmetric energy as in (19), viz. fW (u) := W (�u) (so thatf

W

0(u) = �W

0(�u)), and we denote by fmu(·), pul (·), and p

u

r (·) the functions asso-

ciated with f

W via (32) and (35), it can be easily checked that

nu(u) = fm(�u)(�u), q

u

r (`) = �p

(�u)l (�`), q

u

l (`) = �p

(�u)r (�`). (49)

Hence the obvious analogue of Lemma 4.2 holds.

4.3. Monotone loadings and BV solutions. We apply the notions introducedin §4.1 to characterize BV solutions when ` is monotone.

First of all, we provide an explicit formula yielding BV solutions for an increasingloading `. The case of a decreasing and of a piecewise monotone loading will easilyfollow by applying Propositions 3 and 4.

Theorem 4.3. Let u 2 R and ` 2 C1([a, b]) be a nondecreasing loading such that

`(a) � W

0(u)� ��. (50)

Any map u : [a, b] ! R satisfying

u is nondecreasing, u(t) 2 pu

c (`(t)� �+) for every t 2 [a, b] \ Ju

,

W

0(u(b)) `(b)� �+(51)

is a BV solution of the rate independent system (R,E,⇧) of §2.5. In particular,

(51) yields

u(t) 2 [pul (`(t)� �+), pu

r (`(t)� �+)] for every t 2 [a, b], (52)

and (52) is equivalent to (51) when ` is strictly increasing.

Proof. We apply Theorem 3.1. It is immediate to check that u satisfies (Sloc,R).By continuity, (51) yields

W

0(ul(t)) + �+ = W

0(ur(t)) + �+ = `(t) for every t 2 [a, b] (53)

so that b) and c) are satisfied.To check the jump conditions of d), let us notice that by (44), (53), and the

monotonicity of u we have

p

u

l (`(t)� �+) ul(t) u(t) ur(t) p

u

r (`(t)� �+) for every t 2 [a, b], (54)

which yields (25) and (26) by (42).The inequalities in (54) also show that (51) implies (52). Conversely, when

` is strictly increasing, it is immediate to check that any map satisfying (52) isnondecreasing. Since the jump set of u coincides with {t 2 [a, b] : pul (`(t) � �+) <p

u

r (`(t) � �+)}, we have p

u

l (`(t) � �+) = p

u

r (`(t) � �+) = pu

c (`(t) � �+) at everyt 2 [a, b] \ J

u

. Then, also the second condition of (51) is satisfied.

Applying Proposition 3 and the discussion of §4.2 we have:


Corollary 1. Let u 2 R and ` 2 C1([a, b]) be a nonincreasing loading satisfying

`(a) W

0(u) + �+. Any map u : [a, b] ! R satisfying

u is nonincreasing, u(t) 2 qu

c (`(t) + ��) for every t 2 [a, b] \ Ju

,

W

0(u(b)) � `(b) + ��(55)

is a BV solution of the rate independent system (R,E,⇧). In particular, (55) yields

u(t) 2 [qul (`(t) + ��), qu

r (`(t) + ��)] for every t 2 [a, b], (56)

and (56) is equivalent to (55) when ` is strictly decreasing.

The next result states that, under a slightly stronger condition on the initialdata, any BV solution driven by an increasing loading admits the representation(51).

Theorem 4.4 (Nondecreasing loadings). Let ` 2 C1([a, b]) be a nondecreasing

loading and let u 2 BV([a, b];R) be a BV solution of the rate-independent system

(R,E,⇧) satisfying`(a) � W

0(u(a))� ��, (57)

W

0< W

0(u(a)) in a left neighborhood of u(a) if `(a) = W

0(u(a))� ��. (58)

Then u can be represented as in Theorem 4.3, i.e. it satisfies

u is nondecreasing, u(t) 2 pu(a)c (`(t)� �+) for every t 2 [a, b] \ J

u

, (59)

and

u(t) 2 [pu(a)l (`(t)� �+), pu(a)r (`(t)� �+)] for all t 2 [a, b]. (60)

Proof. We apply Theorem 3.1 and we split the argument in a few steps; as usualwe use the short-hand notation u := u(a).Claim 1: There exists ↵ 2 [a, b] such that `(t)�W

0(u(t)) > �� for all t 2 (↵, b],and u(t) ⌘ u, `(t) ⌘ `(a) in [a,↵].

Let us consider the set

⌃ := {t 2 [a, b] : `(t)�W

0(u(t)) ��} (61)

and observe that by (25) and (26) we also have ⌃ \ (a, b) = {t 2 [a, b] : `(t) �W

0(ulr(t)) = ��}, (with obvious modifications for ⌃ \ {a, b}), so that ⌃ is closedthanks to b) of Theorem 3.1.

If a 62 ⌃ we set ↵ := a and ⌃a

:= ;. If a 2 ⌃ we denote by ⌃a

the connectedcomponent of ⌃ containing a and we set ↵ = max⌃

a

. If ↵ > a then

`(t)�W

0(u(t)) �� = `(t)�W

0(ur(t)) = `(t)�W

0(ul(t)) (62)

for every t 2 [a,↵], so that by c) of Theorem 3.1 u is nonincreasing in [a,↵]. Then,(58), (62) and the jump condition (25) show that a 62 J

u

(otherwise, ur(a) < u(a)by (62) and (26), and (26) would also imply W

0(#) � `(a) + �� = W

0(u) for # 2

[ur(a), u(a)], contrary to (58)), so that ur(a) = u(a) = u. Since ` is nondecreasing,with a similar argument and still invoking (58) we conclude that u(t) = ur(t) ⌘ u

and `(t) ⌘ `(a) in [a,↵]. If ↵ = b the claim is proved.If ↵ < b let us show that ⌃\⌃

a

is empty. Indeed, if not there exists � 2 (↵, b]\⌃and we can consider �

0 := min(⌃ \ [�, b]); since ⌃ is closed, by construction �

0>

� > ↵ and the definition of ⌃ yields �� < `(t) � W

0(ul(t)) < �+ in some leftneighborhood (�00

,�

0) of �0. Lemma 3.2 yields that ul(t) = u(t) ⌘ ul(�0) in (�00,�

0),and this contradicts the fact that ` is nondecreasing and `(�0)�W

0(ul(�0)) = ��.Claim 2: u is nondecreasing on [a, b].


Relation (24) and Claim 1 imply that (u0)��

[a, b)�

= 0, so that u is nondecreasingin [a, b). If b were a jump point, then by (26) it would be u(b) > ul(b).Claim 3: let B := {t 2 [↵, b] : W 0(u) + �+ = `(t)} and � := minB (we set � := b if

B is empty). Then u(t) ⌘ u in [a,�) and

W

0(ul(t)) = W

0(ur(t)) = `(t)� �+, W

0(u(t)) `(t)� �+ for all t 2 (�, b]. (63)

In particular

ul(t) � p

u

l (`(t)� �+) for every t 2 [a, b]. (64)

The first statement follows from Claim 1 and Lemma 3.2. To prove the secondproperty in (63), we argue by contradiction and we suppose that a point s 2 (�, b]exists such that if W 0(u(s)) + �+ > `(s). Since ` is nondecreasing, Claim 1 andLemma 3.2 show that u(t) ⌘ u(s) for every t 2 [↵, s], so that s �.

The first identity in (63) then follows by a continuity argument and the stabilitycondition (Sloc,R). Inequality (64) ensues from (63) and the first characterization in(44).Claim 4: For every t 2 [a, b] we have ur(t) p

u

r (`(t)� �+).If ur(t) = u there is nothing to prove. Otherwise, let t � � and take z 2 (u, ur(t)).Since u is nondecreasing, there exists s 2 [�, t] such that ul(s) z ur(s), so thatby (63) and (26)

W

0(z) `(s)� �+ `(t)� �+,

since ` is nondecreasing. Being z < ur(t) arbitrary, the claim follows from thesecond of (44).Conclusion:

Relation (60) follows from Claim 2, (64), and Claim 4. At every continuity pointt for u we have ul(t) = ur(t) = u(t), so that (59) is due to (60) and (63).

A straightforward consequence of Proposition 3 and the discussion of §4.2 con-cerns the characterization of BV solutions in the case of a decreasing load: it canbe deduced from the analysis of the increasing case.

Corollary 2 (Nonincreasing loadings). Let u 2 R, let ` 2 C1([a, b]) be a nonin-

creasing loading satisfying `(a) W

0(u) + �+, and let us suppose that

W

0> W

0(u) in a right neighborhood of u if `(a) = W

0(u) + �+. (65)

Then every BV solution u 2 BV([a, b];R) with u(a) = u satisfies

u is nonincreasing, u(t) 2 qu

c (`(t)) for every t 2 [a, b] \ Ju

, (66)

and therefore

u(t) 2 [qul (`(t) + ��), qu

r (`(t) + ��)] for all t 2 [a, b]. (67)

We conclude this section with some examples illustrating the previous results.In particular, Example 2 shows that the characterization (60) may not hold if theloading ` does not comply with (58).

Example 1 (BV solutions driven by a double-well energy). The double-well po-tential energy

W (u) =1

4(u2

� 1)2 (68)

clearly fulfills condition (22). Note that W 0(u) = u

3� u, and

W

0(u) > 0 for u < u1 := �

1p

3and u > u2 :=

1p

3, with W

0(u) < 0 for u1 < u < u2.


Let us set

`

⇤ := W

0(u1) =2p

3

9, `⇤ := W

0(u2) = �

2p

3

9,

and, for later convenience,

u⇤ := min{u 2 R : W

0(u) = `⇤}, u

⇤ := max{u 2 R : W

0(u) = `

⇤}. (69)

We also introduce the intervals I1 := (�1, u1], I2 := [u1, u2], I3 := [u2,+1), and,for i = 1, 2, 3, we denote by S

i

the inverse function ((W 0)|Ii )�1. Hence

S1 : (�1, `

⇤] ! (�1, u1), S2 : [`⇤, `⇤] ! [u1, u2], S3 : [`⇤,+1) ! [u2,+1). (70)

Notice that the functions S1 and S3 are strictly increasing, whereas S2 is strictlydecreasing. Let us consider the evolution in the case of an external loading ` 2

[0, 1] ! R such that

` is strictly increasing in [0, 1/2], ` is strictly decreasing in [1/2, 1],

u(0) < u1, `(0) := W

0(u(0)) + �+, `(1/2) > `

⇤ + �+, `(1) < `⇤ � ��.

There are points

t1 2 (0, 1/2) : `(t1) = `

⇤ + �+; t2 2 (1/2, 1) : `(t2) + �� = `(1/2)� �+;

t3 2 (t2, 1) : `(t3) = `⇤ � ��.

All the BV solutions are then given by

u(t) =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

S1(`(t)� �+) in [0, t1),

u

02 [u1, u

⇤] if t = t1,

S3(`(t)� �+) in (t1, 1/2],

S3(`(1/2)� �+) in [1/2, t2],

S3(`(t) + ��) in [t2, t3),

u

002 [u⇤, u3] if t = t3,

S1(`(t) + ��) in (t3, 1].

Example 2 (Bifurcation of BV solutions driven by critical loadings). We considerthe same potential energy (68) as in Example 1, but we suppose now that `(1/2) =`

⇤ + �+ (in this case (65) is not satisfied at a = 1/2). In addition to the solutionconsidered before, we have another family of solutions, among which e.g.

u(t) =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

S1(`(t)� �+) in [0, 1/2),

S2(`(t)� �+) in [1/2, t⇤),

S2(`(t⇤)� �+) in [t⇤, t3],

u

002 [u⇤, u3] if t = t3,

S1(`(t) + ��) in (t3, 1],

where t⇤ 2 (1/2, t3) is defined by `(t⇤)� �� = `(t3) + �+.

5. Energetic solutions of rate-independent systems in R. In this section weprovide a general characterization of energetic solutions to the rate-independentsystem (R,E, ) considered in §2.5.

In order to express the global stability and jump conditions for energetic solu-tions, let us introduce the following one-sided global slopes

W

0i,r(u) := inf

z>u

W (z)�W (u)

z � u

, W

0s,l(u) := sup

z<u

W (z)�W (u)

z � u

, (71)


where the subscripts i r and s l stands for inf-right and sup-left respectively. Theysatisfy

W

0i,r(u) W

0(u) W

0s,l(u), W

0s,l(u) = �W

0i,r(�u) for every u 2 R, (72)

and it not di�cult to check that they are continuous. Indeed, it is su�cient tointroduce the continuous function V : R⇥ R ! R

V (u, z) :=

(

W

0(u) if z = u,

W (z)�W (u)z�u

if z 6= u,

and observe, e.g. for W 0i,r, that W

0i,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 2 [a, b]

� �� `(t)�W

0s,l(u(t)) `(t)�W

0(u(t)) `(t)�W

0i,r(u(t)) �+. (73)

The continuity property of the one-sided slopes also yields for all t 2 [a, b]

�� `(t)�W

0s,l(ur(t)) `(t)�W

0(ur(t)) `(t)�W

0i,r(ur(t)) �+ , (74)

�� `(t)�W

0s,l(ul(t)) `(t)�W

0(ul(t)) `(t)�W

0i,r(ul(t)) �+ . (75)

We can state the main characterization theorem concerning energetic solutions,which the reader may compare with Thm. 3.1 for BV solutions.

Theorem 5.1. A function u 2 BV([a, b];R) is an energetic solution of the rate-

independent system (R,E, ) of §2.5 if and only if the following properties hold:

a) u satisfies the global stability condition (73) (and therefore (74) and (75) as

well).

b) u satisfies the following precise formulation of the doubly nonlinear di↵erential

inclusion (DN)

W

0(ur(t)) = W

0i,r(ur(t)) = `(t)� �+ for every t 2 S+ := supp

�

(u0)+�

, (76)

W

0(ur(t)) = W

0s,l(ur(t)) = `(t) + �� for every t 2 S� := supp

�

(u0)��

. (77)

c) At each point t 2 Ju

, u fulfills the jump conditions:

u(t) = (1� ✓)ul(t) + ✓ur(t),W (u(t)) = (1� ✓)W (ul(t)) + ✓W (ur(t))

�

for some ✓ 2 [0, 1], (78)

and

W

0i,r(z) W

0i,r(ul(t)) =

W (ur(t))�W (ul(t))

ur(t)� ul(t)= `(t)� �+ if ul(t) z < ur(t), (79)

W

0s,l(z) � W

0s,l(ul(t)) =


ur(t)� ul(t)= `(t) + �� if ul(t) > z � ur(t). (80)

In particular, u is locally constant in the open set

I :=n

t 2 [a, b] : �� < `(t)�W

0s,l(u(t)) `(t)�W

0i,r(u(t)) < �+

o

. (81)

Since any jump point belongs either to the support of (u0)+, or of (u0)�, com-bining (78), (76), and (79) and (78), (77), and (80) we also get at every jump point


W

0i,r(ul) = W

0i,r(u) = W

0i,r(ur) = W

0(ur) =W (ur)�W (ul)

ur � ul= `� �+ if ul < ur,

(82a)

W

0s,l(ul) = W

0s,l(u) = W

0s,l(ur) = W

0(ur) =W (ur)�W (ul)

ur � ul= `+ �� if ul > ur.

(82b)

We now develop the proof of Thm. 5.1.

Proof. It is easy to check that, if u 2 BV([a, b];R) satisfies conditions a) – c) thenu is an energetic solution to (R,E, ): we omit the simple details.

We discuss here the converse implication. Hence, let u be an energetic solution of(R,E, ). We have already shown point a); let us first consider point c). The firstproperty of (78) easily follows by summing the identities of the jump conditions(Jener), thus obtaining

(ur(t)� ul(t)) = (ur(t)� u(t)) + (u(t)� ul(t)). (83)

In particular, this implies that min(ul(t), ur(t)) u(t) max(ur(t), ul(t)�

, so thatu(t) is a convex combination of ul(t) and ur(t) with a uniquely determined coe�cient✓ 2 [0, 1]. Conditions (Jener) then yield the corresponding property for W (u(t)).

Let us now consider, e.g., the case ul(t) < ur(t) and prove (79). From the firstof the jump conditions (Jener) and the definition of the right global slope W 0

i,r(·), wefind

`(t)�W

0i,r(ul(t)) � `(t)�


ur(t)� ul(t)= �+.

Combining this inequality with (75) we conclude that the identities in (79) hold. Ifnow z 2 [ul(t), ur(t)) we obtain

W (z)�W (ur(t)) = W (z)�W (ul(t))�W

0i,r(ul(t))(ur(t)� ul(t))

� W

0i,r(ul(t))(z � ul(t))�W

0i,r(ul(t))(ur(t)� ul(t))

= W

0i,r(ul(t))(z � ur(t)),

(84)

where the second inequality ensues from the definition (71) of W 0i,r. Dividing by

z � ur(t) we then have that W

0i,r(z) W

0i,r(ul(t)). The proof of (80) is completely

analogous.Concerning b), we notice that (DN) yields

W

0(u(t)) = `(t)� �+ for (u0co)

+-a.e. t 2 (a, b),

so that (76) holds by continuity and by (74) in supp�

u

0�+\ J

u

. On the other hand,

for every t 2 Ju

\ supp�

u

0�+ we have ul(t) < ur(t) so that, dividing inequality (84)by z � ur(t) and passing to the limit as z " ur(t) we obtain

W

0(ur(t)) W

0i,r(ul(t)) = `(t)� �+, (85)

and applying (74) we conclude the proof of (76). The identities in (77) follow bythe same argument.

It remains to check (81): by (79) and (80) any t 2 I is a continuity point for u;the continuity properties of W 0

i,r(·) and W

0s,l(·) then show that a neighborhood of t

is also contained in I, so that I is open and disjoint from Ju

. Relations (76) and(77) then yield that u

0 = 0 in the sense of distributions in I, so that u is locallyconstant.


6. Energetic solutions of the rate-independent system (R,E, ) with mono-tone loadings.

6.1. Convex envelopes and their subdi↵erentials. This section is devotedto some preliminary convex analysis results, which turn out to be useful for thecharacterization of energetic solutions driven by monotone loadings.

Let W : R ! (�1,+1] be a function with proper non-empty domain D(W) :={u 2 R : W(u) < 1}. For our purposes, we will assume that

D(W) is a closed interval, W is of class C1 in D(W)

and it is bounded from below.(86)

The (convex analysis) subdi↵erential of W is the multivalued map @W : R ◆ Rdefined by

⇠ 2 @W(u) , W (u) + ⇠(z � u) W (z) for every z 2 R. (87)

Clearly, @W(u) is empty if u 62 D(W). In the present one-dimensional setting, wehave a simple characterization of the subdi↵erential in terms of the one-sided slopesdefined in (71):

⇠ 2 @W(u) , W0s,l(u) ⇠ W0

i,r(u). (88)

If u 2 int�

D(W)�

, then @W(u) either coincides with {W0(u)} or it is empty. By(88), the former case can also be characterized by

@W(u) 6= ; , W0s,l(u) = W0(u) = W0

i,r(u) for every u 2 int�

D(W)�

; (89)

if this is the case, the common value on the right-hand side of (89) is the uniqueelement of @W(u).

The Fenchel-Moreau conjugate of W is the function

W⇤(`) := supu2R

(`u�W(u)) , (90)

and a further iteration of the conjugation yields

W⇤⇤(u) := sup`2R

�

u `�W⇤(`)�

. (91)

Indeed, W⇤⇤ coincides with the l.s.c. convex envelope of W, i.e. the maximal convexand l.s.c. function less than W. It can also be defined by

W⇤⇤(u) = sup {au+ b : ay + b W(y) for all y 2 R} . (92)

We introduce the contact set

C(W) :=�

u 2 R : W(u) = W⇤⇤(u)

(93)

and its complement

B(W) := {u 2 R : W(u)�W⇤⇤(u) > 0} = D(W) \ C(W). (94)

It can be shown (see e.g. [11, Lemma 3.3]) that B(W) ⇢ D(W) is open in R, henceit is the disjoint union of a (at most) countable collection of open intervals. Further,for every connected component (↵,�) of B(W) there holds

W⇤⇤(↵) = W(↵), W⇤⇤(�) = W(�),

W⇤⇤((1� ✓)↵+ ✓�) = (1� ✓)W(↵) + ✓W(�)(95)

for all ✓ 2 [0, 1]. Using (95), it can be checked that W⇤⇤2 C1(D(W)). It is well

known (see e.g. [3, Ch. 1, §5.1]) that

@W(u) 6= ; , W(u) = W⇤⇤(u), and in this case @W(u) = @W⇤⇤(u). (96)


Moreover, for every ` 2 R

@W�1(`) = Argminu2R

�

W(u)� `u

�

= C(W) \ @W⇤(`) ⇢ @W⇤(`) (97)

and

@W⇤(`) = (@W⇤⇤)�1(`) = co�

(@W)�1(`)�

, (98)

where the latter set is the convex hull of (@W)�1(`).We apply the above notions to the functions W(·) := W (·; u), u 2 R, defined by

W (u; u) := W (u) + I[u,1)(u) =

(

+1 if u < u,

W (u) if u � u,

for u 2 R. (99)

We consider their conjugates W ⇤(·; u), and the subdi↵erentials

pu(`) := @W

⇤(`; u) = (@W ⇤⇤(·; u))�1(`),

pu

c (`) :=�

@W (·; u)��1

(`) = @W

⇤(`; u) \ Cu

,

(100)

where Cu := C(W (·; u)).We observe that (cf. (35))

pu(`) = [pul (`), pu

r (`)] with

(

pul (`) = min{u � u : @W

⇤⇤(u; u) 3 `},

pur (`) = max{u � u : @W

⇤⇤(u; u) 3 `}.

(101)

Furthermore, the functions pul and pur are nondecreasing and satisfy the obviouscontinuity properties

(`n

# ` as n ! 1 ) ) limn!1

pur (`n) = pur (`),

(`n

" ` as n ! 1 ) ) limn!1

pul (`n) = pul (`);(102)

There is a last interesting property which relates the one-sided slopes W 0s,l(·),W

0i,r(·),

their upper monotone envelopes, and the convex envelope of W(·) = W (·; u).

Lemma 6.1. For u 2 R let us consider the upper monotone envelope

8

>

<

>

:

ru(u) := maxuvu

W

0i,r(v) if u > u,

ru(u) := (�1,W

0i,r(u)],

ru(u) := ; if u < u.

(103)

Then, there holds

@W

⇤⇤(u; u) = ru(u) for every u 2 R. (104)

In particular,

` 2 ru(u) , u 2 pu(`) = [pul (`), pu

r (`)],

` = W

0i,r(u) 2 ru(u) , u 2 pu

c (`),(105)

and

pul (`) = min{u � u : W 0i,r(u) � `}, pur (`) = inf{u � u : W 0

i,r(u) > `}. (106)

Proof. Let us start by proving (104): it is su�cient to consider the case u > u.Let W(u) = W (u; u). If u 2 Cu

\ (u,1) then, by (89) and (96) we have W

0i,r(u) =

W

0(u) = dduW

⇤⇤(u; u) ru(u). Let us consider the case when u 62 Cu and let (a, b)be the connected component of B(W) containing u. Since (W ⇤⇤)0 is constant in


(a, b), then (W⇤⇤)0(u) = (W⇤⇤)0(a) ru(a) ru(u). All in all, we have (W⇤⇤)0(u) ru(u) for every u > u. Since for every connected component (a, b) of B(W) we have

W0i,r(v)

W(b)�W(v)

b� v

W⇤⇤(b)�W⇤⇤(v)

b� v

= (W⇤⇤)0(v) for all v 2 (a, b), (107)

we also deduce that (W⇤⇤)0 � W0i,r(·). Combining the fact that ru(u) is the minimal

monotone map above W0i,r(·) in (u,1), and the monotonicity of (W⇤⇤)0, we conclude

(104).The first equivalence in (105) is a consequence of (100) and (101), and (106)

corresponds to (44). Concerning the second equivalence in (105), if u < u 2 pu

c (`)then (89) yields ` = W0

i,r(u). Conversely, if u 2 [pul (`), pu

r (`)] and W

0i,r(u) = ` then

we get u 2 Cu. In fact, this property obviously holds if u 2 {pul (`), pu

r (`)}; ifu 2 (pul (`), p

u

r (`)) then, combining W

0i,r(u) = ` = (W⇤⇤)0(u) with (107), we conclude

that all inequalities in (107) hold as equalities. Hence, we find

W(u)�W(pur (`))

u� pur (`)=

W

⇤⇤(pul (u))�W(pur (`))

u� pur (`),

whence W(u) = W⇤⇤(u).

6.2. Energetic solutions of (R,E, ) with monotone loadings.

Theorem 6.2. Let u 2 R and ` 2 C

1([a, b]) be a nondecreasing loading such that

W

0s,l(u)� �� `(a) W

0i,r(u) + �+. (108)

Any map u : [a, b] ! R satisfying

u(a) = u, u is nondecreasing, u(t) 2 pu

c (`(t)� �+) for every t 2 (a, b] (109)

is an energetic solution of (R,E, ). In particular, (109) yields

u(t) 2 [pul (`(t)� �+), pu

r (`(t)� �+)]. (110)

Notice that, if ` is strictly increasing, then any selection u(t) of pu

c (`(t) � �+)is also strictly increasing, so that the second condition of (109) is automaticallysatisfied.

Proof. Notice that (108) and (109) yield

`(t)� �+ 2 @W (u(t); u) for every t 2 [a, b]. (111)

Indeed, (111) holds also at t = a in view of (88) applied to W(u) := W (u; u), sincein this case W0

s,l(u) = �1 and W0i,r(u) = W

0i,r(u), cf. (99). A further application of

(88) yields the global stability condition (73). Since the subdi↵erential has a closedgraph, we also have

`(t)� �+ 2 @W (ur(t); u), `(t)� �+ 2 @W (ul(t); u) for every t 2 [a, b]. (112)

Let us set ↵ := inf{t > a : u(t) > u(a)}. Since u(a) satisfies the global stabilitycondition by (108), the function u is clearly a constant energetic solution in [a,↵].Thus, Proposition 4 shows that it is not restrictive to assume that ↵ = a.

In this case, ur(t) > u(a) for every t > a, so that ur(t) belongs to the interior ofthe domain of W (·; u); it follows from (109) that W 0(ur(t)) = W

0i,r(ur(t)) = `(t)� �+

in (a, b], and also at t = a by passing to the limit in the equation as t # a. Theformulation (76) of (DN) is thus satisfied in [a, b].

Let us check now the point c) of Theorem 5.1 concerning the jump conditions. Ift 2 J

u

, then in view of (112) pu

c (`(t)��+) contains two distinct points ul(t) < ur(t),


so that the graph of W ⇤⇤(·; u) is linear on [ul(t), ur(t)]. Since ul(t), u(t), ur(t) belongto the contact set Cu, in view of (95) the jump conditions (78) and (79) are alsosatisfied, and therefore u is an energetic solution.

We are now in the position to state the analogue of Theorem 4.4 for energeticsolutions. We also show a direct link with the notion of BV solution. To do this,we introduce the modified energy

W (u) := W (u) +

Z

u

u

W

0i,r(r) dr, E

t

(u) := W (u)� `(t)u. (113)

Theorem 6.3. Let ` 2 C

1([a, b]) be a nondecreasing loading and let u 2 BV([a, b];R)be an energetic solution of (R,E, ) satisfyingW (z)�W (u(a))

z � u(a)> W

0s,l(u(a)) for every z < u(a) if W

0s,l(u(a)) = `(a) + ��,

(114a)

W

0(z) < W

0(u(a)) in a left neighborhood of u(a) if W

0(u(a)) = `(a) + ��.(114b)

Then

a) u is also a BV solution of the rate-independent system (R, E,⇧).b) u can be represented as in Theorem 6.2, i.e.

u is nondecreasing, u(t) 2 pu(a)c (`(t)� �+) for every t 2 (a, b]. (115)

Proof. Claim 1: Let us suppose that ` and u are constant in the interval (⇢,�) and� 2 J

u

with ur(�) < ul(�). If ⇢ 2 Ju

then also ur(⇢) < ul(⇢).Let us denote by ¯ and u the constant value of ` and u on (⇢,�); by continuity

`(�) = `(⇢) = ¯ and ul(�) = ur(⇢) = u. We argue by contradiction and we supposethat u = ur(⇢) > ul(⇢). The jump conditions (79) and (80) yield

W (u)�W (ur(�)) = (¯+ ��)(u� ur(�)),

W (u)�W (ul(⇢)) = (¯� �+)(u� ul(⇢)).(116)

Taking the di↵erence of the two identities we get

W (ul(⇢))�W (ur(�)) = (�� + �+)(u� ul(⇢)) + (¯+ ��)(ul(⇢)� ur(�)) (117)

= (�� + �+)(u� ur(�)) + (¯� �+)(ul(⇢)� ur(�)) (118)

Clearly, the case ul(⇢) = ur(�) is impossible; if ul(⇢) > ur(�), upon dividing byul(⇢)� ur(�) (117) yields

W

0s,l(ul(⇢)) > ¯+ ��,

which contradicts the first inequality of (75); if ul(⇢) < ur(�), upon dividing byul(⇢)� ur(�) (118) yields

W

0i,r(ul(⇢)) < ¯

� �+,

which contradicts the last inequality of (75).Claim 2: Let a < ⇢ � 2 J

u

with ur(�) < ul(�), and let us assume that ⇢ 62 Ju

and ` and u are constant in the interval (⇢,�) if ⇢ < �. If ur(t) ur(⇢) in a

left neighborhood of ⇢, then there exists " > 0 such that `(t) ⌘ `(⇢) = `(�) and

u(t) ⌘ u(⇢) = ul(�) for every t 2 [⇢� ", ⇢]As in the previous claim, let us denote by ¯ = `(⇢) and u = ul(�) the constant

value of ` and u on (⇢,�). We argue by contradiction and we assume that there


exists a sequence t

n

< ⇢ converging to ⇢ such that un

:= ur(tn) " u, `

n

:= `(tn

) " ¯

and u

n

+ `

n

< u+ ¯. The global stability and the jump condition (82b) yield

(¯+ ��)(u� ur(�))

= W (u)�W (ur(�)) = W (u)�W (un

) +W (un

)�W (ur(�))

W

0(u)(u� u

n

) + o(u� u

n

) + (`n

+ ��)(un

� ur(�)) as n ! 1.

(119)

If for some n 2 N there holds un

⌘ u for every n � n, then we get

(¯+ ��)(u� ur(�)) (`n

+ ��)(u� ur(�)),

which contradicts the assumption `

n

<

¯. Up to extracting a suitable subsequencewe can then assume that u

n

< u for every n 2 N, so that u 2 supp(u0)+ andW

0(u) = ¯� �+ by (76). From (119) we then get

(¯+ ��)(u� u

n

) + (¯+ ��)(un

� ur(�))

(¯� �+)(u� u

n

) + (`n

+ ��)(un

� ur(�)) + o(u� u

n

)

as n ! 1, so that

(�� + �+)(u� u

n

) + (¯� `

n

)(un

� ur(�)) o(u� u

n

) as n ! 1,

which can be satisfied only if for n su�ciently big u

n

⌘ u and `

n

⌘

¯. Hence, wehave a contradiction.Claim 3: If for some t 2 J

u

there holds ul(t) = u(a) and `(t) = `(a), then ul(t) <ur(t).

It is su�cient to notice that (114a) is not compatible with (80).Claim 4:

Let a < �

0< � b be such that

`(t)�W

0(ur(t)) > �� = `(�)�W

0(ur(�)) for every t 2 [�0,�). (120)

Then � 62 Ju

.

We argue by contradiction and assume that � 2 Ju

. In view of (82a), necessarily

ul(�) > ur(�), (121)

and (77) shows that ur is nondecreasing in [�0,�).

Let R := {⇢ 2 [a,�] : ur(t) ⌘ ul(�), `(t) ⌘ `(�) for all t 2 [⇢,�]}. R is clearlyclosed and contains �. Let us show that R is open in [a,�]: it is su�cient to provethat for every ⇢ 2 R\(a,�], the set R contains a left neighborhood of ⇢ (R obviouslycontains also a right neighborhood of ⇢ if ⇢ < �).

When ⇢ > �

0 we can easily check that we can apply Claim 2. Indeed, ur(t) ur(⇢)in a left neighborhood of ⇢ since ur is nondecreasing in [�0

,�). Further, ⇢ cannot be ajump point for u thanks to Claim 1, which prevents a jump point with ul(⇢) < ur(⇢),and the fact that `(⇢) � W

0(ur(⇢)) > ��, which prevents a jump point withul(⇢) > ur(⇢). By the way, this shows that R � [�0

,�].If ⇢ �

0 we can still apply Claim 2, since we have that `(⇢) = `(�0) and ur(⇢) =ur(�0), so that `(⇢) �W

0(ur(⇢)) > ��, u is nondecreasing in a left neighborhoodof ⇢ by (77) and ⇢ 62 J

u

by the same arguments as above.Since R is both open and closed, we conclude that R = [a,�]. Claim 1 and Claim

3 yield that a 62 Ju

, so that ul(�) = u(a) and `(�) = `(a). A further application ofClaim 3 gives the contradiction with (121).Claim 5: There exists ↵ 2 [a, b] such that `(t) �W

0(ur(t)) > �� for all t 2 (↵, b]and u(t) ⌘ u(a), `(t) ⌘ `(a) in [a,↵].


Let us consider the set

⌃ := {t 2 [a, b] : W

0(ur(t)) = `(t) + ��}, (122)

and observe that

t

n

2 ⌃, t

n

# t ) t 2 ⌃. (123)

If a 2 ⌃ we denote by ⌃a

the connected component of ⌃ containing a, and weset ↵ = sup⌃

a

. If ↵ > a, then

W

0(ur(t)) = `(t) + �� for every t 2 [a,↵], (124)

so that by (76) u is nonincreasing in [a,↵]. Assumption (114a) and the jumpconditions (82b) imply that a 62 J

u

and ur(a) = u(a). Since also ` is nondecreasingwe conclude by (114b) and (77) that u(t) ⌘ u(a) and `(t) ⌘ `(a) in [a,↵]; moreover,by the same argument, ↵ 62 J

u

so that ↵ 2 ⌃. When a 62 ⌃ we simply set ↵ := a

and ⌃a

= ;.The Claim then follows if we show that ⌃ \⌃

a

is empty. This is trivial if ↵ = b.If ↵ < b we suppose ⌃ \ ⌃

a

6= ; and we argue by contradiction.We can find points ↵2 > ↵1 > ↵ such that ↵1 62 ⌃, and ↵2 2 ⌃. By (123) we

can consider � := min(⌃ \ [↵1, b]) > ↵1 > ↵; Claim 4 (with �

0 := ↵1) yields that� 62 J

u

, so that we can find " > 0 such that

� �� < `(t)�W

0(ur(t)) < �+ for every t 2 (� � ",�). (125)

Point b) of Theorem 5.1 implies that ur(t) = u(t) ⌘ ul(�) = ur(�) is constantin (� � ",�). Hence, W 0(ur(t)) ⌘ W

0(ur(�)) = `(�) + �� `(t) + �� for everyt 2 (� � ",�), since ` is nondecreasing and � 2 ⌃. This contradicts (125).Claim 6: u is nondecreasing on [a, b].

It follows immediately from (77) and Claim 5.Claim 7: Let B := {t 2 [↵, b] : W 0

i,r(u(a)) + �+ = `(t)} and let � := minB (with the

convention � := b if B is empty). Then u(t) ⌘ u(a) in [a,�) and

W

0i,r(ul(t)) = W

0i,r(ur(t)) = `(t)� �+ for all t 2 (�, b]. (126)

In particular, ul(t) � pul (`(t)� �+) for all t 2 (�, b].The first statement follows from the previous Claim and (76).To prove the second identity in (126) for ur(t), we argue by contradiction and

we suppose that a point s 2 (�, b] exists such that if W 0i,r(ur(s)) + �+ > `(s). Then,

by (82a) s is not a jump point, and therefore in view of point b) of Thm. 5.1, u islocally constant around s. Since ` is nondecreasing, because of (76) we concludethat u(t) ⌘ u(s) for every t 2 [↵, s], so that s �, a contradiction.

The first identity in (126) then follows by continuity. The last statement of Claim7 is a consequence of the first of (106).Claim 8: for all t 2 [a, b] we have ur(t) pur (`(t)� �+).The statement is obvious if ur(t) = u = u(a). If ur(t) > u, let z 2 (u, ur(t)): sinceu is nondecreasing we find s 2 [a, t] such that z 2 [ul(s), ur(s)], so that (79) (in thecase s 2 J

u

) or (76) (in the case ul(s) = ur(s)) yield

W

0i,r(z) `(s)� �+ `(t)� �+.

Since z is arbitrary, we conclude by the second of (106).Claim 9: conclusion.In order to check point a), we apply Theorem 4.3. In fact, due to the previousclaims, u fulfills (51) with the energy W (113).


Recalling Lemma 6.1, it is easy to check that also b) holds. In fact Claims 7 and8 yield

u(t) 2 [pul (`(t)� �+), pu

r (`(t)� �+)] = pu(`(t)� �+) for every t 2 (a, b]. (127)

Identities (126) and (105) show that ur(t), ul(t) 2 Cu for t > a. The jump condition(78) also yields that u(t) 2 Cu, and we conclude by definition (100).

Remark 1. In view of Proposition 3, a characterization of energetic solutionsanalogous to the one provided in Thm. 6.3 could be given in the case of a decreasingloading as well, cf. also Corollary 2.

Finally, we conclude with an example, to be compared with Ex. 1, which illu-strates energetic evolution in the case of a nonconvex double-well potential energy.

Example 3 (Energetic solutions driven by a double-well energy). We consider thevery same setting as in Example 1, viz. the double-well energy

W (x) =1

4(x2

� 1)2 for all x 2 R,

and a strictly increasing loading ` on [0, T ]. We start from an initial datum u 2

(u⇤,�1) with u⇤ as in (69). In order to illustrate the input-output relation on theinterval [0, t1], we consider the convexification of W on [u,+1), viz.

W

⇤⇤(u; u) =

8

>

<

>

:

14 (u

2� 1)2 if u �1,

0 if � 1 u 1,14 (u

2� 1)2 if u � 1.

with

d

duW

⇤⇤(u; u) =

8

>

<

>

:

u

3� u if u �1,

0 if � 1 u 1,

u

3� u if u � 1.

Therefore, taking into account the explicit form of dduW

⇤⇤(·; u)�1

in terms of thefunctions S

i

, i = 1, 3, in (70), we find the energetic solution

u(t)

8

>

<

>

:

= S1(`(t)� �+) if `(t) < �+,

2 [�1, 1] if `(t) = �+,

= S3(`(t)� �+) if `(t) > �+

for all t 2 [0, T ].

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Springer-Verlag, Berlin, 1994.

Received September 2011; revised October 2011.

E-mail address: [email protected] address: [email protected]

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