On rate–independent hysteresis models
Alexander Mielke Florian Theil Mathematisches Institut A Mathematics Institute Universit at Stuttgart University of Warwick D–70569 Stuttgart Coventry CV4 7AL, U.K. [email protected] [email protected]
5 April 2001 / revised 21 September 2001
1 Introduction
This paper deals with a general approach to the modeling of rate–independent processes which may display hysteretic behavior. Such processes play an important role in many applications like plasticity and phase transformations in elastic solids, electromagnetism, dry friction on surfaces, or in pinning problems in superconductivity, cf. [Vis94, BrS96]. The evolution equations which govern those processes constitute the limit problems if the in uence of inertia and relaxation times vanishes, i.e. the system rests unless the external loading is varied. Only the stick–slip dynamics is present in the Cauchy prob- lem, this means that the evolution equations are necessarily non–autonomous. Although the solutions often exhibit quite singular behavior, the reduced framework o ers great advantages. Firstly the amount of modelling can be reduced to its absolute minimum. More importantly, our approach is only based on energy principles. This allows us to treat the Cauchy problem by mainly using variational techniques. This robustness is necessary in order to study problems which come from continuum mechanics like plasticity, cf. [Mie00, CHM01, Mie01]. There the potential energy is invariant under the group of rigid body rotations SO(d) where d 1, 2, 3 is the dimension. This invariance implies that ∈ { } convexity can almost never be expected and more advanced lower semicontinuity results (like polyconvexity) are required to assure the existence of solutions for a time discretized version of the problem. An example which illustrates this remark is a problem from phase transformations in solids, see [MTL00]. Although none of the classical methods from Section 7 can be applied, we are able to prove the existence of solutions by establishing weak lower semicontinuity of certain critical quantities. Here we present an abstract framework which is based on two energy functionals, namely the potential energy I(t, z) and the dissipation (z˙). Here z X, X a separable, re exive Banach space with dual X , is the variable describing the∈ process, and z˙ is the time derivative. The central feature of rate–independence means that a solution z : [0, T ] X remains a solution if the time is rescaled. This leads to a dissipation functional→ : X [0, ) which is homogeneous of degree 1, i.e., ( v) = (v) for → ∞ 0 and v X. Special cases∈ of this situation are well studied in the theory of variational inequalities
1 or as sweeping processes. There we have the potential energy 1 I(t, z) = Az, z g (t), z (1.1) 2h i h i with A Lin(X, X ) symmetric and positive de nite and the dissipation has the form ∈ (v) = max w , v : w F (1.2) {h i ∈ } where F X is a bounded, closed, convex set containing 0 in its interior. The varia- tional inequalit y problem then is to nd z W 1,1([0, T ], X) such that for a.a. t [0, T ] ∈ ∈ Az g, v z˙ + (v) (z˙) 0 for all v X. (1.3) h i ∈ The equivalent sweeping process formulation is
z˙ ∂ (g Az) (or z˙ ∂ (Az)). (1.4) ∈ F ∈ g F Here F is the (convex) characteristic function of F taking the value 0 on F and else, and ∂ denotes the subdi erential. We refer to [Mor77, Mon93, KuM97, KuM98] for∞ more details on this matter. We treat these problems in Section 7 under the assumption, that I satis es certain smoothness assumptions. The purpose of this work is to broaden the model class sig- ni cantly. On the one hand we want to be able to deal with nonconvex or not strictly convex potentials. This leads to the possibility that the solutions are discontinuous, i.e. jumps can occur. Such singularities indicate that our purely energetic formulation is too simplistic to describe physical phenomena since the dynamics is not slow anymore. As it stands, our formulation favours discontinuities, i.e. the state will jump as soon as possible, independently of the shape of the energy landscape between the left hand and the right hand limit. We have to pay this price to obtain energy conservation independently from the regularity of the solutions. On the other hand we want to allow for restrictions of the state space as well, i.e. we want to be able to restrict z(t) to a closed convex set E X. A large part of this work is new even in the case of nite dimensional spaces X. Our point of departure is the following energy formulation of the problem: Find z : [0, T ] E with → (S) For a.a. t [0, T ] we have ∈ I(t, z(t)) I(t, y) + (y z(t)) for all y E, (1.5) ∈ (E) I(t, z(t)) + t (dz) I(s, z(s)) + t ∂ I( , z( ))d for all 0 s t T. s s t The rst condition R(S) can be interpreted Ras global stability of the state z(t) at time t: by changing z(t) E into y E the release of potential energy I(t, z(t)) I(t, y) is never ∈ ∈ larger than the associated dissipated energy. The second condition (E) is the integrated energy balance between nal, dissipated, initial energy and the “work done by external forces” in every time interval [s, t] [0, T ] (often ∂ I(t, z) = g˙ (t), z(t) ). t h i In Section 3 we show that formulation (1.5) is more general than the local formulation (1.3) but both formulations are equivalent if I(t, ) : X R is convex and satis es → 2 further technical conditions. Moreover, for convex problems we establish equivalence to the subdi erential formulation
0 ∂ (z˙) + ∂I(t, z), (1.6) ∈ which was introduced in [CoV90, Vis01] and called “doubly nonlinear problem”. However, these formulations have to be treated with care since in the nonconvex or not strictly convex case we have to allow for jumps of z. Hence, the right solution space is BV ([0, T ], X), the space of left–continuous functions having nite total variation. For an exact formulation of (1.6) we need to replace the derivative z˙ by a reduced derivative rd(z) : [0, T ] v X : v 1 and an associated Radon measure z ([0, T ]) such that → { ∈ k k } ∈ M z(t ) z(t ) = rd(z)(r) (dr). (1.7) 2 1 z Z[t1,t2) holds which means that rd(z)(t) z(dt) plays the role of the time–derivative which is a vector–valued measure. The correct replacement of (1.6) is then
0 ∂ (rd(z)(t)) + ∂I(t, z(t)) for –a.a. t [0, T ]. (1.8) ∈ z ∈ In Section 4 we analyse a general method for obtaining piecewise constant in time ap- proximations which works for nonconvex problems as well. It is based on the incremental problem for the time discretization 0 = t0 < t1 < t2 < . . . < tN 1 < tN = T
Let z = z(0). For k = 1, . . . N nd z E with 0 k ∈ (1.9) I(t , z ) inf I(t , y) + (y z ) : y E . ( k k { k k 1 ∈ } This approach leads in a natural way to discretized versions of energy formulation (1.5) (see Theorem 4.1) and, in the convex case, of the variational inequality (1.3) (see (4.3)). A central object in this study are the sets (t) of stable states at time t, namely S (t) = z E : I(t, z) I(t, y) + (y z) for all y E . S { ∈ ∈ } The major problem arises from (t) not being weakly closed since even for convex I(t, ) the set (t) may not be convex.S S Assuming weak closedness of the set of stable states plus some regularity conditions on I we give an existence proof in Theorem 6.3. ThisSproof relies on the incremental problem and on an abstract version of Helly’s selection principle (cf. [BaP86]). Here we do not require that I(t, ) is smooth, especially the assumption E = X which is central for Theorem 7.1 is not necessary . In Section 7 for E = X and I(t, ) uniformly convex we prove existence, uniqueness (Theorem 7.1) and discuss the question of temporal smoothness of solutions. To this end we show (Theorem 7.3) that the solutions of the incremental problem converge strongly, by generalizing ideas in [HaR95]. As far as the authors know there is no uniqueness result even for nontrivial E X X = R2. We propose a new “structure condition” which implies uniqueness, cf. Section 7.2 and Appendix C. We note that in [HaR95] a convergence result of the incremental problem (for the case (1.1) only) is given only under the assumption z W 2,1((0, T ), X), i.e. z L1((0, T ), X). ∈ ∈ 3 Our above–mentioned convergence result Theorem 7.3 is independent of this assumption. From very simple examples we see that, even for the case E = X, we can only expect z˙ BV((0, T ), X). Our Theorem 7.8 proves this under the additional assumption that ∈ the boundary of F is C2. For many applications (see e.g. [KuM00, MTL00, GoM00]) the assumption of re ex- itivity of X is too restrictive. We note that the formulation (1.5) via energy functionals provides us with the advantage that the time derivative z˙ is not required. Moreover, the existence theorem 6.3 uses only the assumption of weak sequential compactness which is considerably weaker, see Remark 6.4. Throughout this work we have assumed that the underlying geometry is the linear space X, in particular the dissipation functional does not depend on the position z ∈ E X. A more general treatment should involve Banach manifolds and the dissipation is then a Finsler metric. For applications in this context see [Mie00, Mie01].
2 Notations and setup of the problem
Let X be a separable, re exive Banach space and E a closed convex subset. For each z E the (inward) tangential cone T E is de ned via ∈ z T E = w X : r > 0 : z + rw E , (2.1) z { ∈ ∃ ∈ } where A is the closure of A in the strong topology. Later on we will also use the (outward) normal cone N E X de ned via z N E = y X : y , w 0 w T E . z { ∈ h i ∀ ∈ z }
The dissipation is implemented by a function : X [0, ) which is convex, homo- geneous of degree 1 (i.e. ( z) = (z) for 0 and→z ∞X) and satis es, for some ∈ C(2) C(1) 1, C(1) z (z) C(2) z for all z X. (2.2) k k k k ∈ Convexity and homogeneity of degree 1 imply the triangle inequality
(z+z˜) (z) + (z˜) for all z, z˜ X (2.3) ∈ which will be used often subsequently. These assumptions are equivalent to the existence of a convex closed set F in X with z X : z C(1) F z X : z C(2) such that { ∈ k k } { ∈ k k } (z) = max z , z : z F . (2.4) {h i ∈ } We continue to use , for the duality pairing on X X. h i The space of functions of bounded variations is de ned here to be
BV([0, T ], X) = z : [0, T ] X : Var(z, [0, T ]) < { → ∞}
4 N where Var(z, [0, T ]) = sup k=1 z(tk) z(tk 1) : 0 t0 ([s, t)) = t s + dz and z(t) z(s) = rd(z)(r) (dr). z k k z Z[s,t) Z[s,t) Here rd(z) : [0, T ] X is de ned only –almost everywhere (a.e.). Note that our de - → z nition of the di erential measure di ers from that in [Mon93, Sect.0.1] where z([s, t)) = dz and nd(z)(t) = 1 –a.e. in [0, T ] such that dz = nd(z) = rd(z) . The [s,t) k k k k z z z derivative doesn’t distinguish between right and left continuous versions, i.e., rd(z ) = R + rd(z ) and z = z+ . For z = z+ with initial datum z(0 ) = z0 = z(0) we have ( 0 ) = z(0) z =: r > 0 and rd(z)(0) = 1 [z(0) z ]. 6 z { } k 0k r 0 For z BV ([0, T ], X) the –variation on the interval J [0, T ] is de ned by ∈ def Var (z, J) = (dz) = (rd(z)(t)) z(dt). ZJ ZJ This is the same as the supremum over all sums of the form N (z(t ) z(t )) where k=1 k k 1 N N, t0, tN J and t0 convex: I(t, z ) (1 )I(t, z ) + I(t, z ) for all [0, 1]; 0 1 ∈ strictly convex: I(t, z ) < (1 )I(t, z ) + I(t, z ) for all (0, 1); 0 1 ∈ –unif. convex: I(t, z ) (1 )I(t, z ) + I(t, z ) ( 2) z z 2 for all [0, 1]; 0 1 2 k 0 1k ∈ where z = (1 )z + z . These convexity conditions are never assumed without stating 0 1 explicitly. The most general formulation for our rate–independent processes is global in time. We have a set of stable states which depends on time, in addition a solution satis es the energy inequality. This formulation does not rely on convexity assumptions for I and is particularly useful for problems where only generalized convexity notions like quasicon- vexity hold, since weak lower semi–continuity is still valid. Moreover it does not involve derivatives of I with respect to z. (GF) [Global Formulation] Find z BV ([0, T ], X) with z(0 ) = z0 and z(t) E such that conditions (S) and (E) hold: ∈ ∈ (S) for –a.a. t [0, T ] : I(t, z(t)) I(t, y) + (y z(t)) for all y E, ∈ t2 ∈ t2 (E) for all 0 t1 t2 T : I(t2, z(t2)) + (dz) I(t1, z(t1)) + ∂tI(s, z(s))ds, t1 t1 t2 R R where (dz) = (dz) in the case z BV ([0, T ], X) and accordingly if t1 [t1,t2) ∈ z BV ([0, T ], X) ∈ R+ R Here (S) is the condition of global stability in the whole state space E, and denotes the one–dimensional Lebesgue measure. The de nition of the energy inequality (E) is such that it implies the two natural requirements for evolutionary problems, namely restrictions and concatenations of solu- tions remain solutions. To be more precise consider a solution z : [0, T ] E and any → subinterval [s, t] [0, T ], then the restriction z [s,t] solves (GF) with initial datum z(s ). Moreover, if z : [0, t] E and z : [t, T ] E solv| e (GF) on the respective intervals and 1 → 2 → if z (t ) = z (t ) then the concatenation z : [0, T ] E solves (GF) as well. 1 2 → In particular, (S) and (E) imply that if z jumps at time t from z to z+ then I(t, z ) + (z z ) = I(t, z ). (2.7) + + 3 Three alternative formulations of the problem There exist three di erent formulations which are equivalent to (GF) if the potential energy I is convex. In the nonconvex case only certain implications are correct. The global formulation has the big advantage that it doesn’t need the di erentiability of the potential energy I(t, ). The other three formulations involve the derivative DI(t, z(t)). 6 The rst formulation (LF) localizes both, the de nition of stable sets and the energy inequality. It is somehow impractical since the structure of a constrained evolution is still present; in particular, the tangent spaces Tz(t)E depend discontinuously on z(t). The second formulation (VI) is a variational inequality which is a standard rewriting of the local formulation (LF). Hence, it is always equivalent to (LF). The third formulation (SF) is independent from these ideas. It is a single evolution equation without constraints which has the di culty that the time derivative appears inside of a strong nonlinearity. The following formulations will make essential use of knowledge on the reduce deriva- tive which is contained in the following lemma. This result explains why certain of our formulations are more natural for z = z while others are more natural for z = z+. We will always indicate this by adding a subscript or + to the formulation. However, for the global formulation (GF) this is not necessary, as it is easily seen that z = z solves (GF) if and only if w = z+ solves (GF)+. Lemma 3.1 For z BV ([0, T ], X) with z : [0, T ] E there exists a set [0, T ] of full –measure such∈that for all t we have → T z ∈ T rd(z)(t) T E and rd(z)(t) T E. ∈ z (t) ∈ z+(t) Proof. We use the stretched function z CLip([0, T ], X) associated to z which is de ned ∈ in (A.1). De ne the set [0, T ] to be the set of points where we have T b z0( ) = d z( ) = lim 1 [z(b + ) z( )] = lim 1 [z( ) z( )]. d e &b0 &0 By Lebesgue’s lemma has full Lebesgue measure, i.e., ( ) = ([0, T ]) = T , cf. [Mon93]. b 0bT b b 0T b b Moreover, z( ) z ( ) + o( ) = z( ) E implies z ( ) Tb E for all . ∈ ∈ z( ) ∈ T The desired result bis now obtained by undoing the stretcb hing usingb theb mapping t : t andbthe setb = t( ) b[0, T ]. We nd z( )b= ( ) = T = z([0, T ]).b For 7→ T T T T continuity points of z we have rd(z)(t) = z0( (t)), see (A.2). At jump points the desiredb result is obvious as rd(z) bpoinb ts in the jump direction from z(tb) = z b(t) to z+(t). 1 The local formulation needs the assumptionb b that I is a C function. The measure z also appears and thus jump points can be treated suitably. Recall that the set of jump points has Lebesgue measure 0, but each jump point has a positive z–measure. (LF) [Local Formulation] Find z BV ([0, T ], X) with z(0) = z and z(t) E ∈ 0 ∈ such that conditions (Sloc) and (Eloc) hold z–a.e. in [0, T ]: (S ) DI(t, z(t)), v + (v) 0 for all v T E, loc h i ∈ z(t) (E ) DI(t, z(t)), rd(z)(t) + (rd(z)(t)) 0. loc h i Now restrict to the case z = z . Using rd(z)(t) Tz(t)E (from Lemma 3.1) and subtracting the two conditions we are lead to the single∈variational inequality DI(t, z(t)), v rd(z)(t) + (v) (rd(z)(t)) 0 for all v T E h i ∈ z(t) which is equivalent to (Sloc) & (Eloc). Introducing the characteristic function Tz(t)E we can incorporate the side condition into the variational inequality and obtain the third formulation. 7 (VI) [Variational Inequality] Find z BV ([0, T ], X) with z(0) = z and z(t) ∈ 0 ∈ E such that DI(t, z(t)), w rd(z)(t) + (w) (rd(z)(t)) h i + (w) (rd(z)(t)) 0 for all w X Tz(t)E Tz(t)E ∈ holds z–a.e. in [0, T ]. The fourth formulation uses subdi erentials denoted by ∂. For a characteristic function : X [0, ] we nd the normal cone mapping, i.e., ∂ (z) = N E for z X. The E → ∞ E z ∈ subdi erential of the dissipation functional is given via F for v = 0, ∂ (v) = argmax z , v = (3.1) z ∈F h i F : (v) = , v for v = 0. ( { ∈ h i } 6 (SF) [Subdi erential Formulation] Find z BV ([0, T ], X) with z(0 ) = z0 and z(t) E such that ∈ ∈ 0 ∂ (rd(z)(t)) + DI(t, z(t)) + ∂ (z(t)) for –a.a. t [0, T ]. ∈ E z ∈ This formulation is especially useful for general, convex I(t, ) (not necessarily di er- entiable, but lower semicontinuous). In writing IE(t, z) = I(t, z) + E(z) we obtain the short form 0 ∂ (rd(z)(t)) + ∂I (z(t)) efor –a.a. t [0, T ]. (3.2) ∈ E z ∈ Assuming I (t, z) = J(z) g (t), z this is exactly the doubly nonlinear formulation of E h i e Colli & Visintin [CoV90, Vis01], namely g (t) ∂ (rd(z)(t)) + ∂J(z(t)). ∈ e The rst aim of this section is to show that the global formulation (GF) always implies the local formulations (LF) and (LF)+ but not vice versa. For the case that I(t, ) is convex and satis es the above technical assumption we show that all three formulations are equivalent. Second we compare the formulations (SF) and (LF) and at the end of the section we present the associated formulation as a sweeping process. Example 3.2 The formulation (SF) is not very useful, as is seen by this simple example. 2 Take E = [0, l] R and I(t, z) = z /2 tz with > 0. Choosing z0 = 0 the unique solution of (GF), (LF) , (VI) and (SF)+ is given by zsln(t) = max 0, min l, (t 1)/ for t 0. This function is also a solution of (SF) , however, there{ are{ man y more}} solutions: Choose (0, 1] and Z (0, max l, (t+1)/ ). Then, the function z with z (t) = 0 for t and∈ z (t) = max∈Z, z (t){ for t > }solves (SF) as well. { sln } Example 3.3 We consider E = X = R, (v) = v and a general nonconvex, di er- | | entiable potential I : [0, T ] R [0, ). We discuss what conditions are imposed on → ∞ possible jumps from z0 to z1 by the di erent formulations (GF), (LF) and (LF)+, re- spectively. In all cases stability leads to the necessary condition DI(t, z ) 1. For | j | (GF) we nd DI(t, z ) = DI(t, z ) = sign(z z ) together with the global condi- 0 1 1 0 tion I(t, z0) = I(t, z1)+ z1 z0 I(t, y)+ y z0 for all y R. For (LF) we nd | | | | ∈ 8 DI(t, z ) = sign(z z ) and DI(t, z ) 1 whereas (LF) gives DI(t, z ) 1 and 0 1 0 | 1 | + | 0 | DI(t, z1) = sign(z1 z0). The two local formulations (LF) make no statement on the jump of I, so generally (E) cannot be recovered. Theorem 3.4 (Relation between (GF) and (LF)) (a) For general I we have (GF) (LFaver) , where (E ) is replaced by ⇒ loc (Eaver) B (t), rd(z)(t) + (rd(z)(t)) 0 –a.e. on [0, T ], (3.3) loc h i z with B (t) = 1 DI(t, (1 )z (t)+ z (t))d . 0 + (b) for general I we have (LF) (VI) ; R ⇔ (c) for convex I we have (GF) (LF) (LF) . ⇔ ⇔ + Theorem 3.5 (Relation between (LF) and (SF)) (a) For general I we have (SF) (LF) ; + ⇔ + (b) for general I we have (LF) (SF) ; ⇒ (c) for convex I we have (GF) (LF) (SF) . ⇔ ⇔ + The proofs of both theorems use several intermediate results which are developed now. Lemma 3.6 (S) implies (Sloc). Proof. For any v T E 0 we nd a sequence of w X and a sequence r > 0 ∈ z(t) \ { } k ∈ k such that wk = v , wk v, rk 0 and zk = z(t)+rkwk E. From stability (S) and di erentiabilitk ky wekconcludek → → ∈ 1 0 I(t, z(t)+rkwk) + (rkwk) I(t, z(t)) rk = DhI(t, z(t)), w + (w ) + o(r ) . i h ki k k k→∞ With k we obtain (S ). → ∞ loc The next lemma shows that the energy inequality (E) can be replaced by an en- ergy identity under natural circumstances. In fact, considering the energy I(t, z(t)) + (dz) ∂ I( , z( ))d we see that (S) implies that this energy cannot decrease. [0,t) [0,t) t The same conclusion follows from (S ) at continuity points of z. R R loc Lemma 3.7 (Energy conservation) A process z BV ([0, T ], E) satis es for all 0 ∈ s < t T the energy identity I(t, z(t)) + [s,t) (dz) = I(s, z(s)) + [s,t) ∂tI( , z( )) d . (3.4) if one of the following two cRonditions is satis ed: R (1) z satis es (S ) and (E ) and z C([0, T ], E); loc loc ∈ (2) z satis es (S) on all of [0, T ] and (E) holds for t1 = 0 and t2 = T only. 9 Proof. We de ne the quantity e(s, t) = I(t, z(t)) + (dz) I(s, z(s)) ∂ I( , z( )) d . t Z[s,t) Z[s,t) By de nition we have e(r, t) = e(r, s) + e(s, t) for any r < s < t. We use part (c) of Theorem A.1: t I(t, z(t)) I(s, z(s)) = ∂ I(r, z(r)) dr + B (t), rd(z)(r) (dr) (3.5) t h i z Zs Z[s,t) with B (t) = 1 DI(t, (1 )z(t)+ z (t))d . This yields 0 + R e(s, t) = B (t), rd(z)(t) + (rd(z)(t)) z(dt). (3.6) [s,t) h i Z h i If z is continuous we have B (t) = DI(t, z(t)), and (Sloc) & (Eloc) together with Lemma 3.1 give the rst claim. For the second claim we have that e(0, T ) 0. We use (Sloc) to show e(s, t) 0 for all s < t which gives the desired result e(s, t) = 0. Observe that the integrand in (3.6) is nonnegative z–a.e. in [0, T ] by (S). Indeed, if z is continuous at t then this follows from (Sloc), and if z has a jump from z to z+ at time t, then B (t), rd(z)(t) + (rd(z)(t)) (dt) = B (t), z z + (z z ) {t} h i z h + i + h i = I(t, z ) I(t, z ) + (z z ) R + + which is 0 by the global stability of z at time t, see (2.7). In the proof of Lemma 3.7 we saw that it might be more natural to replace in (LF) the local energy condition (Eloc) by an averaged version (3.3). This unsymmetry disappears in the case of convex I(t, ), as the following result implies B (t) = DI(t, z(t)) z–a.e. in [0, T ]. Lemma 3.8 Assume z BV ([0, T ], E) solves (GF) and that I is convex. Then, the map [0, T ] X ; t D∈I(t, z(t)) is continuous. Moreover, if z jumps at time t from z to z , then→ DI(t, 7→) is constant along the straight jump line. ¡ + ¢ Proof. The result is trivial at points where z is continuous. Hence, we consider the case of a jump. Let z = (1 )z + z+ and B = DI(t, z ). On the one hand convexity implies I(t, z ) I(t, z ) + B , z z . On the other hand global stability gives h + i I(t, z ) I(t, z ) + (z+ z ). Together with (2.7) we nd that the function I(t, ) is a ne on the straight line connecting z and z . Since I(t, ) is convex and di erentiable, + at all points of the jump line the tangent planes are the same. This proves the result. We now relate the subdi erential formulations (SF) to the local formulations (LF) . The basic result is obtained from the following lemma which uses simple arguments from convex analysis. 10 Lemma 3.9 Let C X be a closed convex cone and C X the dual cone, see Appendix B. Moreover, assume that : X [0, ) is as ab ove. Then, the following three conditions on w X and X satisfy→ the∞implications (i) (ii) (iii). ∈ ∈ ⇒ ⇒ (i) w C and 0 ∂ (w) + + C ; ∈ ∈ (ii) , v + (v) 0 for all v C and , w + (w) 0; h i ∈ h i (iii) 0 ∂ (w) + + C . ∈ This result has a strange unsymmetry which cannot be avoided. Already the simplest case X = R, C = [0, ) and (v) = v shows this. The values = 1 and w = 1 satisfy ∞ | | the condition (iii) but not (ii). Moreover, = 1 and w = 1 satis es (ii) but not (i). Assuming w C in addition to (iii) doesn’t help to imply (ii). ∈ Proof. Condition (iii) is equivalent to ∂ (w) + C . From the form of the subdif- ∈ ferential ∂ in (3.1) we nd the equivalent formulation (iii)’ v C : , v + (v) 0 and C : , w + (w)+ , w = 0 . ∀ ∈ h i ∃ ∈ h i h i h i h i Adding the condition w C we obtain (i), and the implication (i) (ii) follows directly from , w 0. ∈ → Toh pro ve thati (ii) implies (iii) we consider two cases. If , w + (w) = 0, then we h i choose = 0 and (iii)’ holds. If , w + (w) < 0 then the rst condition in (ii) tells us that w C. Thus there existsha i C with , w > 0. Choosing r > 0 suitably the vector 6∈ = r satis es the second ∈condition inh (iii)’.i After the preparatory work, it is easy to prove the Theorems 3.4 & 3.5. Proof of Theorem 3.4. It is essential that for any solution z of the four formulations we can apply Lemma 3.1 which guarantees rd(z)(t) T E –a.e.. ∈ z (t) z Part (a) and the direction “ ” in parts (b) and (c) are proved above. ⇒ ad (b) “ ”: Inserting w = 0 as a test state in (VI) we obtain (Eloc). Next insert w = v+rd(⇐z)(t), divide by and take the limit . By the lower semicontinuity of → ∞ ( ) and Tz(t) ( ) we arrive at (Sloc). ad (c) “ ”: In the case of convexity the global stability (S) immediately follows from ⇐ the local one (Sloc). Integrating (Eloc) and using (3.5) together with Lemma 3.8 gives the global energy condition (E). Proof of Theorem 3.5. We observe the correspondence between (SF)+, (LF) and (SF) and the conditions (i), (ii) and (iii) in Lemma 3.9, respectively, where rd(z)(t) T E ∈ z+(t) is used for (i) but rd(z)(t) Tz (t)E is of no help in (iii). This proves (a) and (b),∈ where the equivalence in (a) follows as rd(z)(t) T E ∈ z+(t) is known also in (LF)+. Part (c) is a consequence of (a) and Theorem 3.4(c). Finally we connect our formulation to the so–called sweeping processes as discussed in [KuM97, KuM98, Mon93]. Again we assume convexity of I. Our subdi erential for- mulation (SF) is posed in the dual space X and we need to employ duality arguments (see Appendix B) involving the Legendre–Fenchel transform to return to an equation L 11 in the space X. In the case E = X, our equation reads DI(t, z(t)) ∂ (rd(z)(t)), and by Theorem B.1 this is equivalent to ∈ rd(z)(t) ∂ ( DI(t, z(t))) for –a.a. t [0, T ], (3.7) ∈ F z ∈ since is the Legendre transform of . F L In the case of E & X the situation is a little more involved as we cannot insert a set–valued function into a subdi erential. We start from the variational inequal- ity (VI) which can be rewritten as m (w) m (rd(z)(t)) = 0 for all w X where t t ∈ mt(w) = DI(t, z(t)), w + (w) + Tz(t)E(w). This is equivalent to 0 ∂mt(rd(z)(t)) or DI(t, z(ht)) ∂f (rd(zi)(t)) where f (w) = (w) + (w). By the∈ duality theorem ∈ t t Tz(t)E B.1 we have rd(z)(t) ∂( ft)( DI(t, z(t))). Using Proposition B.3 we obtain an explicit form for which leads∈ toLthe nal result. L (SP) [Sweeping Process Formulation] Find z BV ([0, T ], X) with z(0) = z ∈ 0 and z(t) E such that ∈ rd(z)(t) ∂ ( DI(t, z(t))) for –a.a. t [0, T ]. (3.8) ∈ F +Nz(t)E z ∈ 4 Time discretization One of the standard methods to obtain solutions of nonlinear evolution equations is that of approximation by time discretizations. To this end we choose discrete times 0 = t0 < t1 < . . . < tN = T and consider the incremental problem. (IP) For given z E nd z , . . . , z E such that 0 ∈ 1 N ∈ z argmin I(t , z) + (z z ) : z E (4.1) k ∈ { k k 1 ∈ } for k = 1, . . . , N. By weak lower semi–continuity and boundedness from below of I(t, ) we obtain the following result. Theorem 4.1 The incremental problem (4.1) always has a solution. Any solution satis- es for k = 1, . . . , N (i) zk is stable for time tk (ii) ∂tI(s, zk) ds I(tk, zk) I(tk 1, zk 1) + (zk zk 1) ∂tI(s, zk 1) ds [tk 1,tk) [tk 1,tk) (iii) N (z z ) I(0, z ) + C T R k=1 k k 1 0 1 R (iv) z z + (I(0, z ) + C T )/C(1). kPkk k 0k 0 1 Remark: The assertions (i) and (ii) are the best replacements of the conditions (S) and (E) in the time–continuous case. Proof. From I(t, z) 0 we have I(t , z) + (z z ) C(1) z z . k k 1 k k 1k 12 j Hence any minimizing sequence (zk)j∈N is bounded and a subsequence converges weakly to z E X. Convexity of and weak lower semi–continuity of I(t , ) gives k ∈ k j j I(tk, zk) + (zk zk 1) lim inf I(tk, z ) + lim inf (z zk 1) j→∞ k j→∞ k j j lim inf I(tk, z ) + (z zk 1) = inf I(tk, y) + (y zk 1) : y E . j→∞ k k { ∈ } h i This is equivalent to zk argmin I(t, ) + ( zk 1). The stability (i) is obtained∈ by the minimization property and the triangle inequality (2.3) as follows. For all w E we have ∈ I(t , w) + (w z ) = I(t , w) + (w z ) + (w z ) (w z ) k k k k 1 k k 1 I(t , z ) + (z z ) + (w z ) (w z ) I(t , z ). k k k k 1 k k 1 k k The lower estimate in the energy estimate (ii) is deduced from the stability of zk 1 with respect to zk: I(t , z ) + (z z ) = I(t , z ) + (z z ) + ∂ I(s, z ) ds k k k k 1 k 1 k k k 1 t k Z[tk 1,tk) I(t , z ) + ∂ I(s, z ) ds. k 1 k 1 t k Z[tk 1,tk) The upper estimate in (ii) follows since zk is a minimizer: I(t , z ) + (z z ) I(t , z ) = I(t , z ) + ∂ I(s, z ) ds. k k k k 1 k k 1 k 1 k 1 t k 1 Z[tk 1,tk) Adding up (ii) for k = 1, . . . , N we nd N N I(T, zN ) I(0, z0) + (zk zk 1) ∂tI(s, zk 1) ds. [tk 1,tk) Xk=1 Xk=1 Z Using I(t, z) 0 and ∂ I(t, z) C we obtain (iii). Now (iv) follows from C(1) z z | t | 1 k k 0k (z z ) N (z z ) and (iii). k 0 j=1 j j 1 For each discretization P = 0, t , . . . , t , T of the interval [0, T ] and each incre- P { 1 N 1 } mental solution (zk)k=1,...,N of (IP) we denote by ZP a piecewise constant function with Z (0) = z and Z (t) = z for t (t , t ]. (4.2) P 0 P k ∈ k 1 k Summing (iii) in Theorem 4.1 over k = 1, . . . , N we nd the following result. Corollary 4.2 If (zk)k=1...N solves (IP), then ZP satis es the energy inequality T I(T, Z (T )) + (dZ ) I(0, z ) + ∂ I(s, Z (s))ds. P P 0 t P Z[0,T ] Z0 The minimization property of zk leads to a necessary local condition: 13 Proposition 4.3 If zk solves (4.1) then DI(t , z ), w z +z + (w) (z z ) 0 for all w T E. (4.3) h k k k k 1i k k 1 ∈ zk If I(tk, ) is strictly convex and E = X, then (4.3) has a unique solution. Then (4.1) and (4.3) ar e equivalent. Proof. Let z = (1 )z + z where [0, 1]. The minimization property of z gives k k 1 ∈ k I(t , z ) + (z z ) I(t , z ) + (z z ) k k k k 1 k k 1 = I(t , z ) + DI(t , z ), z z ) + o( ) + (1 ) (z z ), k k h k k k 1 k i k k 1 for 0. Subtracting the terms of order 0, dividing by and taking the limit 0 yields→DI(t , z ), z z ) + (z z ) 0. → h k k k k 1 i k k 1 From Theorem 4.1 we know that zk is stable and comparing with z = zk + w, w Tzk E, gives similarly DI(tk, zk), w + (w) 0. Subtracting the previous inequality giv∈es (4.3). h i (j) Let E = X and let zk , j = 1, 2 be two solutions of (4.3). Then, we can use w = (3 j) (3 j) zk zk 1 as test–function in (4.3) and the estimates for j = 1 and 2 to obtain (1) (2) (2) (2) (1) (1) DI(tk, zk ) DI(tk, zk ), zk zk 1 (zk zk 1) 0. Induction over k together with h (1) (2) i strict convexity implies that zk = zk . For the uniformly convex case we obtain a Lipschitz bound for the incremental problem (4.1). Theorem 4.4 If I(t, ) is –uniformly convex then any solution of (4.1) satis es C z z 2 t t for k = 1, . . . , N. k k k 1k | k k 1| Proof. The stability of z at t implies via (S ) and z z T E the estimate k 1 k 1 loc k k 1 ∈ zk 1 DI(t , z ), z z + (z z ) 0. (4.4) h k 1 k 1 k k 1i k k 1 Adding this to (4.3) with w = 0 we have DI(t , z ) DI(t , z ), z z 0. With h k 1 k 1 k k k k 1i the uniform convexity and assumption (2.6) we continue 0 DI(t , z ) DI(t , z ), z z + DI(t , z ) DI(t , z ), z z h k k k k 1 k k 1i h k k 1 k 1 k 1 k k 1i z z 2 C t t z z ; k k k 1k 2| k k 1|k k k 1k and the result is established. 5 Stable sets The sets of stable points play an important role in the analysis. For t [0, T ] we let ∈ (t) = z E : I(t, z) I(t, y) + (y z) for all y E , S { ∈ ∈ } which is the set of all stable points at time t. The condition (S) now reads “z(t) (t)”. ∈ S 14 Lemma 5.1 Let I( , z) : [0, T ] [0, ) be continuous and I(t, ) : E [0, ) be lower semicontinuous. Then, → ∞ → ∞ (a) for each t the set (t) is closed; S (b) if t t and z (t ) with z z, then z (t). n → n ∈ S n n → ∈ S Proof. Let H(t, z, w) = I(t, w) + (w z) I(t, z) and (t, z) = inf H(t, z, w) : w E . H { ∈ } Clearly, (t, z) 0 and z (t) (t, z) = 0. H ∈ S ⇔ H Assume zn (t) and zn z, then H(t, zn, w) 0 for all w. By continuity of I(t, ) and ( ) we nd∈ SH(t, z, w) →0 and conclude z (t). This proves (a). ∈ S Part (b) follows from (a) and the (strong) continuity of H( , z, w) : [0, T ] R. → In the case E = X, I(t, ) convex and in C1(X, R) the stable set is simply characterized by (t) = z : DI(t, z) F . In the general case we have S { ∈ } z (t) DI(t, z) F + N E (5.1) ∈ S ⇒ ∈ z with equivalence if I(t, ) is convex. We will see in the follo wing that weak closedness of (t) is a very desirable property. A natural way to obtain weak closedness of the stable setsS is to show convexity by using that F is su ciently round. We say F is –round if [0, 1] z , z F : ∀ ∈ ∀ 0 1 ∈ (5.2) w X : w ( z + (1 )z ) (1 ) z z 2 F . { ∈ k 1 0 k k 0 1 k } 1 In a Hilbert space the ball BR(z ) is (2R) –round. Theorem 5.2 Assume E = X and that F is –round. Moreover, assume that I(t, ) 3 3 ∈ C (X, R) with D I(t, z) X X X→R M for all z and that I is –uniformly convex. Then the inequalityk M/(2k 2) implies that (t) is convex. S Proof. Take z , z (t). By (5.1) we have = DI(t, z ) F for j = 0 and 1. For 0 1 ∈ S j j ∈ [0, 1] we let z = (1 )z + z , then it su ces to show ∈ 0 1 DI(t, z ) (1 )DI(t, z ) DI(t, z ) (1 ) 2. k 0 1 k k 0 1k The left–hand side takes the form (1 ) D2I(t, z +s(z z )) D2I(t, z +s(z z )) (z z )ds 1 1 0 0 0 1 ° Z[0,1] ° ° £ ¤ ° ° M 2 ° and thus °can be estimated by (1 ) z0 z1 . Uniform convexity gives ° 2 k k DI(t, z ) DI(t, z ), z z h 0 1 0 1i z z k 0 1 k z z k 0 1k k 0 1k and the result is established. An important special case is E = X and I being a convex quadratic functional. 1 Corollary 5.3 Assume E = X and I(t, z) = 2 A(t)z, z g(t), z with A(t) Lin(X, X ) and A(t)z, z > 0 for all z X. Then (t) ishconvex. i h i ∈ h i ∈ S 15 For convenience, we give a direct proof. However, this result is a special case of Theo- rem 5.2 and of Theorem 5.4. Proof. Since I(t, ) is convex and smooth we have z (t) DI(t, z) = A(t)z+g(t) ∈ S ⇔ ∈ F . Since F is convex and DI is linear we conclude convexity of (t). S Theorem 5.4 If one of the following conditions holds, then the stable sets (t) are weakly closed : S (1) (t) is convex. S (2) For all w E the mapping (w ) : E R is weakly continuous. ∈ → (3) E = X and the mapping DI(t, ) : E X is weakly continuous. → Proof. Part (1) is clear since closed convex sets are weakly closed. Part (2) follows by using weak continuity of ( ) and weak lower semi–continuity of I(t, ) as follows. Assume zn * z and zn (t). With the notation as in the proof of Lemma 5.1 we have ∈ S 0 lim sup H(t, w, zn) = I(t, w) + lim (w zn) lim inf I(t, zn) n→∞ n→∞ n→∞ I(t, w) + (w z) I(t, z) = H(t, w, z) for all w E. Thus, z (t). Part (3)∈ follows with∈ (5.1)S where N E = 0 . z { } (1) (2) Note that the condition (2) together with 0 < C C < in (2.2) is rather restrictive; it means that E B (0) is compact for each r > 0. ∞ ∩ r In the following examples we give simple functionals I(t, z) and (z) such that the stable set (t) is nonconvex and not weakly closed. See also Example C.3. S Example 5.5 Let E = X = R H where H is a Hilbert space. Let z = (a, h) X and ∈ 1 I(t, z) = (a2+ h 2)2 (t)a, (z) = a2 + h 2. 4 k k k k p 2 2 a (t) Then z (t) DI(t, z) 1, where DI(t, (a, h)) = (a + h ) h 0 . Now, ∈ S ⇔ k k 1/3 k k assume (t1) = 2, then (a, 0) (t1) if and only if a [1, 3 ]. Consider z = (a , h ) 5 8 1/3 ∈ S 3 16 1/6 ∈ ¡ ¢ ¡ ¢ with a = (3 /2 ) < 1 and h = (3 5 /2 ) , then a straight forward calculation gives DI(t , z ) = 1. In fact,kthesek z are the ones having the smallest a–component. k 1 k Clearly, (t ) cannot be convex, since (a , h ) and (a , h ) are in (t ) but (a , 0) / S 1 S 1 ∈ (t1), see Fig. 1 for a visualization. Moreover, if H is in nite dimensional then (t1) is Snot weakly closed. In fact, take any sequence h with h = h and h * S0, then k k kk k k k (a , h ) (t ) and (a , h ) * (a , 0) / (t ). k ∈ S 1 k ∈ S 1 Example 5.6 In this example I is quadratic but E & X. Consider E = BR(0) X Hilbert space, I(t, z) = z 2 g(t), z , (z) = z . 2 k k h i k k 16 a 1.4 1.2 1 h ±0.6 ±0.4 ±0.2 0 0.2 0.4 0.6 Figure 1: The visualization of the stable set in Example 5.5 in the case H = R clearly shows that convexity can not be expected. Then z with z < R is stable if and only if z g(t) 1. For z with z = R k k k k k k the boundary of E can stabilize; and stability holds if there exists [ , ) such that z g(t) 1. ∈ ∞ k k Thus, in the case g(t) √1+ 2R2 we have the convex stable set (t) = z E : k k S { ∈ z g(t) 1 BR(0), which is the intersection of two balls. In the case g(t) > k√1+ 2R2kw e ha}v∩e k k 1/2 (t) = z E : z g(t) 1 z E : z =R, g(t) 2 1 z Rg(t) R S { ∈ k k } ∪ { ∈ k k k k k k } which always contains a nonconvex part of the boundary¡ of the sphere.¢ However, there are also many nontrivial examples where convexity of the stable set can 1 n be shown directly. For instance, consider X = L ( ; R ), (z) = z 1 = z(x) dx and k k | | E = L1( ; K) where K is a compact and convex subset of Rn. Let W : [0, T ] K R R → be continuous and W (t, x, ) : K R be convex. For I(t, z) = W (t, x, z(x)) dx the → stability of z E can be checked pointwise: z is stable at time t if and only if for a.a. R x the vector∈ z(x) is stable for I(t, ) = W (t, x, ) and ( ) = . We refer to ∈ | | [MTL00] for a nontrivial application of this idea. b b 6 Existence and uniqueness results for general I The existence theory can be approached in a rather general setting even without convexity. We use the incremental method of Section 4. For a discretization P = 0, t , . . . , t , T { 1 N 1 } of [0, T ] the neness is (P ) = max tk tk 1 : k=1, . . . , N and ZP denotes the left– continuous, piecewise constant solution{ associates to a solution} of (IP), see (4.2). By our assumptions all these functions ZP are bounded in BV([0, T ], X), since ZP (0) = z0 and N (z z ) I(0, z ) + C T , see Lemma 4.1 (iii). To extract a convergent k=1 k k 1 0 1 subsequence we use the following generalization of Helly’s selection principle, see [BaP86]. We skPetch the proof for the convenience of the reader. Theorem 6.1 Let (zn)n∈N be a bounded sequence in BV([0, T ], X) with zn(t) E for ∈ ∞ all n N and t [0, T ]. Then there exists a subsequence (nk) and functions ∈ ∈ ∈ BV([0, T ], R) and z∞ BV([0, T ], X) such that the following holds: ∈ 17 (a) (dz ) ∞(t) for all t [0, T ]; [0,t) nk → ∈ (b) z (t) * z∞(t) E for all t [0, T ]; Rnk (c) (dz∞) ∈ ∞(t) ∞(s)∈for all 0 s < t T . [s,t) R Proof. First consider the monotone functions n(t) = [0,t) (dzn). By the real–valued ∞ version of Helly’s principle there exists a subsequence (nl) and a monotone function : R ∞ [0, T ] [0, ) such that for all t [0, T ] we have nl (t) (t) for l . Clearly→ , ∞ ∞ has an at most a coun∈ table jump set J →[0, T ]. Now →cho∞ose a sequence (t ) N which is dense in [0, T ] and contains J. Since there is an R > 0 with z (t) R j j∈ k n k we have weak compactness and can construct a further subsequence (nk) such that for all ∞ ∞ j N we have zn (tj) * z (tj) for k . This is the de nition of z at t = tj. ∈ k → ∞ Using the weak lower semi–continuity of it follows that z∞ must be continuous except at the jump points of ∞(t). Thus, we de ne z∞ on all of [0, T ] by its continuous extension in [0, T ]/J. Moreover, by b) we have weak convergence for all t [0, T ] as follows. The case t J is clear, hence assume t [0, T ]/J and choose y X .∈Then, ∈ ∈ ∈ z (t) z∞(t), y z (t) z (t ) y |h nk i| k nk nk j kk k + z (t ) z∞(t ), y + z∞(t ) z∞(t) y . |h nk j j i| k j kk k Now we can choose tj such that the rst and third term are less that ε/3 for all k k0, since both terms can be estimated by ∞(t) ∞(t ) 2 y . Keeping j xed and increasing | j | k k k if necessary the second term is less than ε/3 as well as for k k . 0 0 In the above and in the following result the function z is a general BV–function where at jump points z+(t), z(t) and z (t) may all be di erent. Combining the a– priori estimates in Section 4 and the previous theorem we arrive at the following result. (j) (j) (j) Proposition 6.2 Let P = 0, t1 , . . . , tN(j) 1, T be a sequence of discretizations whose (j) { (j) } neness (P ) tends to 0. Denote by z = ZP (j) the associated step functions (4.2) of the incremental problems (IP). Then there exists a subsequence (jl)l∈N and functions ∞, i∞ : [0, T ] R and z∞ BV([0, T ], X) such that → ∈ (i) z(jl)(t) * z∞(t), (dz(jl)) ∞(t) for all t [0, T ]; [0,t) → ∈ (ii) (dz∞) ∞(t) ∞(s) for all s < t, [s,t) R ∞ (jl) ∞ (iii) i (t) = liml→∞ I(t, z (t)) I(t, z (t)); R∞ ∞ ∞ (iv) i (t) + (t) = I(0, z0) + [0,t) ∂tI(s, z (s)) ds. R Proof. Applying Theorem 6.1 to the sequence z(j) we immediately obtain (i) and (ii). The weak continuity of ∂ I(s, ) (see assumption (2.6)) implies t ∂ I(s, z(jl)(s)) ds ∂ I(s, z∞(s)) ds t → t Z[0,t) Z[0,t) for all t [0, T ]. Adding the two–sided energy estimate (ii) in Lemma 4.1 we have ∈ (j) (j) (j) (j) (j) (j) ∂tI(s, z (s)) ds I(tn , z (tn )) I(0, z0) + (j) (dz ) [0,tn ) [0,tn ) (j) (6.1) (j) ∂tI(s, z (s)) ds, R [0,tn ) R R 18 b (j) N(j) 1 (j) where z (t) = k=0 i[kk,tk+1)(t)zk is the right–continuous step function. At all conti- nuity points t of z∞ we have z(jl)(t) * z∞(t) and hence the upper and lower estimates in P (jl) ∞ (6.1) bboth converge, for jl and tnl t, to ∂tI(s, z (s)) ds. → ∞ → [0,t) With (i) and (6.1) we concludeb that the limit i∞(t) exists for all t [0, T ] and (iv) R ∈ holds. Lower semi–continuity of I(t, ) implies (iii). The remaining task is to show that the limit function gives rise to a solution of our ∞ problem. The functions z = (z ) always satisfy the global energy inequality (E), which follows from I(t, z(t)) + (dz) i∞(t) + ∞(t) = I(0, z ) + ∂ I(s, z(s)) ds. 0 t Z[0,t) Z[0,t) The major problem is to obtain stability of z(t) for all t [0, T ]. To derive this from the (j) (j) ∈ stability of z = ZP (j) at the times t P we have two choices. Either we improve the weak convergence in Proposition 6.2(i)∈ to strong convergence and use the closedness of the stable sets, see Lemma 5.1. Or we stay with the weak convergence and show that (t) is weakly closed. S Theorem 6.3 Assume that E, and I satisfy the above assumption. Take a hierarchical (1) (2) (j) (l) sequence of discretizations P P . . . with (P ) 0 and de ne z = ZP (jl) and ∞ ∞ → z as in Proposition 6.2. Then z = (z ) is a solution of (GF) if there exists a dense subset of [0, T ] with ∞ P (j) such that one of the following two conditions holds T T j=1 b (1) z(l)(t) z∞(t) for all t ; S (2) (t) is→weakly closed for∈alTl t . S ∈ T b Proof. The two conditions are such that we can conclude z(t) (t) for all t by ∈ S ∈ T using the stability of z(jl)(t) at time t for all su ciently large l and the closedness of (t) in the suitable topology. S Lemma 5.1(b) and the density of then imply that z is stable for all t [0, T ] and T ∈ the theorem is established. Remark 6.4 The above theorem was derived under the assumption of re exivity of X. However, we only used that closed, bounded subsets of E are sequentially weakly compact. 1 Interesting applications involve the choice X = L ( ) and E = z : z ∞ 1 , cf. [MTL00]. { k k } Actually in case (1) the assumption of the weak continuity of ∂ I(t, ) is unnecessary, t strong continuity would su ce. A simple example to which the second part of our theorem applies is the following. Let E = X where X is an in nite dimensional Hilbert space. Let (z) = z and k k I(t, z) = z 2 + z 4 g(t), z 2 k k k k h i for a small > 0 and a suitable smooth function g : [0, T ] X. By Example 5.5 we → know that the stable sets (t) are not weakly closed for an open set of funtions g. Following the spirit ofSthe existence theorem we can also deduce a uniqueness result which requires that the stable sets are convex. 19 Theorem 6.5 If in addition to the above assumptions the function I has the form I(t, z) = J(z) g (t), z where J is strictly convex and the stable sets (t) are convex for all t [0 , Th], then theri e is a unique solution for each initial conditionS z E. ∈ 0 ∈ Proof. The existence of a solution is a consequence of Theorem 6.3, since convexity of (t) implies weak closedness via the strong closedness, cf. Lemma 5.1. S 1 For any two solutions zj : [0, T ] E with zj(0) = z0 (0) de ne z(t) = 2 (z0(t)+z1(t)). By convexity of the stable sets w→e know that z(t) is∈ stableS for all t. Now assume z (t) = z (t) for some t > 0. Using strict convexity of J, the energy identity (3.4) 0 6 1 e and the linearity of ∂tI we obtain e 1 1 I(t, z(t)) + [0,t] (dz) < 2 [I(t, z0(t)) + I(t, z1(t))] + [0,t] 2 [ (dz0)+ (dz2)] = 1 [I(t, z (0)) + I(0, z (0))] t 1 [ g˙ , z + g˙ , z ]ds R 2 0 1 R 0 2 h 0i h 1i e e = I(0, z ) + t ∂ I(s, z(s))ds. 0 0 t R Thus, z also satis es the global energy conditionR (E) and is a solution as well. However, e by Lemma 3.7 every solution satis es the energy equality (3.4) which is not the case here due toethe strict convexity. Hence, we conclude z0 = z1 on [0, T ]. In [CoV90] uniqueness for our situation is obtained only if E = X and I is a quadratic functional, i.e., I(t, z) = Az g (t), z where A : X X is symmetric and positive de nite. Clearly, this is a sph ecial case ofi the above result.→ 7 The good case: I uniformly convex and X = E In this section we establish the well posedness of the time–continuous evolution problem in the convex case. By Theorem 3.4.c and Theorem 3.5.c we see that all our formulations are equivalent. It is obvious that one can only expect that the solutions are unique if the I is strictly convex. In the degenerate cases it is easy to construct examples for nonuniqueness. If we additionally assume that I satis es a smoothness condition we get a pretty complete picture. For every initial value z(0) = z X there exists a unique 0 ∈ continuous process z and z(t) depends continuously on z0. Furthermore the solutions to the incremental problem converge to z as the neness of the discretization tends to 0. This improves the result in [HaR95] where the existence of a solution z W 2,1([0, T ], X) (resp. with z˙ BV([0, T ], X), cf. Section 7.2) has to be assumed before ∈showing conver- ∈ gence of the incremental method. At the end of the section we state the optimal regularity in time of the solutions. This requires that the boundary of F is smooth. Unfortunately our smoothness assumption on I(t, ) rules out the case E = X and we 6 are unable to generalize the methods of this chapter to cases where E has a boundary. Theorem 7.1 (Well posedness in the convex, smooth case) Assume E = X and I(t, ) C3(X) –uniformly convex in z. Then for every z in X there exists a unique ∈ 0 solution z W 1,∞([0, T ], X) of (GF) (or alternatively of (LF), (VI) or (SF)). For each ∈ t [0, T ] the state z(t) depends Lipschitz continuously on the initial value z0. Furthermore ther∈ e exists a constant C > 0 so that z Z ∞ C√ k P kL ([0,T ],X) 20 with ZP from (4.2) and = min ti ti 1 : i=1, . . . , N is the neness of the time discretization. { } Proof. The equivalence of (GF), (LF), (VI) and (SF) is established in the Theorems 3.4 and 3.5. The proof of existence of the solutions to the Cauchy problem is given in Theo- rem 7.3. There also the convergence of the solutions of the incremental problems to the time continuous solution is established. The uniqueness and the continuous dependence on the initial value is established in Theorem 7.4 7.1 Strong convergence We rst prove a stability result for the incremental problem when I is perturbed. This will then be used to compare the incremental solutions for two di erent discretizations. Proposition 7.2 Let E = X, j 1, 2 , I j C3(X, R), Ij –uniformly convex. Then, ∈ { } ∈ there exists a constant C > 0 such that for all discretizations P = 0, t1, . . . , tN 1, T the j { } solutions (zk)k=0,...N , j = 1, 2, of the incremental problem satisfy z1 z2 C 1/2 k k kk 1 2 where = sup DI ( , z) DI ( , z) ∞ . z∈X k kL ([0,T ],X ) Proof. We generalize the idea of the proof of Theorem 2.3 in [HaR95]. We introduce the j j 1 2 notation (t, z) = DI (t, z), ek = zk zk and the di erence operator k = k k 1 where j j l stands for t, z , (t, zk) or e. j j j j Convexity, E = X and (4.3) give (tk, zk), w kz + (w) ( kz ) 0 for all w X. Inserting w = z3 j and addingh the equations fori j = 1 and 2 gives ∈ k 1(t , z1) 2(t , z2), e 0. (7.1) h k k k k k i The nal estimate is derived using the quantity def= 1(t , z1) 1(t , z2), e = DI1(t , z1) DI1(t , z2), z1 z2 k h k k k k ki h k k k k k ki which by uniform convexity controls the error e via e 2 . The increment = k k kk k k can be estimated via (7.1) as follows k k 1 = 1(t , z1) 1(t , z2), e + ( 1(t , z1) 1(t , z2)), e k h k k k k k i h k k k k k k 1i = 2 1(t , z1) 2(t , z2), e + h k k k k k i k 1 1 1 2 1 1 1 2 1 2 where k = k( (tk, zk) (tk, zk)), ek 1 (tk, zk) (tk, zk), ke + 2 (tk, zk) 2(t , z2), he takes the form i h i h k k k i = A e A e , e A e , e + 2 1(t , z2) 2(t , z2), e k h k k k 1 k 1 k 1i h k k k i h k k k k k i = A e, e + (A A )e , e + 2 1(t , z2) 2(t , z2), e . h k k k i h k k 1 k 1 k 1i h k k k k k i The symmetric operators A Lin(X, X ) are de ned via A = D2I1(t , z2 + e ) d k ∈ k [0,1] k k k and satisfy A e = 1(t , z1) 1(t , z2). By convexity and three–times di erentiability k k k k k k R we obtain A y, y 0 and A A C z1 + z2 . h k i k k k 1k 3 k k k k k k ¡ ¢ 21 Together with e z1 + z2 we nd k k k k k k k k k C 2G + 3 z1 + z2 , k k k 1 k k k k k k · ¸ 1 2 ¡ ¢ j where G = sup (t , z) (t , z) . The Lipschitz continuity of z (see Theo- k z∈X k k k kX rem 4.4) gives the existence of a constant C which is independent of the discretization P such that + (t t )C[G + ] for k = 1, . . . , N, k k 1 k k 1 k k 1 where C = C sup [ ∂ 1( , z) + ∂ 2( , z) ]. With = 0 and G we obtain z∈X k t k∞ k t b k∞ 0 k k k b b C (t t ) [1 + C(t t )] C eCT T. k n n 1 j j 1 n=1 j=n+1 X Y b b b Together with z1 z2 2 1 this is the desired result. k k kk k Theorem 7.3 Under the assumptions of Proposition 7.2 there is a unique solution z 1,∞ ∈ W ([0, T ], X) and for each discretization P the incremental solution zk = ZP (tk) satis- es the error estimate z z(t ) C (P )1/2 for t P k k k k k ∈ where C is independent of P . Proof. Uniqueness and Lipschitz continuity are shown in Theorems 7.4 and 7.5 below. The existence part is based on the global de nition of solutions via stability (S) and energy inequality (E) and strong convergence. We use the discretizations P (j) = kT 2 j : j (j) { k = 0, . . . , 2 with the associated incremental solutions z = Z (j) . } P The idea is to consider z(j 1) as an incremental solution on P (j) of a slightly modi ed problem. Then z(j 1) and z(j) can be compared using the previous proposition. We let I(j)(kT 2 j, ) = I(kT 2 j, ) for even k and I (j)(kT 2 j, ) = I((k + 1)T 2 j, ) for odd k. Between the points in P (j) the potential I(j) is assumed to be linear in t. Thus, we have T DI(j) DI ∂ DI and ∂ DI(j) 2 ∂ DI , k k∞ 2j 1 k t k∞ k t k∞ k t k∞ where DI = sup DI(t, z) : t [0, T ], z X . k k∞ {k kX ∈ ∈ } Since z(j 1) is the incremental solution on P (j) with we can apply Proposition 7.2 and obtain z(j)(t) z(j 1)(t) C2 j/2 for t P (j). (7.2) k k ∈ n n n (j) Thus, keeping t = kT 2 : k N, k 2 xed, the sequence (z (t))j∈N is a Cauchy sequence.∈ItsT limit{z∞(t) provides∈ a Lipsc hitz} continuous solution by Theorem 6.3 (the incremental solutions satisfy a uniform Lipschitz bound, cf. Theorem 4.4). The estimate for general discretizations P = P (1) is obtained by successive bisection of the previous discretization P (j) such that (P (j+1)) = (P (j))/2. Doing the same trick as (j) (j) (j) above we again obtain DI DI ∞ 2 (P ) and ∂tDI ∞ 2 ∂tDI ∞. Estimate (7.2) is replaced by k k k k k k z(j 1)(t) z(j)(t) C (P (j))1/2 for t P (j). k k ∈ 22 Restricting to t P = P (1) and adding all these estimates we nd ∈ ∞ 1/2 z(1)(t) z∞(t) C (P )2 j+1 = C˜ (P )1/2. k k j=1 X ¡ ¢ (1) ∞ Since z (tk) = zk is the original incremental solution and z (t) = z(t) the time– continuous one the result is established. 7.2 Uniqueness results It is easy to construct examples with I(t, ) (not strictly) convex such that the solution is not unique, see e.g. Example 7.6. A rst uniqueness result was obtained in Theorem 6.5. Theorem 7.4 Assume that E = X, that I is –uniformly convex and that I(t, ) ∈ C3(X, R). Then the solutions are unique and depend Lipschitz continuously on the initial value. Proof. Let z1 and z0 be two solutions. De ne (t) = , z (t) z (t) with = DI(t, z (t)), h 1 0 1 0 i j j then z (t) z (t) 2 (t)/ by –uniform convexity. Moreover, by Theorem 7.5 below k 1 0 k we know that z˙j = (1+ z˙j )rd(zj) exists a.e. in [0, T ] and satis es z˙j(t) C1/ . Thus, we have k k k k ˙ (t) = ∂ DI(t, z ) ∂ DI(t, z ), z z + r , z˙ + r , z˙ h t 1 t 0 1 0i h 1 1i h 0 0i where r = D2I(t, z )[z z ] + = 2( ) + b . Using the estimates j j j 1 j 1 j j j 1 j j b = DI(t, z ) DI(t, z ) D2I(t, z )[z z ] C z z 2 k j k k 1 j j j 1 j j k 3k 1 0k and ∂ DI(t, z ) ∂ DI(t, z ) C z z we nd k t 1 t 0 k 2k 1 0k C ˙ C z z 2 + 2C z z 2 1 + 2 , z˙ z˙ + 2 , z˙ z˙ . (7.3) 2k 1 0k 3k 1 0k h 0 0 1i h 1 1 0i To estimate the sum of the last two terms we use the variational inequality (VI) for the solutions z = zj where, by homogeneity, we can replace rd(zj) by z˙j. We insert the test functions w = z˙1 j and subtraction of the two equations yields , z˙ z˙ 0. (7.4) h 1 0 1 0i Here the characteristic functions vanish as Tzj (t)E = X (recall E = X) and the terms involving annihilate. Thus, (7.3) gives ˙ C with C = (C + 2C C / )/ and, hence, z (t) z (t) 2 5 5 2 1 3 k 1 0 k eC5t (0)/ which implies uniqueness and Lipschitz continuity. Theorem 7.4 can be generalized when we nd a replacement for (7.4). Such a gener- alization is given via the structure condition (C.1) in Appendix C. 23 7.3 Temporal regularity In the general case solutions z : [0, T ] E will not be continuous but just lie in → BV([0, T ], X). This includes the case of I being convex in z, when it is not strictly convex. We obtain continuity if I(t, ) is strictly convex and Lipschitz continuity if I(t, ) is uniformly convex. Even in the simplest case z˙ will have jumps due to the intrinsic nondi erentiability of : X R. Under suitable strong assumptions we will show z˙ BV([0, T ], X). → ∈ Theorem 7.5 (a) If I : [0, T ] E R is continuous and if I(t, ) : E R is strictly convex for all t [0, T ], then any solution→ z : [0, T ] E is continuous. → ∈ → (b) If additionally I(t, ) is –uniformly convex, then z : [0, T ] E is Lipschitz continuous with → C z(t) z(s) 2 t s k k | | where C2 is de ned in assumption (2.6)(b). Proof. (a) Take t [0, T ], then the left and right limits z and z exist and satisfy energy ∈ + identity (2.7). For (0, 1) let z = z+ + (1 )z and assume z+ = z , then strict convexity gives ∈ 6 I(t, z ) + (z z ) < I(t, z ) + (1 )I(t, z ) + (z z ) = I(t, z ). + + This contradicts the stability of z and we conclude the desired continuity z+ = z . (b) Using uniform convexity, for t > s, we obtain z(t) z(s) 2 I(s, z(t)) I(s, z(s)) DI(s, z(s)), z(t) z(s) k k h i ∂ I( , z(t)) d + I(t, z(t)) I(s, z(s)) + (z(t) z(s)) [s,t) t = [∂ I( , z( )) ∂ I( , z(t))] d (dz) + (z(t) z(s)) [s,tR) t t [s,t) ∂ I( , z( )) ∂ I( , z(t)) d C z( ) z(t) ds. R[s,t) | t t | R [s,t) 2k k Here we have used (SlocR) at s for the second estimate, the energyR balance (2.6) on [s, t], (z˙( )) d (z(t) z(s)) for the fourth estimate and nally assumption (2.6). [s,t) Let t be xed and de ne (s˜) = max z(s) z(t) : s [s,˜ t] , then we have shown R z(t) z(s) 2 C t s (s) C t s˜{k (s˜) for allk s˜ ∈s }t. Thus, we conclude k k 2| | 2| | (s˜) C t s˜ which is the desired result. 2| | After having established the well–posedness of the evolution problem (i.e. existence and uniqueness) in certain cases it is desirable to check whether any of the assumptions can be dropped. Unfortunately we do not have an example for nonexistence, except in pathological cases. We can, however, give an example which illustrates that the assump- tion of uniform convexity in Theorem 7.4 can not be dropped. Example 7.6 Let X = R, E = [ 1, 1], (v) = v and I(z) = z2 g z. The existence | | 2 of solutions is clear by trivial compactness in nite dimensions. Since E ( X we can not apply Theorem 7.4 directly but one can easily show that for E satis es the structure condition (C.1). Therefore, by Proposition C.1 we have uniqueness if > 0. For every R the solutions are monotone in time as long as g is monotone, see Fig. 2 on the left ∈ side. If 0 we lose uniqueness and Lipschitz continuity. For < 0 every solution z takes only values within 1, 1 , see Fig. 2 on the right side. { } 24 6z(g) 6z(g) ¢ ¢ ¢ ¢ ¢ g˙ > 0 6g˙ > 0 ¢ ¢ 1 ¢ 1 ¢ 1+ ¢ - - ¢ g 1 g ¢ 1+ ¢ 1+ ¢ g˙ < 0 ¢¢ g˙ < 0 ? ¢ ¢ ¢ ¢ Figure 2: Solutions in Example 7.6 are unique and Lipschitz if > 0 (to the left) and discontinuous and nonunique if < 0 (to the right). In [HaR95] a convergence result for the incremental problem is given under the as- 2 sumption that the time continuous solution lies in z W 2,1([0, T ], X), that is z = d z ∈ dt2 ∈ L1([0, T ], X). In fact, their error estimate is k 2 1 zk z(tk) C max tj tj 1 z˙(tj) (z(tj) z(tj 1)) , (7.5) k k j=1,...,k{ } k tj tj 1 k j=1 X where the constant C does not depend on k and the discretization 0 = t0 < t1 < . . . < tk. The last sum can be estimated by dz˙ , thus we only need z˙ BV([0, T ], X) [0,tk] k k ∈ and not z˙ W 1,1([0, T ], X) as stated in [HaR95]. This di erence is crucial since the R latter inclusion∈ is false for typical cases whereas the former can be shown under natural additional assumptions. Example 7.7 We consider a very simple example with E = X = R, (v) = v , I(t, z) = 1 2 2 | | 2 z (4t t )z and the initial condition z(0) = 0. A simple calculation gives the unique solution 0 for t 2 √3, 4t t2 1 for t [2 √3, 2], z(t) = ∈ 3 for t [2, 2+√2], ∈ 4t t2+1 for t 2+√2. The boundary of the set F = [1, 1] is 1, 1 , and z˙ jumps upon hitting it (t = 2 √3 or 2+√2) but not upon leaving it (t = 2{ ). } Theorem 7.8 Assume E = X = X and that the boundary of F X is of class C2. Let the functional I(t, z) be uniformly convex and C3. Then, the unique solution z satis es z˙ BV([0, T ], X). ∈ Proof. According to Theorems 7.5, 7.4 and 7.3 we know that the solution z : [0, T ] X exists, is unique and Lipschitz continuous. Thus, we have z˙ L∞([0, T ], X). → Since the boundary ∂F is smooth there is for each z ∈∂F a unique outward unit ∈ normal N(z ) X. The mapping N : ∂F X; z N(z ) is Lipschitz continuous and ∈ → → ∂ (z ) = N(z ) : [0, ) . F { ∈ ∞ } 25 For the given solution z let (t) = DI(t, z(t)) F . Since z is Lipschitz and I is ∈ smooth we know that : [0, T ] X is Lipschitz continuous. Moreover (Eloc) and (Sloc) imply → 0 if (t) int(F ), z˙(t) = ∈ (7.6) (t)N( (t)) for a.a. t ( ∈ T where = t [0, T ] : (t) ∂F is closed. Now take any t with (t) > 0 T { ∈ ∈ } ∈ T in (7.6). Since ˙ (t) = A(t)z˙(t)+∂tDI(t, z(t)) is perpendicular to N( (t)), we nd (t) = h∂tDI(t,z(t)),N( (t))i , where A(t) = D2I(t, z(t)) Lin(X, X ). By the implicit hA(t)N( (t)),N( (t))i ∈ function theorem and the uniqueness (forward in time) it can be shown that there exists ε > 0 such that z : [t, t + ε] X satis es the di erential equation → ∂ DI(t, z), N˜(t, z) z˙ = F (t, z) := h t i N˜(t, z) (7.7) D2I(t, z)N˜(t, z), N˜(t, z) h i where N˜(t, z) = N( DI(t, z)). By our smoothness assumptions F (t, z) is continuous in t and Lipschitz in z (t) = z X : DI(t, z) ∂F which is a smooth manifold. De ne the functions∈ Z { ∈ ∈ } (t) for t , N( (t)) for t , h(t) = ∈ T and N(t) = ∈ T ( 0 else; ( linear interpolant else. b Then N : [0, T ] X is Lipschitz. The argument with (7.7) shows that h(t) is Lip- schitz continuous→from the right with a xed Lipschitz constant independent of t and ε. In particular,b h can only be discontinuous at points t where limt%t h(t) = 0 and limt&t h(t) > 0. Between such points h is Lipschitz with a xed constant. Hence, h can be written as a sum of a Lipschitz function h˜ and a piecewise constant function with at most countably many positive jumps jl > 0 at times tl . By Theorem 7.5 z˙(t) and hence h(t) is bounded by C / , which gives k k 2 ∞ C j + h(0) T Lip(h˜) 2 . l Xl=1 This implies that h is of bounded variation: ∞ C Var(h; [0, T ]) T Lip(h˜) + j 2T Lip(h˜) h(0) + 2 . l Xl=1 Now the formula z˙(t) = h(t)N(t) gives the desired result, since Var(h( )N( ); [0, T ])b Var(h; [0, T ]) N( ) ∞ + h( ) ∞Var(N; [0, T ]) k k k k C Var(h; [0, T ]) + 2 T Lip(N). b b b b 26 A The reduced derivative As above we consider z BV ([0, T ], X). Our aim is to de ne a substitute for the derivative which works w∈ell for rate–independent processes. It will be called reduced derivative and its properties are (i) it is a multiple of the derivative if it exists and (ii) at jump points it is a multiple of the jump vector. For z BV ([0, T ], X) we de ne : [0, T ] [0, ) via ∈ → ∞ (t) = t + b dz = t + Var(z, [0, t]). k k Z[0,t] Then, = (left–sidedb limit) and + (right–sided limit) are strictly increasing and coincide except at the (at most countable) jump points of z. With T = (T ) we de ne the conbtinuousb inverse b b b [0, T ] [0, T ], t : → ( max t [0, T ] : (t) b 7→ { ∈ } b and the stretched function z C0([0, T ], X) via ∈ b z( ) = (1 )z (t) + z (t) for = (1 ) (t) + (t). (A.1) b b + + Thus, we have z(t) = z( (t)) and z is linearly interpolated at jump points, i.e. at points b b b where +(t) > (t). By construction webhab ve, for 0b 1 < 2 T , b b z( ) z( ) (t t ) , k 2 1 k 2 1 b 2 1 2 1 where t t satis es = (1 ) (t ) + (t ). Thus, 1 2 jb b j j j + j 1,∞ d z W ([0b, T ], X), b z ∞ b 1. ∈ kd kL ([0,T ],X) d b The derivative vz( ) =b d z( ) is de ned almostbeverywhere in the sense of Lebesgue measure ( a.e.) and the reduced derivative is the pullback of v via : [0, T ] [0, T ], z → more precisely b b 1 rd(z)(t) def= v ( [ (t)+ (t)]) for t [0, Tb]. b (A.2)b z 2 + ∈ Associated with this pullback of the derivative is the pullback z of the Lebesgue measure on [0, T ]: b b b b ([t , t )) def= ([ (t ), (t )) = (t ) (t ) = t t + dz . z 1 2 1 2 2 1 2 1 k k Z[t1,t2] By the general constructionbof pullbacb ks theb reducedb derivative rd(z) is de ned a.e. in z [0, T ]. In particular, if t is a jump point, we have z( t ) = z+(t) z (t) and rd(z)(t) = X { } k k 1 Sign(z+(t) z (t)) ∂B1 (0). If z has a derivative z˙, then rd(z)(t) = (1+ z˙(t) ) z˙(t) ∈t2 k k and ((t , t )) = [1+ z˙(s) ]ds. z 1 2 t1 We will need the follokwingkresults. R 27 ∞ Theorem A.1 Let z BV ([0, T ], X) with z ([0, T ]) and rd(z) L ([0, T ], z) as above. ∈ ∈ M ∈ (a) Let f : [0, T ] X R be continuous and f(t, ) : X R homogeneous of degree 1, then → → T T f( , dz) = f(t, rd(z)(t)) (dt). z Z0 Z0 (b) For any A C0([0, T ], X ) we have ∈ T T A( ), dz = A(t), rd(z)(t) (dt). h i h i z Z0 Z0 (c) For any 0 s < t T and any g C1([0, T ] X, R) we have ∈ t g(t, z(t)) g(s, z(s)) = ∂ g(r, z(r))dr + G(r), rd(z)(r) (dr) t h i z Zs Z[s,t) where G(r) = 1 Dg(r, (1 )z (r) + z (r))d . =0 + Proof. ad (a): Let tj,nR = jT/n, j,n = (tj,n) and j,n = j,n j 1,n. Then T n f( , dz) = lim fb(tj,n, z(tj,n) z(tj 1,n)) n→∞ 0 j=1 Z X n 1 = lim f(t( j,n), [z( j,n) z( j 1,n)]) j,n n→∞ j=1 j,n b X T bd b b = f(t( ), z( ))d . d Z0 The latter convergence holds, since we may introduce further points on jump intervals b b where +(t) > (t) without changing the sum (use t = t on this interval and homogeneity of f(t, )). By pullback (transformation formula) we obtain the desired result. ad b(b). Forbgeneral A C0([0, T ], X ) we obtainb via pullback ∈ b T d T bA( ), z(b ) d = B(t), rd(z)(t) (dt) h d i h i z Z0 Z0 with B(f) = 1 A((1 b ) (t) + (t))d . =0 b + As in theRproof of (a) we obtain A( ) = A(t( )) which gives B(t) = A(t) and the assertion follows. b b b 1,∞ ad (c). We replace z BV ([0, T ],bX) by thebstretched function z W ((0, T ), X) and let = (s) and =∈ (t). We nd ∈ g(t, z(t)) g(s, z(s)) = g(t( ), z( )) g(t( ), z( )) b b db = [g(t( ), z( )]d d b b b b Z 0 0 = ∂tg(t(b ), z(b ))t ( )d + Dg(t( ), z( )), z ( ) d h i Z t Z = ∂ g(rb, z(r))db r +b G(r), rd(z)(br) b(dr),b t h i z Zs Z[s,t) 28 where we have used (b) for the last step. A simple consequence of (a) is rd(z)(r) (dr) = dz = z(t) z(s). (A.3) z Z[s,t) Z[s,t) B Duality and Cones For a function f : X ( , ] we de ne the Legendre–Fenchel transform f via → ∞ ∞ L f : X ( , ]; v sup v , v f(v) : v X . L → ∞ ∞ 7→ {h i ∈ } For a function f : X ( , ] we de ne the inverse Legendre–Fenchel transform f via → ∞ ∞ L f : X ( , ]; v sup v , v f (v ) : v X . L → ∞ ∞ 7→ {h i ∈ } The functions f and f are always lower semicontinuous and convex. If f was lower semicontinuousLand conLvex from the beginning then f = f. For these results and the L L proof of the following theorem we refer to [EkT76]. For convex functions f the subdi erential ∂f is de ned via ∂f(z) = v X : w W : f(w) f(z) + v , w z . { ∈ ∀ ∈ h i } Theorem B.1 Let f : X R be convex and lower semicontinuous. Set f = f. → ∪ {∞} L Then the following statements are equivalent: 1. v ∂f(v) ∈ 2. f(v+w) f(v) + v , w for all w X 3. v argmax v , wh f(iw) : w X∈ 4. v∈ , v = f({hv) + f i(v ) ∈ } h i 5. v argmax w , v f (w ) : w X 6. f (∈v +w ) {hf (v )i+ w , v for al∈l w } X h i ∈ 7. v ∂f (v ). ∈ Let C be a closed convex cone in X, i.e. v C implies v C for all [0, ). The dual cone C is de ned as ∈ ∈ ∈ ∞ C = v X : v , v 0 for all v C . { ∈ h i ∈ } This duality can also be expressed by the characteristic functions of the cones, namely C = C . The sum C1 + C2 and the intersection C1 C2 of convex cones are again Lconvex cones. Moreover, (C +C ) = C C and (C ∩C ) = C +C . 1 2 1 ∩ 2 1 ∩ 2 1 2 In particular, for convex sets E X the (inward) tangent cone TzE (see (2.1)) is closed and convex and its dual cone is the normal cone NzE = ∂ E (z). The following lemma is used in Section 3. Lemma B.2 Let E X be closed and convex and take z , z E. For (0, 1) let 0 1 ∈ ∈ z = (1 )z + z ; then T E = T E + T E for all (0, 1). 0 1 z z0 z1 ∈ 29 Proof. We set C = T E and show C C for j = 0 and 1. By convexity of C this z j implies C0 + C1 C . To show C0 C we consider rn > 0 and vn X such that w = z +r v E and v v C . Then z + (1 )r v = (1 )w ∈+ z E and n 0 n n ∈ n → ∈ 0 n n n 1 ∈ v C follows. ∈ The opposite inclusion follows by duality from showing C0 C1 C . Choose ∩ ∈ C0 C1 and consider the hyperplanes Hj X which have the same (not opposite) normal ∩ and contain zj. As the set E touches in Hj in zj and lies on one side of it, we conclude that H = H . In particular, z H which implies C . 0 1 ∈ 0 ∈ Proposition B.3 Let F X be closed convex set, C X a closed convex cone and f(v) = ( )(v) + (v). Then, f = . L F C L F +C Proof. Each v F + C has the form v = w + y with w F and y C . Then, ∈ ∈ ∈ ( f)(v ) = sup w , v ( F (v) + y , v C (v) : v X L {h i L h i ∈ } ( )(w ) + ( )(y ) = (w ) + (y ) = 0 + 0 = 0. L L F L C F C Now assume v / F + C . By Mazur’s Lemma there exists R and v X such that v , v > z ∈, v for all z F + C . This implies ∈ ∈ h i h i ∈ ( f)(v ) v , v ( )(v) (v) L h i L F L C = v , v sup w , v sup y , v h i w ∈F h i y ∈C h i = v , v sup w + y , v v , v > 0. h i w ∈F ,y ∈C h i h i Testing with v rather than v we conclude ( f)(v ) [ v , v ] for all > 0 and obtain ( f)(v ) = . This concludes the proLof. h i L ∞ C Structure condition Many of the above results need the assumption E = X. This is due to the fact that the stability condition (Sloc) changes discontinuously near or on the boundary ∂E via the tangent cone TzE. If we want to compare two di erent solutions z1 and z2, then the local formulation (LF) involves two di erent tangent cones Tzj (t)E such that we cannot guarantee z˙1(t) Tz2(t)E and vice versa. However, this feature was essential in the uniqueness and con∈ vergence result, see (7.4) and (7.1). We propose a new structure condition. For zj E we de ne j = DzI(t, zj) which lies ∈ in the set (F +Nzj E) whenever zj is stable at time t. Our structure condition involves E, F and I(t, z) simultaneously and reads as follows. (SC) [Structure Condition] For all R > 0 there exists a constant Cstruc > 0 such that for all t [0, T ], all z , z (t) z R and all v V the estimate ∈ 1 2 ∈ S ∩ {k kX } j ∈ zj , v v C z z 2( v + v ) (C.1) h 1 2 1 2i struck 1 2k k 1k k 2k holds, where Vz = N( )(F +NzE) is the set of possible velocities. 30 Note that we always have N (F +N E) T E (cf. also the sweeping process ( j ) zj zj formulation (SP) in (3.8)). The intuition behind the structure condition is that, for uniqueness, it is su cient that the terms which involve the di erentials of I and the velocities have a sign. Proposition C.1 Let I C3(E [0, T ]) and let I be –uniformly convex. If additionally (SC) is satis ed, then every∈ solution of (GF) depends continuously on the initial data and, in particular, uniqueness holds. Proof. The claim follows by repeating the proof of Theorem 7.4 where (SC) takes the role of the estimate (7.4) which is just (SC) with Cstruc = 0. We illustrate now one case, where (SC) holds and one case where (SC) is violated. 2 1 2 t Example C.2 Let X = R , E = [0, ) R, I(t, z) = 2 z 2 + 0 , z and (v) = v + v or equivalently F = [ 1, 1]2.∞Then, the structurek conditionk h (SC)i holds with | 1| | 2| ¡ ¢ Cstruc = 0. We show this by direct calculation. The problem is especially simply as X and X 2 can be identi ed by D I(t, z) = idR2 . The stable sets are convex and given by (t) = [0, max 0, 1 t ] [ 1, 1] = z E : DI(t, z) F . S { } { ∈ ∈ } The sets V = N (F +N E) of possible velocities satis es V = 0 in the interior of z z z { } (t) as well as on the (open) line 0 ( 1, 1) ∂E. For the other boundary points of S∂ (t) we nd V = N (t). In particular,{ } S z zS 0 : 0 for z = with [0, 1 t); { } 1 ∈ 1 t ¡ ¢ : , 0 for z = ¡ 1¢ ; { } V = 1 t z ¡ 0 ¢ : 0 for z = ¡ ¢ with < 1; { } | | 1 t ¡ ¢ : , 0 for z1 =¡ ¢ ; { } 1 0 ¡ ¢: 0 for z = ¡ ¢with [0, 1 t). { } 1 ∈ ¡ ¢ t ¡ ¢ Using = DI(t, z) = z + 0 the structure condition with constant Cstruc = 0 reduces to z z , v v 0 for all v V . However, by convexity of (t) we have h 2 1 2 1i ¡ ¢ j ∈ zj S z3 j zj Tzj (t) whereas vj Nzj (t), see Fig. 3. This implies zj z3 j, vj 0 and adding these∈ tSwo relations giv es∈the desiredS result. h i Example C.3 We take the same X, I and as in the previous example, but now E = z R2 : z, 1 0 . Now, the stable sets may be nonconvex: { ∈ h 1 i } ¡ ¢ t 1 (t) = +F E z = : [ 2 t, 2 t] . S 0 ∩ ∪ { 2 1 ∈ } ³£¡ ¢ ¤ ´ ¡ ¢ Now choose z = 1 and z = 1 ε with a small positive ε. Then, we nd V : 1 1 ε 2 1+ε z1 { 0 0 and V = 0 . With v = and v = 0 we obtain , v v = ε = } z2 ¡ { ¢} 1¡ ¢0 2 h 1 2 1 2i ¡ ¢ 5 1/2 z z ( v v ), see Fig. 3. For ε 0 we see that (SC) cannot hold for any k 1 2k k 1k k 2k ¡ ¢ → nite Cstruc. 31 E = z, e 0 E = z, e +e 0 {h 1i } @ {h 1 2i } z @ s 1 @ @@ z1 (v1 = 0) KAA syX @@XXXz1 z2 A z1 z2 @ X Xs v A @@ z 1 ? @@ 2 A @ A @@ @@ A @ A @@ @ As @ (t) z2 @ v2 v2 @S ? @@ (t) @@@ 1 S @ DI (F ) 1 DI (F ) @ @ Figure 3: Illustration for the structure condition in Examples C.2 (to the left) and Exam- ple C.3 (to the right). In both cases we have v V and the structure condition reduces j ∈ zj to z1 z2, v1 v2 0. In the situation on the left–hand side it is satis ed but for the righht–hand side thei structure condition can’t be satis ed. Acknowledgements F.T. thanks the Deutscher Akademischer Austauschdienst for generous support. The research was partially supported by DFG through the SFB 404 Multi eld Problems. References [BaP86] V. Barbu & T. Precupanu. Convexity and optimization in Banach spaces, D. Reidel Pub. Co, Dordrecht, 2nd edition, 1986. [BrS96] M. Brokate, J. Sprekels. 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