On Uniqueness in Evolution Quasivariational Inequalities

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1 Journal of Convex Analysis Volume 11 24), No. 1, On Uniqueness in Evolution Quasivariational Inequalities Martin Brokate Zentrum Mathematik, TU München, D Garching b. München, Germany Pavel Krejčí Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, CZ Praha 1, Czech Republic Hans Schnabel Zentrum Mathematik, TU München, D Garching b. München, Germany Received July 4, 22 Revised manuscript received March 31, 23 We consider a rate independent evolution quasivariational inequality in a Hilbert space X with closed convex constraints having nonempty interior. We prove that there exists a unique solution which is Lipschitz dependent on the data, if the dependence of the Minkowski functional on the solution is Lipschitzian with a small constant and if also the gradient of the square of the Minkowski functional is Lipschitz continuous with respect to all variables. We exhibit an example of nonuniqueness if the assumption of Lipschitz continuity is violated by an arbitrarily small degree. Keywords: Evolution quasivariational inequality, uniqueness, sweeping process, hysteresis, play operator 2 Mathematics Subject Classification: 49J4, 34C55, 47J2 1. Introduction In 1973, Moreau has introduced the sweeping process, [1, 11]. It describes the movement ξ = ξt) of a point in a Hilbert space X induced by a time-dependent closed convex set C = Ct) according to ξt) N Ct) ξt)), ξ) = ξ, 1) where N K x) denotes the normal cone to a convex set K at a point x. The evolution variational inequality vt), w vt) ft), w vt) w Γ, vt) Γ, v) = v, 2) Supported by the Academy of Sciences of the Czech Republic. Supported by the DFG through SFB 438. Supported by the GAČR under Grant 21/2/158. Supported by the DFG under Grant BR 15/7 1. ISSN / $ 2.5 c Heldermann Verlag

2 112 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... Γ H closed and convex, constitutes a special case of 1), if we set ξt) = vt) fs) ds, ξ = v, Ct) = Γ In the same manner, the evolution quasivariational inequality fs) ds. 3) vt), w vt) ft), w vt) w Γvt)), vt) Γvt)), v) = v, 4) becomes a special case of the implicit or state-dependent sweeping process if we set ξt) = vt) ξt) N Ct,ξt)) ξt)), ξ) = ξ, 5) fs) ds, ξ = v, Ct, ξ) = Γ ξ + ) fs) ds fs) ds. 6) An appropriate meaning has to be given to the time derivative if discontinuous processes are taken into consideration. The Young integral formulation was investigated in [5]. Another approach to nonsmooth evolution differential inclusions based on energy considerations was recently proposed by Mielke and Theil in [9]. While the sweeping process 1) has been an object of extensive study, see the survey [7], much less is known about the implicit process 5). The paper [2] seems to be the first result in this direction, in a more general setting actually, but restricted to the case dim X = 1. Kunze and Monteiro Marques [8] have proved existence for 5), if C satisfies a Lipschitz condition with respect to the Hausdorff distance, d H Ct, ξ), Cs, η)) L 1 t s + L 2 ξ η, 7) if L 2 < 1 and give examples for nonexistence if L 2 > 1. However, no matter how small L 2 is chosen, uniqueness may fail to hold; Ballard [1] has given an example in the context of quasi-static friction problems. Indeed, it is a general feature of quasivariational inequalities that the loss of monotonicity, as caused by the state dependence of the constraint C, is accompanied by a loss of uniqueness of their solution. On the other hand, there is no a priori reason why it should be impossible to enforce uniqueness if the constraint behaves in a sufficiently regular manner. We will show in this paper that uniqueness holds if the normal vectors to the constraint Ct, ξ) satisfy a Lipschitz condition with respect to t, ξ). Under the assumptions below this means that we require the gradient of the square of the Minkowski functional M of C to be Lipschitz continuous as a function of t, ξ), whereas 7) is under suitable hypotheses, see Lemma 3.2 below, equivalent to the Lipschitz continuity of M itself. Our proof of uniqueness and, incidentally, of existence at the same time) is based on the contraction principle, thus the possible loss of monotonicity does not play any role here. In order to obtain a contraction, results concerning time regularity for the solution operator f ξ of the standard variational inequality 1) are required which are stronger than the basic estimate ξ 1 t) ξ 2 t) ξ 1 ξ 2 + f 1 s) f 2 s) ds. 8)

3 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Such stronger estimates have been provided within the context of hysteresis operators, see [4] for a recent exposition. However, to apply those results we need the constraint to be sandwiched uniformly between balls with fixed radius around zero. In particular, constraints with empty topological interior are excluded. In addition, we will relate the nonuniqueness in the quasivariational inequality to the generic nonuniqueness phenomenon for scalar ordinary differential equations. More precisely, to every concave increasing function ψ : R R with ψ) = which gives rise to a positive solution of the scalar problem y = ψy), y) =, 9) besides the trivial one, there corresponds a quasivariational inequality with nonunique solution, where the gradient with respect to the state ξ) of the Minkowski functional varies like ψ ξ ). 2. Problem statement We consider a separable Hilbert space X endowed with a scalar product,, a norm x = x, x for x X, and a family of bounded convex closed sets Zϱ) X parametrized by ϱ R Y, where Y is a reflexive Banach space endowed with a norm Y and R is a convex closed set with non-empty interior R. By Y we mean the dual of Y,, )) is the duality pairing between Y and Y, and Y, LX,Y ) denote natural norms in the corresponding spaces. Throughout the paper, with the exception of Section 4, we will assume that there exist < c C such that B c ) Zϱ) B C ) ϱ R. 1) We consider two problems, namely a variational inequality where the constraint Z depends on an additional given function Problem P )), and a quasivariational inequality Problem I )). For given functions u W 1,1, T ; X), r W 1,1, T ; R) and an initial condition x Zr)) we look for a function ξ W 1,1, T ; X) such that P ) i) ut) ξt) Zrt)) t [, T ], ii) u) ξ) = x, iii) ξt), ut) ξt) y y Zrt)) for a. e. t ], T [. Besides serving as the crucial tool for the solution of problem I ), problem P ) is of interest in itself consider, for example, a constitutive stress-strain law where the yield function depends on the temperature). Actually, problem P ) is a reformulation of 1). We reduce P ) to 1) if we set u =, rt) = t, x = ξ and C = Z, and we reduce 1) to P ) if we set Ct) = ut) Zrt)) and ξ = u) x. We now formulate problem I ). Let g : [, T ] X X R be a mapping satisfying Hypothesis G) stated below at the beginning of Section 6. For a given function u W 1,1, T ; X) and an initial condition x Zg, u), u) x )) for instance, any x B c ) satisfies this inclusion) we look for a solution ξ W 1,1, T ; X) of the implicit problem

4 114 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... I ) i) ut) ξt) Zgt, ut), ξt))) t [, T ], ii) u) ξ) = x, iii) ξt), ut) ξt) y y Zgt, ut), ξt))) for a. e. t ], T [. The implicit sweeping process 5) becomes a special case of I ), if we set u =, gt, u, ξ) = t, ξ), x = ξ and C = Z. On the other hand, I ) becomes a special case of 5), if we set Ct, ξ) = ut) Zgt, ut), ξ)) and ξ = u) x. The quasivariational inequality 4) is subsumed under I ) either going through 6), or by setting ξt) = vt)+ut), ut) = fs) ds, x = v, Z = Γ, gt, u, ξ) = ξ u. 11) The formulation 5) certainly looks more compact than I ). However, in applications a driving function like u or its derivative f) often appears, and it is useful to study the dependence of ξ on u with respect to standard function spaces. In 5) one has to deal with metric properties of the set-valued mapping t Ct, ξ).) Our main results include the existence, uniqueness, and Lipschitz continuous input-output dependence for both Problems P ) and I ) in Sections 4 7. Before, we recall some basic notions from convex analysis as presented in [12, 4]. 3. Preliminaries: Convex sets Consider a Hilbert space X as in the previous section and a convex closed set Z X. The mapping M Z : X R + defined by the formula M Z x) = inf {s > ; x } s Z 12) is called the Minkowski functional associated with Z. The polar set Z to Z is defined by the formula Z = {x X ; x, y 1 y Z}. 13) We first summarize in Lemma 3.1 below some results of [4, Sections 3 and 6]. Lemma 3.1. Assume that there exist C > c > such that i) It holds B c ) Z B C ). 14) B 1/C ) Z B 1/c ), 15) x C M Zx) x c, 16) c x M Z x) C x. 17) ii) Let the derivative x M Z x) X exist for each x X \ {}, and set J Z ) =, J Z x) = M Z x) x M Z x) for x. Then the unit outward normal n Z x) to Z is uniquely determined at each point x Z, and we have n Z x) = J Zx) J Z x) x Z, 18)

5 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution M Z J Z x)) = M Z x) x Z. 19) Let moreover L > such that n Z x) n Z y) L x y x, y Z. 2) Then we have J Z x) J Z y) 1 c + L 2 c 1 + C ) ) 2 x y x, y X. 21) c We can measure the distance of two sets Z 1, Z 2 in the system C of all closed convex subsets Z of X either as the Hausdorff distance or the Minkowski distance d H Z 1, Z 2 ) = max{ sup z 1 Z 1 dist z 1, Z 2 ), sup z 2 Z 2 dist z 2, Z 1 )}, 22) d M Z 1, Z 2 ) = sup M Z1 x) M Z2 x). 23) x =1 We first show that these concepts are equivalent in the class of sets satisfying 14). Lemma 3.2. Let Z 1, Z 2 C be such that 14) holds for Z = Z i, i = 1, 2. Then we have Proof. c 2 d M Z 1, Z 2 ) d H Z 1, Z 2 ) C 2 d M Z 1, Z 2 ). 24) Assume first that there exists x Z 1 \ Z 2. Using 16) we obtain dist x, Z 2 ) x x M Z2 x) x 2 M Z1 x)m Z2 x) C 2 d M Z 1, Z 2 ), ) )) x x M Z2 M Z1 x x and reversing the roles of Z 1 and Z 2 we obtain the right inequality in 24). To prove the left estimate in 24), we divide the unit sphere B 1 ) into the sets A = {x B 1 ) ; M Z1 x) = M Z2 x)}, A 1 = {x B 1 ) ; M Z1 x) > M Z2 x)}, A 2 = {x B 1 ) ; M Z1 x) < M Z2 x)}. For x A 2 set x = x/m Z1 x), and let Q Z2 x be the orthogonal projection of x onto Z 2, that is, Q Z2 x Z 2, P Z2 x = dist x, Z 2 ), 25) where we denote P Z2 x = x Q Z2 x. We have M Z2 x) > M Z1 x) = 1, hence x / Z 2 and d := P Z2 x >. Put m = 1 + d/c. Then the vector 1 m x = c c + d Q Z 2 x + d cp Z2 x c + d d

6 116 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... is a convex combination of elements of Z 2, hence M Z2 x) m. This yields M Z2 x) M Z1 x) m 1 1 c dist x, Z 2) 1 c d HZ 1, Z 2 ). Using 16) we conclude that M Z2 x) M Z1 x) 1 c 2 d HZ 1, Z 2 ), and arguing similarly for x A A 1 we complete the proof. Another distance criterion for convex sets involving the mapping J Z introduced in Lemma 3.1 is used in Sections 5 and 7. Here, we derive the following estimate. Lemma 3.3. Let Z 1, Z 2 C be such x M Z x) exists for each x X \ {} and that 14) and 2) hold for Z = Z i, for Z = Z i, i = 1, 2. Let L J be the Lipschitz constant on the right-hand side of 21). Then J Z1 x) J Z2 x) 2 2 c d M Z 1, Z 2 ) cl J + d M Z 1, Z 2 ))) 1/2 x B 1 ). 26) Proof. Let x X with x = 1 and let J Z1 x) J Z2 x). We define x s = x + s J Z 2 x) J Z1 x) J Z2 x) J Z1 x) for s. 27) We may assume that x s x, x, otherwise we interchange Z 1 and Z 2. The functions λ i s) := 1 2 M 2 Z i x s ) are convex and satisfy Thus, λ i ) + sλ i) λ i s) λ i ) + sλ is) for s. 28) λ 2 s) λ 1 s) λ 2 ) λ 1 ) + s λ 2) λ 1s)) 29) = λ 2 ) λ 1 ) + s λ 2) λ 1)) + s λ 1) λ 1s)). Note that, because s M Zi x s ) is differentiable, the chain rule implies that λ J Z2 x) J Z1 x) is) = J Zi x s ), for s. J Z2 x) J Z1 x) Hence λ 2) λ 1) = J Z2 x) J Z1 x), 3) λ 1s) λ 1) J Z1 x s ) J Z1 x) sl J. 31) We further have by 16) for all s that recall that we have assumed x s x, x ) λ 2 s) λ 1 s) x s c M Z 2 x s ) M Z1 x s ) x s 2 c d M Z 1, Z 2 ) 1 + s2 c d M Z 1, Z 2 ). 32)

7 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Combining 29) 32) we obtain for all s > that J Z2 x) J Z1 x) 2 + s2 sc d M Z 1, Z 2 ) + sl J. 33) The right-hand side attains its minimum for s = 2 d M Z 1, Z 2 )/cl J + d M Z 1, Z 2 )), and the assertion follows. We conclude this section with the following result as a simplified variant of [6, Lemma 3.2]. Lemma 3.4. Let Z X be an arbitrary convex closed set, and for θ > set Z θ = Z + B θ ). Then the outward unit normal n θ x) to Z θ is uniquely determined at each point x Z θ, and we have n θ x) n θ y) 1 θ x y x, y Z θ. 34) Proof. Let Q Z : X Z be the orthogonal projection onto Z. Putting P Z x = x Q Z x we have P Z x = dist x, Z) for each x X, and P Z x, Q Z x z x X, z Z. 35) For x Z θ and ˆx Z θ this yields P Z x = θ, P Z ˆx θ, and P Z x, x ˆx = P Z x, Q Z x Q Z ˆx + P Z x 2 P Z x, P Z ˆx. Let ñ be an arbitrary unit vector such that ñ, x x for every x Z θ. Putting x = Q Z x + θñ we obtain ñ, P Z x θ, hence ñ = 1/θ)P Z x = n θ x). From 35) it follows for all x, y X that P Z x P Z y, Q Z x Q Z y, hence P Z x P Z y x y, and 34) follows. 4. Existence and uniqueness for the explicit problem Since Problem P ) is equivalent to the classical sweeping process 1), the existence and uniqueness result for P ) can be deduced from the results of Moreau [11], if the set-valued function Z is Lipschitz continuous with respect to the Hausdorff metric, that is, if holds for some L Z >. d H Zϱ), Zσ)) L Z ϱ σ ϱ, σ R 36) It is obvious that if Problem P ) has a solution, it is unique. Indeed, if ξ, η are two solutions, then ξ, η ξ, η, ξ η, hence ξ η, ξ η and the assertion follows. Proposition 4.1. Let there exist L Z > such that 36) is satisfied. Then Problem P ) admits a unique solution ξ W 1,1, T ; X) for any given functions u W 1,1, T ; X), r W 1,1, T ; R) and any initial condition x Zr)).

8 118 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... Proof. We consider the sweeping process 1) with Ct) = ut) Zrt)), ξ = u) x, which is equivalent to P ). By 36) we have d H Ct), Cτ)) ut) uτ) + L Z rt) rτ). Since u and r are absolutely continuous, so is C. By Proposition 3c in [11], 1) then has a solution ξ W 1,1, T ; X). Note that in Proposition 4.1 there is no assumption regarding the interior of the sets Zρ), in particular 1) is not needed. Concerning the dependence of u upon C in the sweeping process 1), the basic general result is of Hölder type, see Proposition 2g in [11] or Theorem 4 in [7]. Let ξ, η be the solutions of Problem P ) for the data u, r, x ) respectively v, s, y ). Then Theorem 4 in [7] implies, if u, r, v, s are Lipschitz continuous, ξt) ηt) 2 x y + u) v) ) 2 +L where uτ) vτ) +L Z rτ) sτ) dτ, 37) L = 2 u + v + L Z ṙ + ṡ )). 38) Because of the square in 37), this formula cannot serve as the basis of a contraction mapping argument for problem I ). 5. Lipschitz estimates Under the assumption 1), we denote by Z ϱ) the polar set to Zϱ) defined in 13) for ϱ R, and by M ϱ, ) its Minkowski functional. By Lemma 3.1 we have B 1/C ) Z ϱ) B 1/c ) for every ϱ R, and the inequalities x C Mϱ, x) x c, 39) c x M ϱ, x) C x 4) hold for every x X and ϱ R. The following conditions are assumed to hold: L1) The partial derivatives ϱ Mϱ, x) Y, x Mϱ, x) X exist for every x X \{} and ϱ R, and the mappings Jϱ, x) = Mϱ, x) x Mϱ, x) : R X \ {} X, 41) Kϱ, x) = Mϱ, x) ϱ Mϱ, x) : R X \ {} Y 42) admit continuous extensions to x = and ϱ R. L2) For every x, x B C ) and ϱ, ϱ R we have Kϱ, x) Y K, 43) Jϱ, x) Jϱ, x ) C J ϱ ϱ Y + x x ), 44) Kϱ, x) Kϱ, x ) Y C K ϱ ϱ Y + x x ) 45) with some fixed constants K, C J, C K >.

9 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Lemma 5.1. Let 1), L1) and 43) hold. Then holds for all ϱ, σ R. Proof. By Lemma 3.2, By 39), if x = 1 we have 1 2 M 2 ϱ, x) 1 2 M 2 σ, x) d H Zϱ), Zσ)) C 3 K ϱ σ 46) d H Zϱ), Zσ)) C 2 sup Mϱ, x) Mσ, x). x =1 1 2 Mϱ, x) Mσ, x) Mϱ, x) + Mσ, x)) 1 Mϱ, x) Mσ, x), C and 46) follows from 43). Thus, Problem P ) has a unique solution by Proposition 4.1. The following two lemmas constitute the main steps towards the desired Lipschitz estimates. Lemma 5.2. Let L1) hold, let r, u) W 1,1, T ; R) W 1,1, T ; X) and x Zr)) be given, and let ξ W 1,1, T ; X) be the corresponding solution to Problem P ). For t ], T [ set A[r, u]t) = ξt), Jrt), xt)), B[r, u]t) = 1 2 M 2 rt), xt)), G[r, u]t) = ut), Jrt), xt)) + Krt), xt)), ṙt))), with xt) = ut) ξt). Then for a. e. t ], T [ we have either i) ξt) =, d B[r, u]t) = G[r, u]t) dt or ii) ξt), xt) Zrt)), A[r, u]t) = G[r, u]t) >, B[r, u]t) = max[,t ] B[r, u] = 1/2, d B[r, u]t) =, and dt ξt) = A[r, u]t) Jrt), xt)). 47) Jrt), xt)) 2 d Proof. Let L ], T [ be the set of Lebesgue points of all functions u, ṙ, ξ, B[r, u]. dt Then L has full measure in [, T ], and for t L we have d B[r, u]t) = ẋt), Jrt), xt)) + Krt), xt)), ṙt))). 48) dt

10 12 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... If ξt) =, then ẋt) = ut), and i) follows from 48). If ξt), then xt) Zrt)), hence Mrt), xt)) = 1 = max s [,T ] Mrs), xs)). We therefore have B[r, u]t) = d 1/2 = max [,T ] B[r, u], B[r, u]t) =. As a consequence of P ) iii) we have ξt) = dt k nrt), xt)) with a constant k >, where nrt), xt)) is the unit outward normal to Zrt)) at the point xt), hence k = ξt), nrt), xt)), and 47) follows from Lemma 3.1 ii). Furthermore, 48) yields ẋt), Jrt), xt)) = Krt), xt)), ṙt))), hence ξt), Jrt), xt)) = ut), Jrt), xt)) ẋt), Jrt), xt)) and the proof is complete. = ut), Jrt), xt)) + Krt), xt)), ṙt))), In the situation of Lemma 5.2, we always have G[r, u]t) ut) Jrt), xt)) + K ṙt) Y, 49) ξt) ut) + CK ṙt) Y. 5) Indeed, 5) is trivial if ξt) = ; otherwise we have ξt) = A[r, u]t)/ Jrt), xt)) = G[r, u]t)/ Jrt), xt)) with xt) Zrt)). Lemma 3.1 then yields C Jrt), xt))) M rt), Jrt), xt))) = Mrt), xt)) = 1, and 5) follows from 49). Lemma 5.3. Let L1) and 43) hold, let r, u), s, v) W 1,1, T ; R) W 1,1, T ; X) and x Zr)), y Zs)) be given, let ξ, η W 1,1, T ; X) be the respective solutions to Problem P ), and set x = u ξ, y = v η. Then for a. e. t ], T [ we have A[r, u]t) A[s, v]t) + d B[r, u]t) B[s, v]t) G[r, u]t) G[s, v]t), 51) dt ξt) ηt) C ut) + CK ṙt) Y ) Jrt), xt)) Jst), yt)) 52) Proof. + C A[r, u]t) A[s, v]t). The assertion follows directly from Lemma 5.2 if ξt) = ηt) =. Assume now ξt), ηt). Then 51) is again an immediate consequence of Lemma 5.2. To prove 52), we use 47) and the elementary vector identity z z z 2 z 2 = 1 z z z z for z, z X \ {}, to obtain ξt) ηt) A[r, u]t) Jrt), xt)) Jrt), xt)) Jst), yt)) 2 Jst), yt)) 2 = + 1 A[r, u]t) A[s, v]t) Jst), yt)) 1 G[r, u]t) Jrt), xt)) Jst), yt)) Jrt), xt)) Jst), yt)) + 1 A[r, u]t) A[s, v]t). Jst), yt))

11 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Lemma 3.1 yields Jrt), xt)) 1/C, Jst), yt)) 1/C. inequalities with 49) we obtain the assertion. Let us consider now the case ξt), ηt) =. Combining the above Then A[r, u]t) A[s, v]t) = A[r, u]t), B[r, u]t) B[s, v]t) = 1/2 B[s, v]t), hence A[r, u]t) A[s, v]t) + d dt B[r, u]t) B[s, v]t) = A[r, u]t) d B[s, v]t) dt hence 51) is fulfilled. We further have similarly as above that = G[r, u]t) G[s, v]t), ξt) ηt) = ξt) C A[r, u]t) = C A[r, u]t) A[s, v]t), hence 52) holds. The remaining case ξt) =, ηt) is analogous, and Lemma 5.3 is proved. We are now ready to prove the following crucial estimate. Proposition 5.4. Let L1), L2) hold, let r, u), s, v) W 1,1, T ; R) W 1,1, T ; X) and x Zr)), y Zs)) be given, let ξ, η W 1,1, T ; X) be the respective solutions to Problem P ), and set x = u ξ, y = v η. Then for a. e. t ], T [ we have ξt) ηt) + C d dt B[r, u]t) B[s, v]t) C c ut) vt) + CK ṙt) ṡt) Y 53) ) + C 2C J ut) + C K + CC J K ) ṙt) Y rt) st) Y + xt) yt) ). Proof. By Lemma 3.1 we have c Jrt), xt)) M rt), Jrt), xt))) = Mrt), xt)) 1, hence Jrt), xt)) 1/c for every t [, T ]. By Lemma 5.3, we can estimate the left-hand side of 53) by C ) G[r, u]t) G[s, v]t) + ut) + CK ṙt) Y ) Jrt), xt)) Jst), yt)) which together with the assumptions 43) 45) yields the assertion. 6. Implicit model We now consider Problem I ) under the following hypothesis on the mapping g. G) g : [, T ] X X Y is continuous and gt, u, ξ) R for each t, u, ξ)

12 122 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... [, T ] X X. Its partial derivatives t g, u g, ξ g exist and satisfy the inequalities ξ gt, u, ξ) LX,Y ) γ, 54) u gt, u, ξ) LX,Y ) ω, 55) t gt, u, ξ) Y at), 56) ξ gt, u, ξ) ξ gt, v, η) LX,Y ) C g u v + ξ η ), 57) u gt, u, ξ) u gt, v, η) LX,Y ) C u u v + ξ η ), 58) t gt, u, ξ) t gt, v, η) Y bt) u v + ξ η ) 59) for every u, v, ξ, η X and a. e. t ], T [ with given functions a, b L 1, T ) and given constants γ, ω, C g, C u > such that where C, K are as in Hypotheses 1) and L2). δ = CK γ < 1, 6) Let us start our analysis with the following necessary condition. Lemma 6.1. Let Hypotheses L1), L2), and G) hold, and let ξ W 1,1, T ; X) be a solution to Problem I ) with some u W 1,1, T ; X) and x Zg, u), u) x )). Then we have ξt) 1 1 δ 1 + CK ω) ut) + CK at)) a. e. 61) Proof. Inequality 61) is an easy consequence of 5) with rt) = gt, ut),ξt)). Indeed, using 54), 55) we obtain ṙt) Y at) + ω ut) + γ ξt) and 61) follows. We now prove the converse as the main result of this section. Theorem 6.2. Let Hypotheses L1), L2), and G) hold. Then for every u W 1,1, T ; X) and every x Zg, u),u) x )) there exists a unique solution ξ W 1,1, T ; X) to Problem I ) in the set Ω = { η W 1,1, T ; X) ; ηt) CK 1 δ ω) ut) + CK at)) a. e. η) = u) x }. Proof. Let S : Ω W 1,1, T ; X) be the mapping which with each η Ω associates the solution ξ to Problem P ) with rt) = gt, ut), ηt)). By 5) we have ξt) ut) + CK ṙt) Y 1 + CK ω) ut) + CK at) + δ ηt) 62) 1 1 δ 1 + CK ω) ut) + CK at)), hence SΩ) Ω. The set Ω is convex and closed in W 1,1, T ; X). We now check that S : Ω Ω is a contraction with respect to a suitable norm in W 1,1, T ; X).

13 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Let η 1, η 2 Ω be given. By Proposition 5.4, the functions ξ i = Sη i ) for i = 1, 2 satisfy almost everywhere the inequality ξ 1 t) ξ 2 t) + βt) δ η 1 t) η 2 t) 63) + C δ ut) + at) + bt)) η 1 t) η 2 t) + ξ 1 t) ξ 2 t) ) with βt) = C B[g, u,η 1 ), u]t) B[g, u,η 2 ), u]t), β) =, and with a constant C δ > independent of η 1, η 2. Let now ε > be chosen so small that and let us define an auxiliary function wt) = e 1 ε δ + εc δ 1 εc δ = δ < 1, 64) R t uτ) +aτ)+bτ)) dτ for t [, T ]. 65) We have wt) > for every t [, T ] and ẇt) a. e. We test the inequality 63) by wt) and integrate over [, T ]. Taking into account the relations βt) wt) dt = [βt) wt)] T βt) ẇt) dt, wt) ut) + at) + bt)) η 1 t) η 2 t) + ξ 1 t) ξ 2 t) ) dt ε ẇt) η 1 τ) η 2 τ) + ξ 1 τ) ξ 2 τ) ) dτ dt [ ] T = ε wt) η 1 τ) η 2 τ) + ξ 1 τ) ξ 2 τ) ) dτ + ε wt) η 1 t) η 2 t) + ξ 1 t) ξ 2 t) ) dt ε wt) η 1 t) η 2 t) + ξ 1 t) ξ 2 t) ) dt, we obtain from 63) that wt) ξ 1 t) ξ T 2 t) dt δ+εc δ ) wt) η 1 t) η 2 t) dt+εc δ wt) ξ 1 t) ξ 2 t) dt hence wt) ξ 1 t) ξ T 2 t) dt δ wt) η 1 t) η 2 t) dt. 66) We thus checked that S is a contraction on Ω with respect to the weighted norm η = η) + wt) ηt) dt, hence S admits a unique fixed point ξ which is a solution of I ).

14 124 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Lipschitz continuity In this section we prove local Lipschitz continuity estimates for both Problems P ) and I ) as a combination of Proposition 5.4 and a Gronwall-type argument. Theorem 7.1. Let the assumptions of Proposition 5.4 be fulfilled. Then there exist positive constants C, C 1 such that for every R >, every r, u), s, v) W 1,1, T ; R) W 1,1, T ; X) with u + ṙ Y )dt R, v + ṡ Y )dt R, and every x Zr)), y Zs)), the respective solutions ξ, η W 1,1, T ; X) of problem P ) satisfy the inequality ξ η dt C 1 e C R x y + r) s) Y + ) u v + ṙ ṡ Y )dt. 67) Proof. By Proposition 5.4, there exists a constant C > such that ẋt) ẏt) + βt) C ut) vt) + ṙt) ṡt) Y 68) ) + ut) + ṙt) Y ) xt) yt) + rt) st) Y ) with βt) = C B[r, u]t) B[s, v]t). We argue similarly as in the proof of Theorem 6.2 and test 68) by the function This yields d w 1 t) dt Note that w 1 t) = e C R t u + ṙ Y ) dτ. ) ẋ ẏ dτ + w 1 t) βt) ) C w 1 t) ut) vt) + ṙt) ṡt) Y ẇ 1 t) x y + r) s) Y + 69) ) ṙ ṡ Y dτ. w 1 t) βt) dt = [w 1 t) βt)] T ẇ 1 t) βt) dt w 1 ) β) 7) C 2 M 2 r), x ) M 2 s), y ) CK r) s) Y + C ) c x y. On the other hand, integrating 69) from to T and using the fact that for every t [, T ] we have 1 w 1 t) w 1 T ) e C R we obtain e C R ẋ ẏ dt w 1 t) βt) dt + x y + r) s) Y 71) + C + 1) and the assertion follows from 7), 71). u v + ṙ ṡ Y ) dt,

15 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution Theorem 7.2. Let the assumptions of Theorem 6.2 be fulfilled. Then there exist positive constants C 2, C 3 such that for every R >, every u, v W 1,1, T ; X) with u dt R, v dt R, and every x Zg, u), u) x ), y Zg, v), v) y ), the respective solutions ξ, η W 1,1, T ; X) of problem I ) satisfy the inequality ξ η dt C 3 e C 2R ) x y + u) v) + u v dt. 72) Proof. We use Lemma 6.1 and Proposition 5.4 with rt) = gt, ut), ξt)), st) = gt, vt), ηt)), and find a constant C > such that 1 δ) ξt) ηt) + βt) C ut) vt) 73) ) + ut) + at) + bt)) ut) vt) + ξt) ηt) ) with βt) = C B[g, u, ξ), u]t) B[g, v, η), v]t). Repeating the procedure from the proof of Theorem 7.1 with we easily obtain the assertion. 8. Nonuniqueness C 2 = C 1 δ, w 2t) = e C R t 2 uτ) +aτ)+bτ)) dτ 74) Let ψ : [, 1] [, 1] be an increasing concave function with ψ) =. Then the initial value problem ẇ = δψw), w) =, 75) has a positive solution for any δ > in addition to the trivial one, if and only if Let us assume that 76) holds. Then 1 1 dε <. 76) ψε) ψε) lim ε + ψ ε) = lim ε + ε =. 77) Moreover assume that there exists K > such that ψ ε) K ε for a. e. ε ], 1[. 78) Then A typical example is ψε) = ε β, 1/2 β < 1. We denote ψε) 2K ε ε [, 1]. 79) θ = K 2 8)

16 126 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... with K from 78), and consider for α and η 1 θ the system of convex sets in X = R 2, where Hη, α) is the half-plane V η, α) = B 1 θ ) Hη, α), 81) Hη, α) = {z R 2 ; z, ν α 1 θ η}, ν α = We further define two one-parametric families of convex sets ) 1. 82) α Zε) = V ε + θ 1 + ψ 2 ε) 1), ψε)), 83) Zε) = Zε) + B θ ), 84) for ε [, ε ], where ε ], 1] is chosen in such a way that for every ε ], ε ] we have θ ψ 2 ε)) ψ2 ε) ε + 2θ ε + θ 1 + ψ 2 ε) < 1, 85) 1 ε) ψ2 ε) ε + 1 ε 1 )) 1 ε 1 + ψ2 ε) 2 ε. 86) Let Λθ, ε) denote the left-hand side of 86). To see that condition 86) is meaningful, it suffices to use 79) which entails that ) ) 4K lim sup Λθ, ε) 4K 2 θ 2 + 2θ 1 + 6K ) = 2 1 < 2. 87) ε K 2 The set Zε) can also be characterized as z Zε) z 1 θ, z, νψε) 1 ε θ 1 + ψ2 ε), 88) see Figure 1. As the next step, we check that the segment { ) } x Sε) = z = ; x ψε)y = 1 ε, y θ ψε) y 89) satisfies Sε) Zε) ε [, ε ]. 9) The limit case ε = is easy: we have Z) = B 1 ), and S) contains only one element ) 1 e 1 =. For ε > and z Sε) we define z θ = z θ 1 + ψ2 ε) ν ψε). Then z θ z = θ, 91) zθ, ν ψε) = 1 ε θ 1 + ψ2 ε), 92)

17 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution z θ 2 = z 2 + θ 2 1 ε 2θ 1 + ψ2 ε), 93) where z 2 1 ε + θψ 2 ε)) 2 + θ 2 ψ 2 ε) by 89). Developing the computation and using 86) we obtain from 93) that z θ 2 1 ε) 2 + θ ψ 2 ε) + ψ 4 ε)) + 2θ 1 ε) ψ 2 1 ε ε) 2θ 1 + ψ2 ε) )) = 1 θ) 2 + θ ψ 2 ε)) ψ 2 ε) + 2θ 1 ε) ψ 2 1 ε ε) ε 2 2ε 1 + ψ2 ε) 1 θ) 2, hence z θ Zε) and z Zε). To see that z Zε), we notice using 88) that for arbitrary z Zε) we have z z 1 z z, νψε) θ. 1 + ψ2 ε) In other words, dist z, Zε)) = θ, and 9) is verified. y Zwt)) 1 θ Zwt)) vt) 1 x Figure 1. Trajectory of the nontrivial solution vt). We now construct the example of nonuniqueness for the evolution quasivariational inequality 4). Let w : [, t ] R be the positive solution to the initial value problem 75) for a fixed δ >, where we choose t sufficiently small such that We define vt) = wt ) ε, ) v1 t) with v 2 t) wt ) ψwt )) δ θ, δψwt )) < 1. 94) v 1 t) = 1 wt) + 1 δ wt)ψwt)), v 2t) = 1 wt), 95) δ

18 128 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... and set Then Γv) = Zδv 2 ). 96) v 2 t) ψwt)) v 2t ) ψwt )) < θ, 97) hence vt) Swt)) Zwt)) = Γvt)) for all t [, t ], see Figure 1. We set ft) = f 1 t), ), f 1 t) = 1 + v 1 t), 98) then ft) vt) = ν ψwt)) vt)) N Γvt)) vt)), and f 1 t) > for t t. Therefore, both ) 1 vt) as defined and ˆvt) e 1 = are solutions of 4). Thus, there is nonuniqueness, despite the fact that the outward normal vectors n Zε) z) are Lipschitz continuous in z for every ε with a Lipschitz constant independent of ε as a consequence of Lemma 3.4. Moreover, the orthogonal projections Q Zε) of X onto Zε) and therefore also n Zε) by virtue of the argument in the proof of Lemma 3.4) depend Lipschitz-continuously on ε in each interval [ε 1, ε ] with < ε 1 < ε. For ε = there is indeed no problem in any direction different from e 1, as the boundary of Zε) does not move here for ε small. However, the dependence of J Zε) z) on ε is not globally Lipschitz and the Lipschitz constant behaves like ψε)/ε for z = e 1 near ε =. Indeed, for fixed ε > and z Sε) we have M Zε) z) = z, ν ψε) /1 ε). We easily find qε) > which guarantees that for every y B qε) ) we can find pε, y) > such that pε, y)e 1 + y) Sε). For all such y we thus have M Zε) e 1 + y) = e 1 + y, ν ψε) /1 ε), hence JZε) e 1 ) = ν ψε) /1 ε) 2. Obviously, J Z) x) = x for every x X, and we conclude that J Zε) e 1 ) J Z) e 1 ) = ψ2 ε) + ε 2 2 ε) 2 1 ε) 2, 99) where the right-hand side is of the order of ε β if for example ψε) = ε β, 1/2 β < 1. On the other hand, we will see that Zε) does depend Lipschitz-continuously on ε in the sense of the Hausdorff distance. This implies in turn that the map v Γv) is Lipschitz with an arbitrarily small constant similarly as in the example of [1] provided δ is chosen sufficiently small in fact, to make Figure 1 more transparent, we have chosen a trajectory of the solution vt) corresponding to a very large δ). We conclude the paper by establishing this Lipschitz continuity result as a separate lemma. Lemma 8.1. There exists a constant R > such that for every ε 2 < ε 1 ε we have i) Zε 1 ) Zε 2 ), ii) d H Zε 1 ), Zε 2 )) R ε 1 ε 2 ). It is interesting to compare formula 99) with Lemmas 3.2, 3.3, and 8.1. We see that the hypothesis 79) is sharp in the sense that if one sets ψε) = ε β with β < 1/2 in the example above, one cannot expect that d H Zε), Z)) remains Lipschitz continuous in ε. Proof. To prove i), let us consider an arbitrary z = ) x y Zε 1 ) and assume that

19 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution z / Zε 2 ). Then x ψε 2 )y > 1 ε 2 θ 1 + ψ 2 ε 2 ), 1) x ψε 1 )y 1 ε 1 θ 1 + ψ 2 ε 1 ), 11) For α [, ψε )] we define an auxiliary function x 1 θ. 12) Ψα) = ψ 1 α) + θ 1 + α 2 1) and set α i = ψε i ), i =, 1, 2. From 1) 12) it follows that Ψα 2 ) α 2 > y > Ψα 1) Ψα 2 ) α 1 α 2. 13) Since Ψ is convex, increasing, and Ψ) =, 13) is contradictory and we conclude that Zε 1 ) Zε 2 ), hence i) holds. It remains to prove the Lipschitz estimate in ii). Let z Zε 2 )\ Z ε 1 ) be arbitrary. With the above notation we have We find α [α 2, α 1 [ such that z, ν α2 1 θ Ψα 2 ), 14) z, ν α1 > 1 θ Ψα 1 ). 15) z, ν α = 1 θ Ψα ), 16) and put ε = ψ 1 α ), z 1 = z σν α1, where σ > is chosen in such a way that z 1, ν α1 = 1 θ Ψα 1 ), 17) that is, We have σ = 1 z, ν 1 + α1 2 α1 1 + θ + Ψα 1 )). z 1 2 = z 2 σ 2 ν α1 2 2σ 1 θ Ψα 1 )) < z 2 1 θ) 2, hence z 1 Zε 1 ). Our goal now is to find R > independent of ε 1, ε 2 such that z z 1 R ε 1 ε ), 18) and it will suffice to pass from Zε i ) to Zε i ), i = 1, 2, to complete the proof. Inequality 18) will follow from a series of estimates, where C 1, C 2,... will denote arbitrary positive constants depending only on K. First of all, we have z 1 θ)ν α ν α 2 ) 1 θ) 2 21 θ) 1 θ Ψα )) 19) 1 θ) α 2 + 2ψ 1 α ) ) C 1 ε.

20 13 M. Brokate, P. Krejčí, H. Schnabel / On Uniqueness in Evolution... On the other hand, we have z z 1 = σ ν α1 σ ν α1 2 = z z 1, ν α1 11) = z 1 θ)ν α, ν α1 ν α + 1 θ) ν α, ν α1 ν α + Ψα 1 ) Ψα ) C 1 ε ν α1 ν α + 1 θ να1 2 ν α 2) + Ψα 1 ) Ψα ) 2 C 2 ε α 1 α ) + ε 1 ε ) + α 2 1 α 2 ) ), where, by 78) 79), we have α 1 α 2K ε 1 ε ) and and 18) immediately follows. ε1 α1 2 α 2 = 2 ψε) ψ ε) dε 4K 2 ε 1 ε ), ε Acknowledgements. The authors thank one of the referees for the suggestion that Proposition 4.1 might follow directly from Moreau s results. This turned out to be right, as the exposition above shows. References [1] P. Ballard: A counter-example to uniqueness in quasi-static elastic contact problems with friction, Int. J. Eng. Sci ) [2] V. V. Chernorutskii, M. A. Krasnosel skii: Hysteresis systems with variable characteristics, Nonlinear Anal. TMA ) [3] R. E. Edwards: Functional Analysis, Holt, Rinehart and Winston, New York 1965). [4] P. Krejčí: Evolution variational inequalities and multidimensional hysteresis operators, in: Nonlinear Differential Equations, P. Drábek, P. Krejčí, P. Takáč eds.), Research Notes in Mathematics 44, Chapman & Hall/CRC, London 1999) [5] P. Krejčí, Ph. Laurençot: Generalized variational inequalities, J. Convex Analysis 9 22) [6] P. Krejčí, J. Sprekels: Singular limit in parabolic differential inclusions and the stop operator, Interfaces Free Bound. 4 22) [7] M. Kunze, M. Monteiro Marques: An introduction to Moreau s sweeping process, in: Impacts in Mechanical Systems - Analysis and Modelling, B. Brogliato ed.), Lecture Notes in Physics 551, Springer 2) 1 6. [8] M. Kunze, M. Monteiro Marques: On parabolic quasi-variational inequalities and statedependent sweeping processes, Topol. Methods Nonlinear Anal ) [9] A. Mielke, F. Theil: On rate-independent hysteresis models, Preprint #21/7, SFB 44, Universität Stuttgart, 21. [1] J.-J. Moreau: Problème d évolution associé à un convexe mobile d un espace hilbertien, C. R. Acad. Sci. Paris Sér. A-B ) A791 A794. [11] J.-J. Moreau: Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Eq ) [12] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton 197).