Regularity and approximation for strong solutions to rate ... - arXiv

REGULARITY AND APPROXIMATION OF STRONG SOLUTIONS TO RATE-INDEPENDENT SYSTEMS

arXiv:1702.01427v1 [math.AP] 5 Feb 2017

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

Abstract. Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In contrast with existing approaches, in this work we directly prove the existence of H¨ older-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates. In the second part of the paper we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established. MSC (2010): 49J40 (primary); 47J20, 47J40, 74H30. Keywords: Rate-independent systems, quasistatic evolution, regularity theory, finite element methods. Date: February 7, 2017.

1. Introduction Rate-independent systems are used to model a plethora of physical phenomena for which the speed of evolution does not influence the extent of energy dissipation, including elasto-plasticity, damage and delamination in solids, crack propagation, and shape-memory alloys. We refer to the recent monograph [MR15] for an up-to-date overview of the literature on both the theory and the applications of rate-independent systems. In this work we consider only purely dissipative systems, i.e. those without elastic variables (which, by definition, are those quantities that can be changed without dissipating energy). A formal prototype of a rate-independent system of this kind is u˙ − ∆u + DW0 (u) = f, |u| ˙

u : [0, T ] × Ω → Rm ,

where W0 is a smooth potential defined on Rm and f : [0, T ] × Ω → Rm is a given external force. For the mathematical analysis, the crucial feature of such systems of equations is the quasi-static nature of their evolution, namely that solutions simply rescale when scaling the external forces. In a sense, quasi-static evolutions follow the total (energetic and external) forces with infinite speed and hence the evolution should, at least in sufficiently convex situations, “inherit” regularity from the external force. Questions about the regularity of solutions were in fact already raised in the early stages of development of the modern energetic theory of rate-independent systems; see in particular [MTL02] and [MT04] (Section 7.3 in the second reference discusses temporal regularity for the uniformly convex case). Further investigations in that direction were carried out recently in [TM10, MZ14]. The theory of rate-independent systems is dominated by several notions of weak solution concepts (see [Mie11, MR15]), and the question how “realistic” particular notions of solution are from the point of view of applications has been the source of considerable discussion in the literature. A central question is whether there is “foresight” in the system, i.e. whether the system will dissipatively jump over potential energy barriers in order to realize an ultimate drop in energy. The Mielke–Theil energetic solutions do this, but more recent concepts, such as bounded-viscosity solutions [MRS09, MRS12], do not. A more substantial discussion of this point can be found in [MR15] and Section 1.2 of the present paper. Our objective here is to show that in a “mildly nonconvex” setting with a quadratic gradient regularizer, one may focus right from the start on strong solutions, thereby avoiding much of the ambiguity. We will show that: • strong solutions exist; • they have additional regularity and satisfy quantitative H¨older estimates in space and time; • they are unique within their natural regularity class;

2

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

• and they can be numerically approximated by a convergent finite element method. Thus, in this setting, there is no need to consider weak solution concepts. We refer to Section 1.2 for a motivating scalar example, from which it can be seen that weak solutions are nonunique, jump “too early”, and do not agree with strong solutions, even if both exist. This motivates our focus on strong solutions (where they exist), which seem to be the physically realistic ones. Our assumptions apply for instance to some double-well energies with “mild” nonconvexities, where “mild” refers to the fact that the admissible extent of nonconvexity depends on the L2 -Poincar´e embedding constant of the domain. The class of problems considered here involves an elliptic regularization, which ensures that the regularized functional is convex. This framework is frequently used in the literature, as some regularization is already needed in order to prove the existence of solutions; see [MR15]. Partial results in the direction of regularity have already been presented in [TM10, MPPS10, MZ14, KRZ15], but those papers deal with the Mielke–Theil energetic solution concept. In contrast with those works, we are able to guarantee more regularity in both time and space, and with the natural dependence of the regularity of the solution on that of the data. We also establish quantitative estimates and develop a direct, unified approach, avoiding abstract hypotheses. Some remarks on uniqueness in this framework can be found in Sections 3.4.3 and 3.4.4 in [MR15]. As for the numerical approximation of rate-independent systems, one concrete application was considered by Han and Reddy [HR00], who provide the mathematical analysis of semidiscrete and fully discrete approximations to the primal problem in elastoplasticity. They proved the strong convergence of the sequence of numerical approximations defined by their method(s) under conditions of minimal regularity of the solution, that is, without any assumptions on the regularity of the solution over and above those established in the proof of well-posedness of the problem. More recently, Mielke et al. [MPPS10] have developed error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, they formulated a fully discrete approximation and developed a-priori error estimates. Bartels [Bar14] derived quasioptimal error bounds for implicit discretizations of a class of rate-independent evolution problems, without imposing regularity assumptions on the exact solutions, but the load functionals involved in the problem were assumed to be twice continuously differentiable in time. The new contribution in this paper to the mathematical analysis of finite element approximations of rate-independent systems is that the orders of convergence established in Section 5 are in agreement with the regularity results for the analytical solution developed in Section 3. 1.1. Strong solutions. For technical reasons we only consider the physically relevant case of Ω ⊂ Rd being a bounded open set, with d ∈ {2, 3}. Some of our results are also true in higher dimensions, but additional (more restrictive) assumptions are then necessary. Specifically, we will investigate the class of rate-independent systems that can be formulated as follows: for Ω ⊂ Rd a bounded open Lipschitz domain, d = 2, 3, and T > 0, consider the following (formal) system:  ˙ − Lt u(t) + DW0 (u(t)) ∋ f in (0, T ] × Ω,   ∂R1 (u(t)) u(t)|∂Ω = 0 for t ∈ (0, T ], (1.1)   u(0) = u0 .

Here, R1 : L1 (Ω; Rm ) → R ∪ {+∞} is the rate-independent dissipation potential, which is assumed to be proper (in the sense that it is not identically +∞), convex, and positively 1-homogeneous; ∂R1 is its subdifferential; f ∈ W1,a (0, T ; Lp (Ω; Rm )), for a ∈ (1, ∞) and p ∈ [2, ∞), is the external loading (force), u0 ∈ (W01,2 ∩ Lq )(Ω; Rm ), q ∈ (1, ∞) is the initial value, and W : Lq (Ω; Rm ) → R denotes the (elastic) energy functional; we assume that DW0 (v) ∈ Lq/(q−1) (Ω; Rm ) and kDW0 (v)kLq/(q−1) ≤ C(1 + kvkLq−1 q )

for all v ∈ Lq (Ω; Rm ) and some q ∈ (1, ∞).

Finally, the regularizer Lt is a (possibly time-dependent) second-order linear partial differential operator in the spatial variables (most commonly, Lt = ∆). Our precise assumptions on Lt are detailed below. We note that our choice of a homogeneous Dirichlet boundary condition is made merely to simplify the exposition; analogous results for nonhomogeneous Dirichlet boundary data necessitate only unessential technical alterations to our arguments.

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

3

We call a map u ∈ L∞ (0, T ; (W01,2 ∩ L∞ )(Ω; Rm )), such that its weak time derivative u˙ has regularity u˙ ∈ L1 (0, T ; W01,2 (Ω; Rm )), a strong solution to (1.1) provided that Lt u(t) − DW0 (u(t)) + f (t) ∈ L∞ (0, T ; L2 (Ω; Rm )) and

(

Lt u(t) − DW0 (u(t)) + f (t) ∈ ∂R1 (u(t)) ˙

(1.2)

for a.e. t ∈ (0, T ],

u(0) = u0 .

By recalling the definition of the subdifferential, the above means that

  R (u(t)) ˙ + Lt u(t) − DW0 (u(t)) + f (t), ξ(t) − u(t) ˙ ≤ R1 (ξ(t))   1 1,2 for all ξ ∈ L1 (0, T ; W0 (Ω; Rm )) and a.e. t ∈ (0, T ],   u(0) = u0 ,

(1.3)

where h q , q i denotes the L2 -duality pairing. We note here that the attainment of the initial condition makes sense since our regularity assumptions on u, u˙ imply that u ∈ C([0, T ]; L2 (Ω; Rm )).

Remark 1.1. The above regularity class for u, u˙ is not minimal from the point of view of ensuring that (1.3) is meaningful, but it turns out to be a suitable class of functions for establishing both the existence and the uniqueness of a strong solution. An alternative concept of solution would be to require that u ∈ L∞ (0, T ; (W01,2 ∩ Lq )(Ω; Rm ))

with

u˙ ∈ L1 (0, T ; (W01,2 ∩ Lq )(Ω; Rm ))

such that (1.3) is satisfied for all ξ ∈ L1 (0, T ; (W01,2 ∩ Lq )(Ω; Rm )) (with h q , q i now also denoting the (Lq × Lq/(q−1) )-duality pairing). Note that here we do not assume (1.2) as part of the solution concept. If d = 3 and 1 ≤ q ≤ 6, by the Sobolev embedding theorem we have that (W01,2 ∩ Lq )(Ω; Rm ) = W01,2 (Ω; Rm ), so that the solution we construct in Theorem 1.3 below is also a solution in this sense. However, if q > 6, then our existence proof does not provide a solution in the modified sense. Still, uniqueness holds in this larger class, as will follow from the proof of Theorem 1.5, also see Remark 1.12. 1.2. Motivating example. Our aim in this section is to motivate and discuss why the existence and regularity of strong solutions should not merely be seen as a theory of “improved regularity” for weak solutions (for instance in the energetic sense), and why strong solutions should be considered from the outset as a suitable solution concept. To see this, we consider the following zero-dimensional double-well setup: W0 (z) = W0 (z) := min{z(z + 2), z(z − 2)},

R1 (z) = R1 (z) := |z|.

For f (t) := t,

t ∈ [0, ∞),

an energetic weak solution uweak with initial value u(0) = −1 exists on the infinite time interval [0, ∞). Indeed, it can be checked in an elementary fashion that ( −1 if t ∈ [0, 1), weak u (t) := t+1 if t ∈ [1, ∞) 2 satisfies the energy balance E(t, u(t)) − E(0, u(0)) =

Z

0

t

f ′ (s)u(s) ds − VarR1 (u; [0, t)),

(1.4)

where E(t, z) := W (z) − f (t)z for all t ∈ [0, ∞), and VarR1 denotes the total R1 -variation, as well as the global stability inequality E(t, u(t)) ≤ E(t, z) + R1 (z − u(t)) for all z ∈ R. On the other hand, over the time interval [0, 3) also a strong solution ustrong exists, namely ( −1 if t ∈ [0, 1), ustrong (t) := t−3 if t ∈ [1, 3). 2

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

4

We see that uweak and ustrong coincide on the interval [0, 1), but they differ on the interval [1, 3) where both a weak and a strong solution exist. This fact is closely related to the nonuniqueness of weak solutions: both uweak and ustrong are in fact weak solutions (as can be easily verified), but only one of them is strong. However, uweak jumps “too early” and should be regarded as unphysical since it implies “foresight” in the system, i.e. the system jumps as early as possible to minimize z 7→ E(t, z)+R1 (z−u(t)) at all times. These facts are of course well-known in the theory of rate-independent systems (see e.g. [MR15] and also [MPPS10]). Furthermore, ustrong can be extended to t ∈ [3, ∞), as a weak solution only, by setting   −1 if t ∈ [0, 1), ext u (t) := t−3 if t ∈ [1, 3), 2   t+1 if t ∈ [3, ∞). 2 This function satisfies the energy balance (1.4) as well as the local stability condition −DE(t, u(t)) ∈ ∂R1 (0). We remark that uext can be understood as a strong solution in [0, 3) and (3, ∞) with regard to the convex potentials z 7→ z(z + 2) and z 7→ z(z − 2), respectively. Only at the jump point t = 3 do we need to use the concept of energetic solution. In fact, our existence and uniqueness results, Theorems 1.3 and 1.5, can be used to show that on [0, 3) ∪ (3, ∞) the strong solution exists (which we already know from our explicit construction) and that it is unique within the class of strong solutions. We conclude from this discussion that a physically realistic notion of solution for a rate-independent system could be that of a “maximally-strong” energetic solution, that is, a Mielke–Theil energetic solution such that on every open interval I ⊂ [0, T ] where it agrees with a strong solution, this strong solution cannot be extended beyond I. Clearly, uext satisfies this definition and the jump in uext occurs precisely when the solution moves between regions of convexity of E. In higher dimensions, this definition has to be refined, which we postpone to future work. In the present paper we aim to prepare this approach by establishing maximal regularity, uniqueness, and approximation properties of strong solutions. Related examples can be found in Section 1.8 of [MR15]. 1.3. Assumptions. Unless otherwise stated, the following conditions will be assumed to hold in the rest of the paper: (A1) Let d ∈ {2, 3} and let Ω ⊂ Rd be open, bounded and with boundary of class C1,1 . (A2) The rate-independent dissipation (pseudo)potential R1 : L1 (Ω; Rm ) → R ∪ {+∞} is given as Z R1 (v(x)) dx, v ∈ L1 (Ω; Rm ), R1 (v) = Ω

Rm

with R1 : → R convex, lower semicontinuous, and positively 1-homogeneous, i.e. R1 (αw) = αR1 (w) for any α ≥ 0 and w ∈ Rm . As any convex function defined on Rm is locally Lipschitz continuous, the assumed positive 1-homogeneity of R1 then implies that R1 is in fact globally Lipschitz continuous on Rm . (A3) The energy functional W0 : Lq (Ω; Rm ) → [0, ∞], where q ∈ (1, ∞), has the form Z W0 (u(x)) dx, W0 (u) = Ω 1 m C (R ; [0, ∞))

with W0 ∈ and all v, w ∈ Rm :

satisfying the following assumptions for suitable constants C, µ > 0,

C −1 (|v|q − 1) ≤ W0 (v) ≤ C(|v|q + 1); q−1

|DW0 (v)| ≤ C(1 + |v|

);

−µ|v − w|2 ≤ (DW0 (v) − DW0 (w)) · (v − w); µCP (Ω)2 < κ.

(1.5) (1.6) (1.7) (1.8)

Here, κ is defined via the ellipticity of the regularizer below, and CP (Ω) > 0 is the (best) L2 Poincar´e embedding constant of Ω, i.e. the smallest constant C > 0 such that kvkL2 ≤ Ck∇vkL2

for all v ∈ W01,2 (Ω; Rm ).

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

5

(A4) The regularizer Lt is a second-order linear partial differential operator of the form X X α,β Ai,j (t, x) ∂i v α (x) [Lt v]β (x) := divx [A(t, x) : ∇v(x)]β = ∂j j

α,i

α,β for 1 ≤ α, β ≤ m, 1 ≤ i, j ≤ d. We assume that the fourth-order tensor A = (Aα,β i,j )i,j satisfies the following continuity, ellipticity and symmetry conditions: β,α 0,1 Aα,β for α, β ∈ {1, . . . , N }, i, j ∈ {1, . . . , d}; i,j = Aj,i ∈ C ([0, T ] × Ω) X β 2 ξ : A(t, x) : ξ = ξiα Aα,β for ξ ∈ Rm×d , (t, x) ∈ [0, T ] × Ω, i,j (t, x)ξj ≥ κ|ξ|

(1.9) (1.10)

i,j,α,β

where κ > 0 is the modulus of ellipticity. (A5) The external force has regularity f ∈ W1,a (0, T ; Lp (Ω; Rm )), for some a ∈ (1, ∞] and p ∈ [2, ∞]. (A6) The initial value satisfies u0 ∈ (W01,2 ∩ Lq )(Ω; Rm ) and u0 is stable, i.e. u0 minimizes the functional Z A(0, q ) R1 (v − u0 ) + ∇v : v 7→ : ∇v + W0 (v) − f (0) · v dx, v ∈ (W01,2 ∩ Lq )(Ω; Rm ). 2 Ω

(A7) For the uniqueness result stated in Theorem 1.5 and the convergence result in Theorem 1.8, we also assume that there is a σ ∈ (0, 1) such that for every K > 0 there exists an M > 0, such that |D3 W0 (v)| ≤ max{|v|−σ , M }

for |v| ≤ K.

(1.11)

These assumptions are for instance satisfied by the nonconvex double-well potential W0 (v) = γ(|v|2 − 1)2

for 0 < γ < κ/(4CP (Ω))2 (and q = 4, µ = 4γ).

Further examples of potentials satisfying the above hypotheses are W0 (v) = |v|q

for q ≥ 2,

and

W0 (v) = (|v| + 1)q−2 |v|2

for q > 1.

Remark 1.2. We note that we are allowing nonconvexity in W0 , but, by (1.7) and (1.8), this nonconvexity cannot be too strong. It can be shown that our conditions imply that the combined energy R : ∇u dx + W0 (u) is convex. As a matter of fact, a certain amount of functional W(u) := Ω ∇u : A(t,·) 2 convexity of W0 is necessary, since for strongly nonconvex W0 counterexamples to regularity exist, see the example above or [MT04]. 1.4. Main results. In the course of this work we will prove the following result concerning the existence, uniqueness, and regularity of a strong solution to (1.1). Theorem 1.3. Under the assumptions (A1)–(A6), there exists a strong solution u ∈ L∞ (0, T ; (W01,2 ∩ L∞ )(Ω; Rm ))

with

u˙ ∈ L1 (0, T ; W01,2 (Ω; Rm )),

such that (1.2) holds, and u has the following additional regularity properties: (i) ∇2 u ∈ L∞ (0, T ; Lp (Ω; Rm×d×d )); (ii) ∇u˙ ∈ La (0, T ; L2 (Ω; Rd×m )); (iii) u ∈ C0,γ ([0, T ] × Ω; Rm ) for some γ ∈ (0, 1); (iv) If p > d, then ∇u ∈ C0,ζ ([0, T ] × Ω; Rd×m ) for some ζ ∈ (0, 1). Moreover, we have the quantitative estimates   k∇2 ukL∞ (Lp ) ≤ C 1 + kf kL∞ (Lp ) + kf kLq−1 ∞ (L2 ) ,   k∇uk ˙ La (L2 ) ≤ C 1 + kf kW1,a (L2 ) .

Here, the constant C > 0 depends on all constants in the stated assumptions, as well as on p, a, T, |Ω|. older seminorm of u, the oscillation estimates stated in (iii) and (iv) above Denoting by [u]C0,γ the γ-H¨ are quantified in the following manner:   [u]C0,γ ([0,T ]×Ω) ≤ C 1 + kukL∞ (W2,p ) + kuk ˙ La (W1,2 ) , where γ ∈ (0, 1), and, if p > d,   [∇u]C0,ζ ([0,T ]×Ω) ≤ C k∇ukL∞ (W1,p ) + k∇uk ˙ La (L2 ) ,

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

6

where ζ ∈ (0, 1). These estimates are explained in Remark 3.2 below. Remark 1.4. If W0 satisfies (A3) only in a neighborhood of the initial value u0 , then a similar existence and regularity result holds, but only with a finite time of existence (in Theorem 1.3 we can show existence and regularity for any T > 0 as long as f satisfies (A5) on (0, T )). In fact, the solution can be extended up to the time when u exits the region within which (A3) holds (the “boundary of convexity”). We will construct our solutions via a Rothe-type time-discretization scheme and iterative minimization of incremental functionals (see (3.1)), as in the theory of Mielke–Theil energetic solutions. In fact, at this time-discrete level the functionals used coincide with the ones in the Mielke–Theil theory. It is only for the corresponding time-continuous limits that differences in the solution concepts are seen to emerge. In connection with this, we refer the reader to the remarks concerning the approximability of rate-independent solutions in [MR09]. Our regularity results follow from “elliptic” estimates at the discrete level, see Lemma 3.1. We give a brief formal overview of the relevant estimates in Section 2, which will then be followed by their rigorous proofs in Section 3. Theorem 1.5. Under the assumptions (A1)–(A7), the (strong) solution to (1.1), whose existence is guaranteed by Theorem 1.3, is unique among all strong solutions. Moreover, let u be the solution from Theorem 1.3 and assume that v ∈ L1 (0, T ; (W01,2 ∩ Lq )(Ω; Rm )) where

  b = dp/(dp + p − d) b>1   b=1

with

v˙ ∈ L1 (0, T ; W01,b (Ω; Rm )),

if p ∈ [2, d), if p = d, if p > d,

satisfies v(0) = u0 and assume that Z Z R1 (v˙ − ϕ) − R1 (v) ˙ + f · ϕ dx, ∇v : A : ∇ϕ + DW0 (v) · ϕ dx ≤ Ω

(1.12)

Ω

m for all ϕ ∈ C∞ 0 (Ω; R ). Then, v = u.

Remark 1.6. The condition (1.12) is in particular satisfied if (1.3) holds for solutions v and all ξ ∈ L1 (0, T ; (W01,2 ∩ Lq )(Ω; Rm )), see Remark 1.1. In the above statement, however, we require even less regularity of v and v. ˙ Remark 1.7. Unlike strong solutions, which are unique within their natural regularity class, weak solutions do not have to be unique, as the example in Section 1.2 demonstrates. We will also show under an additional assumption that a continuous piecewise linear finite element approximation of the model under consideration converges, with a rate, to the unique strong solution. Our main result here is the following theorem. Theorem 1.8. Assume (A1)–(A6) with p = 2 and in the case of d = 3 additionally that q ≤ 4. Then the sequence of numerical solutions (uτh )h>0,τ >0 generated by the fully discrete approximation scheme constructed in Section 5, on a quasiuniform family of triangulations of Ω parametrized by the spatial discretization parameter h, and with time step τ , converges to the unique strong solution u of (1.1), which, for some β˜ ∈ (0, 1), satisfies, uniformly in h, τ , [uhτ ]Cβ˜([0,T ]×Ω) + k∇uhτ kL∞ (L6 (Ω)) + k∇u˙ hτ kLa (L2 (Ω)) ≤ C. In Section 6 we will also prove that the sequence of numerical approximations converges to the unique strong solution u with a rate; see Theorem 6.1. Acknowledgments. The authors would like to thank Alexander Mielke, Tom´ aˇs Roub´ıˇcek, Ulisse Stefanelli, and Florian Theil for helpful discussions related to this work. Filip Rindler gratefully acknowledges the support from the EPSRC Research Fellowship on “Singularities in Nonlinear PDEs” (EP/L018934/1); Sebastian Schwarzacher thanks the program PRVOUK P47; Sebastian Schwarzacher and Endre S¨ uli thank the ERC-CZ program MORE of the Charles University Prague.

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

7

2. Formal a-priori estimates We first illustrate in a non-rigorous way what can be gained from a-priori estimates. These arguments will be made precise in subsequent sections. For the sake of simplicity, we restrict ourselves here to the special case Lt = ∆. We shall assume therefore that we have a smooth u : [0, T ] × Ω → Rm satisfying the initial condition u(0) = u0 , the homogeneous Dirichlet boundary condition u|(0,T ]×∂Ω = 0, and Z Z R1 (ξ) dx (2.1) R1 (u(t)) ˙ − ∇u(t) · ∇(ξ − u(t)) ˙ + [−DW0 (u(t)) + f (t)] · (ξ − u(t)) ˙ dx ≤ Ω

Ω

for all t ∈ (0, T ] and all smooth ξ : Ω → Rm such that ξ|(0,T ]×∂Ω = 0. 2.1. Estimates in space. The weak formulation (2.1) implies that for almost every (t, x) ∈ (0, T ] × Ω there is a z(t, x) ∈ ∂R1 (u(t, ˙ x)) such that z(t) − ∆u(t) + DW0 (u(t)) = f (t). As R1 is globally Lipschitz continuous on Rm (cf. Assumption (A2)), there exists a positive constant C such that |z(t, x)| ≤ C, uniformly in (t, x) ∈ (0, T ] × Ω. Hence, −∆u(t) ∈ L∞ (Ω; Rm ) if f and DW0 are globally bounded. Moreover, as Ω is assumed to be bounded with a C1,1 boundary, elliptic regularity theory implies that ∇2 u(t) ∈ Ls (Ω; Rd×d×m ) for all s ∈ (1, ∞), and k∇2 u(t)kLs ≤ Ck∆u(t)kLs

for all s ∈ (1, ∞),

where the constant C depends only on Ω, s, d; see [LU68, (11.8)], or [Bro60, Bro61]. Thus, if for the moment DW0 is assumed bounded, we find that k∇2 u(t)kLs ≤ C(1 + kf (t)kLs )

for all s ∈ (1, ∞).

Morrey’s embedding theorem then yields ∇u ∈ L∞ (0, T ; C0,α (Ω; Rd×m ))

for all α ∈ [0, 1).

The above derivation and the extension to the case when DW0 is unbounded but satisfies (1.6), (1.7) is made precise in Lemma 3.1. m 2.2. Testing with u. Choosing ξ = u(t) ˙ − ϕ for any ϕ ∈ C∞ 0 (Ω; R ) implies that Z R1 (u(t)) ˙ − R1 (u(t) ˙ − ϕ) + ∇u(t) · ∇ϕ + [DW0 (u(t)) − f (t)] · ϕ dx ≤ 0. Ω

Since R is homogeneous of degree 1 and convex, it is subadditive, i.e. R1 (a + b) ≤ R1 (a) + R1 (b). Hence, Z Z R1 (−ϕ) dx. (2.2) ∇u(t) · ∇ϕ + [DW0 (u(t)) − f (t)] · ϕ dx ≤ Ω

Ω

Using ϕ = u(t), and noting DW0 (u(t)) · u(t) ≥ −µ|u(t)|2 + DW0 (0) · u(t), which follows from (1.7), we get that k∇u(t)k2L2 ≤ µku(t)k2L2 + kf (t)kL2 ku(t)kL2 + Cku(t)kL1 , which implies by Poincar´e’s inequality that k∇u(t)kL2 ≤ kf (t)kL2 + C(1 + ku(t)kL2 ). This holds for almost every t ∈ (0, T ]. When W0 is convex, the term (DW0 (u(t)) − DW0 (0)) · u(t) is nonnegative. In this case, or in the case when DW0 is globally bounded, we arrive at an upper bound on k∇u(t)kL2 that is independent of u(t).

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

8

2.3. Testing with u. ˙ An a-priori estimate in a stronger norm can be derived by choosing ξ = 0. Then,   Z Z   |∇u(t)|2 R1 (u(t)) ˙ + ∂t f (t) · u(t) ˙ dx. + ∂t W0 (u(t)) dx ≤ 2 Ω Ω

We integrate this over the time interval (0, τ ) ⊂ (0, T ) and, assuming for simplicity that u(0) = 0 and W0 (0) = 0, we find that Z Z τZ |∇u(τ )|2 + W0 (τ ) dx R1 (u) ˙ dx dt + 2 Ω 0 Ω Z τZ f · u˙ dx dt ≤ 0 Ω Z τZ Z ˙ f (τ ) · u(τ ) dx =− f · u dx dt + 0

Ω

Ω

≤ kf˙kL1 (L2 ) kukL∞ (L2 ) + kf kL∞ (L2 ) kukL∞ (L2 ) .

This implies, by taking the supremum over all τ ∈ (0, T ], assuming additionally that R1 (z) ≥ c|z|, and absorbing via the Poincar´e inequality, that Z   2 kuk ˙ L1 (L1 ) + k∇ukL∞ (L2 ) + sup W0 (u(τ )) ≤ C kf˙k2L1 (L2 ) + kf k2L∞ (L2 ) dx τ

Ω

≤ Ckf k2W1,1 (L2 ) .

2.4. Testing with u ¨. The following estimate is the crucial one. Let t ∈ (0, T ) and assume that τ ∈ (0, T − t]. We will use (2.1) at t and at t + τ . At time t we take ξ = u(t) ˙ + u(t ˙ + τ ) and at time t + τ we take ξ = 0. By adding the two inequalities we find Z R1 (u(t ˙ + τ )) + R1 (u(t)) ˙ + ∇(u(t + τ ) − u(t)) · ∇(u(t ˙ + τ )) Ω

+ [DW0 (u(t + τ )) − DW0 (u(t)) + f (t) − f (t + τ )] · u(t ˙ + τ ) dx Z R1 (u(t) ˙ + u(t ˙ + τ )) dx. ≤ Ω

By the subadditivity of R1 we get   Z ∇(u(t + τ ) − u(t)) DW0 (u(t + τ )) − DW0 (u(t)) · ∇(u(t ˙ + τ )) + · u(t ˙ + τ ) dx τ τ Ω Z f (t + τ ) − f (t) ≤ · u(t ˙ + τ ) dx. τ Ω Letting τ → 0, and estimating, we find Z Z 2 2 f˙(t) · u(t) ˙ dx D W0 (u(t)) [u(t), ˙ u(t)] ˙ dx ≤ k∇u(t)k ˙ L2 + Ω

Ω

for any fixed t ∈ (0, T ). We remark that this is exactly the point where the mild convexity assumption (1.8) is essential. Indeed, (1.7) implies (via difference quotients) in the case where W0 is twice differentiable that D2 W0 (u)[u, ˙ u] ˙ ≥ −µ|u| ˙ 2 . Hence, by H¨older’s inequality and Poincar´e’s inequality, 2 2 ˙ ˙ ˙ k∇u(t)k ˙ L2 + µku(t)k L2 L2 ≤ kf (t)kL2 ku(t)k 2 2 ˙ ˙ ≤ CP (Ω)kf˙(t)kL2 k∇u(t)k 2. L2 + µCP (Ω) k∇u(t)k L

Thus, invoking (1.8) with κ = 1 (recall that we took Lt = ∆ in this section, for simplicity), we have k∇u(t)k ˙ L2 ≤

CP (Ω) kf˙(t)kL2 . 1 − µCP (Ω)2

Another heuristic approach to the derivation of the above estimate is to use −(uη)˙ ˙ as test function, where η has compact support in (0, T ).

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

9

2.5. Optimality in the scalar case. In the scalar case, corresponding to m = 1, the following example shows that even if the right-hand side of (1.1) is smooth, not more than Lipschitz regularity of the solution can be expected. Example 2.1. The uniqueness result from Theorem 1.5 implies that, for t ∈ [0, 2], the function v defined by v(t) = max{t − 1, 0} is the unique strong solution to the rate-independent system ∂|v(t)| ˙ ∋ t − v(t)

for t ∈ (0, 2]

and

v(0) = 0.

Now let −∆ϕ + ϕ = 1

in Ω

and

ϕ=0

on ∂Ω.

By decomposing ϕ as ϕ = ϕ+ + ϕ− , where ϕ+ := max{ϕ, 0}, ϕ− := min{ϕ, 0}, noting that ϕ+ ≥ 0 and R ϕ− ≤ 0, and testing the equation with ϕ− , we find that k∇ϕ− k2L2 + kϕ− k2L2 = Ω ϕ− dx ≤ 0, whereby ϕ− = 0; hence ϕ = ϕ+ ≥ 0. Thus, letting u(t, x) := v(t)ϕ(x), we have that t − u(t) + ∆u(t) = t − v(t) ∈ ∂|v(t)| ˙ ⊆ ∂|u(t)| ˙

for t ∈ (0, 2]

and

u(0) = 0.

Hence u is a strong solution (in the sense of (1.3)) to ∂|u| ˙ ∋ t − u + ∆u in (0, 2] × Ω Moreover, clearly u˙ ∈

L∞ ([0, 2]; C∞ (Ω))

and

u(0) = 0.

\ C0 ([0, 2]; C∞ (Ω)).

3. Existence and regularity of solutions In this section we will prove Theorem 1.3. We will do so by the Rothe method and discrete analogues of the estimates from Sections 2.1 and 2.4. It will transpire that our a-priori information on solutions is quite strong and thus we obtain compactness in a variety of spaces. The difficulty is to identify the limiting equation, and for this we will need H¨older continuity of the solution. 3.1. Time discretization. We consider a sequence of partitions N N 0 = tN 0 < t1 < · · · < tN = T,

where

N tN k − tk−1 = τ :=

T , N

N ∈ N,

and seek a sequence of approximations 1,2 q m (uN k )k=0,...,N ⊂ (W0 ∩ L )(Ω; R ),

which solve a suitable temporally discretized version of (1.1). As a discretization of the source term f we set fkN := f (tN k )

for k = 0, . . . , N ,

and

f N :=

N X k=1

Observe that our assumption f ∈

W1,a (0, T ; Lp (Ω; Rm ))

1(tNk−1 ,tNk ] fkN .

for some a ∈ (1, ∞) implies that

f N ∈ L∞ (0, T ; Lp (Ω; Rm )). Further, we define the following approximations of the elliptic operator Lt : X X α,β α β N β q Ai,j (tN ∂j [LN k , ) ∂i v . k v] = div([Ak v] ) := j

i,α

Next, we set uN −1 := u0 and, successively, for each k = 0, 1, . . . , N , minimize the functional Z AN N k R1 (v − uN Fk (v) := : ∇v + W0 (v) − fkN · v dx k−1 ) + ∇v : 2 Ω

(3.1)

over all v ∈ (W01,2 ∩ Lq )(Ω; Rm ). Here, we have used the notation X α,β α β ∇v : AN : ∇w := Ai,j (tN k k , x) ∂i v ∂j w . α,β,i,j

First of all, thanks to (A6), we may set uN 0 = u0 . Further, since R1 is convex and lower semicontinuous and W0 is of lower order, we may deduce by the usual Direct Method that a minimizer exists, which we 1,2 q m N call uN k . More precisely, we take a minimizing sequence (vj )j≥1 ⊂ (W0 ∩ L )(Ω; R ) with Fk (vj ) →

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

10

min FkN . Then, by the coercivity of W0 (see (1.5)), the strong ellipticity of Lt (see (1.10)) and R ≥ 0, we get the estimate k∇vj k2L2 + kvj kqLq ≤ C(1 + kfkN kL2 kvj kL2 ) for a j-independent constant C > 0. Thus, using the Poincar´e and Young inequalities, k∇vj kL2 ≤ C(1 + kfkN kL2 )

kvj kqLq ≤ C(1 + kfkN k2L2 ).

and

That is, we have shown coercivity in (W01,2 ∩ Lq )(Ω; Rm ). Hence, we may assume after selecting a nonrelabeled subsequence that vj ⇀ v in (W1,2 ∩ Lq )(Ω; Rm ). Furthermore, by the compact embedding c W01,2 (Ω; Rm ) ֒→ L2 (Ω; Rm ) and selecting another (not relabeled) subsequence, vj → v pointwise almost everywhere. Now, for the convex terms in FkN we get lower semicontinuity immediately, and for W0 (≥ 0) we deduce by Fatou’s lemma that Z Z W0 (v(x)) dx. lim inf W0 (vj (x)) dx ≥ j→∞

Ω

Ω

Hence, the Direct Method applies and yields the existence of a minimizer, which we call uN k , and which satisfies the bounds N k∇uN k kL2 ≤ C(1 + kfk kL2 )

q N 2 kuN k kLq ≤ C(1 + kfk kL2 ).

and

(3.2)

The minimizer uN k satisfies the Euler–Lagrange equation N N N N N 0 ∈ ∂R1 (uN k − uk−1 ) − Lk uk + DW0 (uk ) − fk

in a weak sense. That is, for any test function ξ ∈ W01,2 (Ω; Rm ) we have that Z N N N N N R1 (uN k − uk−1 ) − ∇uk : Ak : ∇(ξ − (uk − uk−1 )) Ω Z   N N N N + −DW0 (uk ) + fk · (ξ − (uk − uk−1 )) dx ≤ R1 (ξ) dx.

(3.3)

Ω

To see this, we observe that for ξ ∈ (W01,2 ∩ Lq )(Ω; Rm ) we have
N N N N FkN uN k + ε(ξ + uk−1 − uk ) − Fk (uk ) 0≤ , ε > 0. (3.4) ε First, since R1 is homogeneous of degree 1 and convex, it is subadditive, i.e. R1 (a + b) ≤ R1 (a) + R1 (b), and so

N N N N N R1 u N k + ε(ξ + uk−1 − uk ) − uk−1 − R1 uk − uk−1

N N N = R1 εξ + (1 − ε)(uN k − uk−1 ) − R1 uk − uk−1 N ≤ εR1 (ξ) − εR1 (uN k − uk−1 ).

N For the regularizer, using the symmetry of the coefficients in Lt , see (1.9), and setting η := ξ+uN k−1 −uk , we have that Z N AN 1 N N Ak k [∇uN + ε∇η] : : [∇u + ε∇η] − ∇u : : ∇uN k k k k dx ε Ω 2 2 Z N as ε ↓ 0 ∇uN → k : Ak : ∇η dx Ω

by the

L2 -bounds

on all of the quantities involved. Finally, note (using the notation Z Z ε Z 1 N N − DW0 (uN W0 (uk + εη) − W0 (uk ) dx = k + τ η) · η dτ dx ε Ω Ω 0 Z DW0 (uN → k ) · η dτ dx

Rε − 0

:=

1 ε

Rε

0)

that

Ω

by the continuity of DW0 and the estimate (1.5) on the growth of DW0 . Thus, letting ε ↓ 0 in (3.4), we arrive at (3.3) for ξ ∈ (W01,2 ∩ Lq )(Ω; Rm ). A density argument allows us to conclude (3.3) also for ξ ∈ W01,2 (Ω; Rm ).

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS N Actually, since uN 0 = u−1 = u0 , we find that Z Z   R1 (ξ) dx. −∇u0 : A0 : ∇ξ + −DW0 (u0 ) + f (0) · ξ dx ≤

11

(3.5)

Ω

Ω

3.2. A-priori estimates in space. For k = 0, . . . , N , we define the temporal difference quotient N uN k − uk−1 N tN , τ := tN −1 := −τ, k − tk−1 , τ N with the convection that uN 0 = u−1 = u0 , as before. Then, upon dividing (3.3) by τ and replacing ξ/τ by ξ, we get Z   N N N N N R1 (δkN ) − ∇uN k : Ak : ∇(ξ − δk ) + −DW0 (uk ) + fk · (ξ − δk ) dx Ω Z R1 (ξ) dx for all ξ ∈ W01,2 (Ω; Rm ). (3.6) ≤

δkN :=

Ω

Now, we may further replace ξ − δkN by ξb and use the subadditivity of R1 to get Z Z   N N N N b dx b b R1 (ξ) for all ξb ∈ W01,2 (Ω; Rm ). −∇uk : Ak : ∇ξ + −DW0 (uk ) + fk · ξ dx ≤ Ω

R

Ω

b dx ≤ Ckξk b L1 . Consequently, By the 1-homogeneity of R1 , we find that Ω R1 (ξ) Z   N N N b b sup −∇uN k : Ak : ∇ξ + −DW0 (uk ) + fk · ξ dx ≤ C. b ∞ (Ω;Rm ) : kξk b 1 } Ω {ξ∈C 0 L ≤1

Hence, there is a wkN ∈ L∞ (Ω; Rm ), with kwkN kL∞ ≤ C (uniformly in k, N ), such that N N N N −LN k uk + DW0 (uk ) = fk − wk

with the differential operator LN k interpreted in a weak sense. By Lemma 3.1 below we thus get (recall that p ∈ [2, ∞) from the assumption on the external force f stated in (A5)) N q−1 N k∇2 uN k kLp ≤ C(1 + kfk kLp + kfk kL2 ).

(3.7)

The next lemma is an elliptic regularity result that is specifically tailored to our situation. We wish to point out that it allows for rather general, physically-motivated, assumptions on the elastic energy functional W0 , namely (1.5)–(1.7), but no other structural hypotheses, such as symmetry, are needed. Lemma 3.1. With our usual hypotheses from Section 1.3, but excluding the mild convexity assumption (1.8), let u ∈ (W01,2 ∩ Lq )(Ω; Rm ) be any weak solution to ( −Lt u + DW0 (u) = g in Ω, (3.8) u|∂Ω = 0. If g ∈ Ls (Ω; Rm ) for s ∈ [2, ∞), then k∇2 ukLs ≤ C(1 + kgkLs + kgkLq−1 2 ).

(3.9)

Note that the inhomogeneity with exponent q − 1 is due to the (q − 1)-growth of DW0 via (1.6). Proof. The existence of a solution u ∈ (W01,2 ∩ Lq )(Ω; Rm ) is guaranteed by the same variational argument as the one above. Analogously to (3.2) we find that k∇ukL2 ≤ C(1 + kgkL2 ).

(3.10)

Next, we recall that by regularity theory for elliptic systems, if −Lt u = g − DW0 (u) ∈ Ls (Ω; Rm ) then k∇2 ukLs ≤ CkLt ukLs

(3.11)

since the coefficients are assumed to be Lipschitz continuous and the boundary of Ω is of regularity class C1,1 . For these results see [ADN59, ADN64] and also [GM12, Theorem 7.3] (the standard scalar case is better known and treated for instance in [Eva10]).

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

12

Therefore, we are left to establish a bound on kDW0 (u)kLs . In the case d = 2 we find by Sobolev embedding that kukLs ≤ CkukW1,2 for all s ∈ [1, ∞). Therefore, by (1.6), (3.10) and the Poincar´e inequality, q−1 kDW0 (u)kLs ≤ C(1 + kukLq−1 s(q−1) ) ≤ C(1 + kgkL2 ).

(3.12)

This implies, via (3.11), the desired estimate. In the following we will obtain the same estimate for d = 3, which we therefore assume from now on until the end of the lemma. We will achieve this goal in several steps. Our first estimate concerns local W2,2 -regularity: for any ball B5R = B5R (x0 ) ⊂ Ω (x0 ∈ Ω, R > 0) we will show that Z  Z |∇u|2 2 2 2 |∇ u| dx ≤ C + |g| dx . (3.13) 2 B5R R BR Since the structure of the system (3.8) is invariant under translation and scaling of coordinates, we may assume that R = 1 and that the ball is centered at the origin, x0 = 0. Indeed, if u solves (3.8) in the scaled and translated ball B(z, r), then u ˜(y) = u(z + Ry) solves (3.8) in the unit ball with ˜ 0 := R2 W0 . g˜(y) = R2 g(z + Ry) and W Thus, in the following, with no loss of generality, we will consider u to be a solution in the ball B5 = B5 (0). We take a cut-off function η ∈ C10 (B2 ), where B2 = B2 (0), such that η ≡ 1 on B1 = B1 (0). We define the difference quotient of v : Ω → Rm in the direction of the kth unit coordinate vector ek , k ∈ {1, 2, 3} and h ∈ R by Z h v(x + hek ) − v(x) h (3.14) = − ∂k v(x + sek ) ds. Dk v(x) := h 0 1,2 m 2 h Now, for k ∈ {1, 2, 3} and h ∈ (0, 1/2), we take −D−h k (η Dk (u)) ∈ W0 (B3 , R ) as a test function in (3.8), with B3 = B3 (0), and employ the summation-by-parts rule for difference quotients, to get Z Z X α,β h α 2 h β Dhk (DW0 (u)) · Dhk u η 2 dx Dk (Ai,j ∂i u ) ∂j (η Dk u ) dx + (I) + (II) := Ω

Ω i,j,α,β

=−

Z

Ω

2 h g · D−h k (η Dk u) dx =: (III).

We begin by deriving a lower bound on (I). By the product rule for difference quotients (Dhk (vw)(x) = Dhk v(x)w(x + hek ) + v(x)Dhk w(x)) and (3.14) we find that Z X α q 2 h β Dhk (Aα,β (I) ≥ i,j ) ∂i u ( + hek ) ∂j (η Dk u ) dx Ω i,j,α,β

+2 +

Z

Z

X

Ω i,j,α,β

X

Ω i,j,α,β

h α h β Aα,β i,j Dk (∂i u ) η (∂j η) Dk u dx

β α 2 h h Aα,β i,j Dk (∂i u ) η Dk (∂j u ) dx

Z

|∇u( q + hek )||Dhk u|k∇ηkL∞ η + |∇u( q + hek )||Dhk ∇u|η 2 dx Z ZΩ h h |Dhk ∇u|2 η 2 dx. |Dk ∇u||Dk u|η dx + κ − k∇ηkL∞ kAk∞

≥ −CkAkC0,1 ([0,T ]×Ω)

Ω

Ω

Young’s inequality then implies, recalling that supp η ⊂ B2 , Z Z κ h 2 2 2 2 (I) ≥ |D ∇u| η dx − Cκ (1 + k∇ηkL∞ )kAkC0,1 ([0,T ]×Ω) |∇u|2 + |Dhk u|2 dx. 2 B2 k B3 Next, we estimate (II) by (1.7) to find that Z |Dhk u|2 η 2 dx. (II) ≥ −µ B2

(3.15)

(3.16)

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

13

Finally, choosing 0 < ε < κ/2, we deduce by Young’s inequality and (3.14) that Z 2 −h 2 (|g|2 η 2 + |D−h (III) ≤ Cε k η| |Dk u| ) dx B2

Z −h Z + ε− 0

B2

|Dhk ∂k u(x + sek )|2 η 2 dx ds.

(3.17)

Observe that, as h ∈ (0, 12 ), we have that Z Z −h 2 −h 2 −h 2 |Dk η| |Dk u| dx ≤ kDk ηkL∞ B2

≤ k∂k ηk2L∞ ≤ k∇ηk2L∞ and

Z −h Z − 0

B2

Z −h 2 − ∂k u(x + sek ) ds dx B2 0 Z sup |∂k u(x − sek )|2 dx

s∈(0,h) B2

Z

|∇u|2 dx B3

|Dhk ∂k u(x + sek )|2 η 2 dx ds ≤ sup

Z

s∈(0,h) B2

|Dhk ∂k u(x − sek )|2 dx ≤

Z

B3

|Dhk ∂k u|2 dx.

Combining (3.15), (3.16) and (3.17) with the above, we arrive at Z Z Z Z κ 2 2 h 2 |Dhk ∇u|2 dx. |g| dx + ε |∇u| dx + Cε |D ∇u| dx ≤ Cκ,ε 2 B1 k B3 B2 B3

To conclude the estimate, we need to absorb the ε-term into the left-hand side. To this end, we note that the previous inequality implies, for s ∈ (0, h) and τks v(x) := v(x + sek ) − v(x), that Z Z Z Z κ 2 2 s 2 2 2 |τ ∇u| dx ≤ Cκ,ε h |∇u| dx + Cε h |τks ∇u|2 dx. |g| dx + ε 2 B1 k B3 B3 B2 If we transform this back to the translated and scaled ball, we arrive at (now for the original u) Z Z Z Z h2 κ 2 s 2 2 2 − |τ ∇u| dx ≤ Cκ,ε h − |∇u| dx + Cε 2 − |g| dx + ε − |τks ∇u|2 dx 2 BR k R B2R B3R B3R

for every R > 0. Hence we can apply the interpolation result of Giaquinta–Modica type stated in [DE08, Lemma 13] (with γ∇u (R, h) = h2 and γg (R, h) = h2 R−2 , hence in the result we need to fix h0 as a constant multiple of R, e.g. R/10). Thus, with a different constant Cκ,ε > 0, we obtain  Z Z 2 h 2 2 |Dk ∇u| dx ≤ Cκ,ε |∇u| + |g| dx . B1

B5

By letting h → 0 we then deduce the desired local estimate (3.13). Moreover, Sobolev embedding then implies that   1 2 k∇ukL2 (B5R ) + kgkL2 (B5R ) + kukL1 (BR ) (3.18) kukLs (BR ) ≤ Ck∇ ukL2 (BR ) + kukL1 (BR ) ≤ C R for all B5R ⊂ Ω and s ∈ (1, ∞) (recall that we have assumed that d = 3). The aim of the next step is to obtain estimates near the boundary of Ω. More precisely, we will show tangential differentiability up to the boundary, for which we will use a flattening argument. By the same scaling argument as before we assume the center point 0 to be a boundary point, and that we have a one-to-one diffeomorphism Ψ : B5 → R3 , such that Ψ(B5 ∩ [R2 × {0}]) = ∂Ω ∩ B5

and

Ψ(B5+ ) = Ω ∩ B5 ,

+ := BR ∩ [R2 × (0, ∞)]. By a straightforward transformation, we find that on the half-ball where BR B5+ we have that u ˜ = u ◦ Ψ : B5+ → Rm is a weak solution of

where

Let (˜ u) + DW0 (˜ u) = g ◦ Ψ,

[Let ]β =

X j

∂j

X

α,i,k,l

α Jjk Aα,β kl Jli ∂i u ,

Jij := ∂i Ψj .

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

14

We can assume that det ∇Ψ > κ1 for some κ1 > 0 depending on the prescribed boundary alone. Thus, (1.10) holds with the modulus of ellipticity κκ21 . By (3.13) we get for the half-balls that  Z Z Z 2 2 2 |g ◦ Ψ| dx (3.19) |∇˜ u| dx + |∂k ∇˜ u| dx ≤ C B5+

B1+

B5+

for k ∈ {1, 2}. Recalling once again that d = 3, we will now show that u ∈ Ls (Ω; Rm ) for all s ∈ (1, ∞) and that kukLs ≤ C(1 + kgkL2 ),

(3.20)

where C depends on s, Ω and the constants of our assumptions. Rz For z ∈ (0, 1/2) we define the function U z (x, y) := 0 ∂z u ˜(x, y, s) ds. We first observe that U z ∈ W1,2 (B1/2 ∩ [R2 × {0}]). Indeed, Z z 2 2  Z 1/2 Z Z ∂x ∂z u ˜(x, y, s) ds dx dy ≤ |∂x ∂z u ˜| dz dx dy. B1/2 ∩[R2 ×{0}]

B1/2 ∩[R2 ×{0}]

0

0

The last estimate holds for ∂y as well and is in both cases controlled by (3.19). Sobolev embedding further implies kU z kLs (B1/2 ∩[R2 ×{0}]) ≤ CkU z kW1,2 (B1/2 ∩[R2 ×{0}]) for any s ∈ (1, ∞). Rz From u ˜(x, y, z) = 0 ∂z u ˜(x, y, s) ds we therefore deduce that k˜ ukLs (B +

1/2

)

≤ Ck˜ ukW1,2 (B + ) + kg ◦ ΨkL2 (B + ) . 5

5

To finish, we cover Ω with finitely many balls. For x ∈ Ω, there exists either BR (x), such that + (0). Since Ω is compact we can choose a B5R (x) ⊂ Ω or such that B5R (x) ∩ Ω is diffeomorphic to B5R finite subfamily of balls for which either (3.13) or (3.18) holds. This enables us to complete the proof for d = 3 as in case d = 2 and we get (3.9) via (3.11) and (3.12) and a (countable) covering of Ω with balls B(y, R) such that every x ∈ Ω is in at most N ∈ N (with N independent of x) balls B(y, 5R). In fact, this holds with N = 11d . This standard covering lemma can be proved by setting Ω0 := Ω, choosing xk+1 ∈ Ωk and setting Ωk+1 := Ωk \ B(xk , R). Then, |xi − xj | ≥ R if i 6= j. If x ∈ Ω lies in B(xk , 5R) for all k ∈ J ⊂ N, then the reduced balls B(xk , R/2) are disjoint and all lie in B(x, 5R + R/2). Thus, J can contain at most 11d elements.  3.3. Estimates for the time derivatives. For k = 1, . . . , N we test the kth inequality of (3.3) with N ξ = 0 and the (k − 1)st inequality with ξ := uN k − uk−2 . Then we add the two resulting inequalities and divide by τ to find Z N N N N N N ) + ∇uN R1 (δkN ) + R1 (δk−1 k : Ak : ∇δk − ∇uk−1 : Ak−1 : ∇δk Ω  N  N N N N N N + DW0 (uN k ) − DW0 (uk−1 ) · δk − (fk − fk−1 ) · δk − R1 (δk + δk−1 ) dx ≤ 0. This can be transformed into Z  N  N N N N N N ) + (∇uN R1 (δkN ) + R1 (δk−1 k − ∇uk−1 ) : Ak : ∇δk + DW0 (uk ) − DW0 (uk−1 ) · δk Ω

N N N N N ) + ∇uN ) · δkN − R1 (δkN + δk−1 − (fkN − fk−1 k−1 : (Ak − Ak−1 ) : ∇δk dx ≤ 0.

We divide by τ > 0 and use the subadditivity of R1 (to cancel the R1 -terms) and the ellipticity (1.10) to get Z Z Z N | N |fkN − fk−1 |AN k − Ak−1 | N κ |∇δkN |2 dx ≤ |δkN | dx + |∇uN k−1 | |δk | dx τ τ Ω Ω Ω Z  1 N N N − ) · (uN ) − DW (u DW (u 0 0 k − uk−1 ) dx. k−1 k 2 τ Ω The first term on the right-hand side can be estimated as follows:   Z tk   Z tk Z N | |fkN − fk−1 N N ˙ ˙ |δk | dx ≤ − kf kL2 dt kδk kL2 ≤ C − kf kL2 dt k∇δkN kL2 . τ tk−1 tk−1 Ω

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

15

For the second term we use (1.9), the Poincar´e inequality and (3.2) to find Z N |AN k − Ak−1 | N N N |∇uN k−1 | |δk | dx ≤ CkAkC0,1 ([0,T ]×Ω) k∇uk−1 kL2 k∇δk kL2 τ Ω N ≤ C(1 + kfk−1 kL2 )k∇δkN kL2 .

For the third term we use (1.7) and the Poincar´e inequality to get Z Z   1 N N N N − 2 DW0 (uk ) − DW0 (uk−1 ) · (uk − uk−1 ) dx ≤ µ |δkN |2 dx τ Ω Ω Z 2 ≤ µCP (Ω) |∇δkN |2 dx, Ω

where we recall that CP (Ω) > 0 denotes the Poincar´e constant of Ω. Hence, by combining these inequalities, we get   Z tk C N N k∇δk kL2 ≤ (3.21) 1 + − kf˙kL2 dt + kfk kL2 . κ − µCP (Ω)2 tk−1 By (1.8), the constant on the right is greater than zero. 3.4. H¨ older continuity of the gradient. In this section only we additionally assume that p > d (see the statement of Theorem 1.3). For Borel subsets E ⊂ Rd with positive and finite Lebesgue measure we will use the notation Z Z 1 f dx. hf iE := − f dx = |E| E E We also define t − tk−1 N tk − t N for t ∈ (tk−1 , tk ], k = 0, . . . , N , and t−1 := −τ . uN (t) := uk + uk−1 τ τ N N By taking k = 0 and recalling that uN 0 = u−1 := u0 , we have that u (0) = u0 for all N ∈ N. For p ∈ [2, ∞), a ∈ (1, ∞], and f ∈ W1,a (0, T ; Lp (Ω, Rm )) we find by embedding that f ∈ C0,(a−1)/a ([0, T ]; Lp (Ω, Rm )). Therefore, (3.7) and (3.21) imply that k∇2 uN kL∞ (Lp ) + k∇u˙ N kLa (L2 ) ≤ C,

(3.22)

uniformly in N . Thus, uN ∈ L∞ (0, T ; W2,p (Ω; Rm )),

u˙ N ∈ La (0, T ; W1,2 (Ω; Rm )),

(3.23)

and their respective norms are uniformly bounded with respect to N . Now, W2,p (Ω; Rm ) ֒→ C1,α (Ω; Rm ) for some α ∈ (0, 1) if p ∈ (d, ∞); hence uN ∈ L∞ (0, T ; C1,α (Ω; Rm )) and, in fact, the embedding L∞ (0, T ; W2,p (Ω; Rm )) ∩ W1,a (0, T ; W1,2 (Ω; Rm )) ֒→ Lr (0, T ; C1,α (Ω; Rm )) is compact for any r ∈ [1, ∞) and any smaller α ∈ (0, 1). We will show that if a ∈ (1, ∞) (where the assumption that a < ∞ was made with no loss of generality and merely for the sake of simplicity of the exposition) and p ∈ (d, ∞), then ∇uN is uniformly Lipschitz continuous with respect to the metric
ρ (t, x), (s, y) := |t − s|ζ + |x − y|α , for any α ∈ (0, 1) such that uN ∈ L∞ (C1,α ), and where d  a α α(a − 1)
= , b= +α . ζ= d b 2 a−1 2 +α a

By Campanato’s integral characterization of H¨older continuity [Cam63] (see also Sec. III.1 in [Gia83] or Sec. 2.3 in [Giu03]), we need to show that Z t+rbZ |∇uN − h∇uN i(t,t+rb )×Br (x) | dx dt ≤ Cr α − − t

Br (x)∩Ω

(t, t + r b ) × B

d for all r (x) ⊂ [0, T ] × R , r > 0. We remark first that this “parabolic” version follows from the usual one via the transformation g(s, x) = bsb−1 f (s, x). Moreover, from the boundary regularity we

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

16

infer that Ω has no internal or external cusps, hence the boundary version of Campanato’s Theorem is applicable (see Section 2.3 in [Giu03]). In the following we only prove the statement for internal points, but will comment on the differences for boundary points afterwards. For ease of exposition we assume that (t, x) = (0, 0) and estimate Z rbZ − − |∇uN − h∇uN i(0,rb )×Br | dx dt Br 0 Z ≤ sup − |∇uN (t, x) − h∇uN (t)iBr | dx 0≤t≤r b

Br

Z rb + − |h∇uN i(0,rb )×Br − h∇uN (t)iBr | dt 0

Z rb Z ≤ Cr α + C − r b − |∇u˙ N | dx dt . Br |0 {z } =:(I)

Here we have used the a-priori L∞ (C1,α ) regularity result on the first integral and the Poincar´e inequality in the time direction on the second integral. To bound (I), we use the La (L2 ) bound on ∇u˙ N and H¨older’s inequality to get with a′ = a/(a − 1) that a/2 1/a  Z rb Z 1/2 Z rb Z b/a′ N 2 N 2 dt dt ≤ r (I) ≤ − |∇u˙ | dx − |∇u˙ | dx 0

= Cr

0

Br

b/a′ −d/2

Br

k∇u˙ N kLa (L2 ) ≤ Cr α.

For boundary points x ∈ ∂Ω, in all of the above estimates Br needs to be replaced by Br (x) ∩ Ω. In the very last estimate we additionally need to use |Br (x) ∩ Ω| ≥ α|Br (x)| for some fixed α ∈ (0, 1) (depending on the bounds on the boundary regularity). Thus we may apply Campanato’s Theorem to conclude H¨older regularity. Finally, Lipschitz continuity in the metric ρ implies H¨older continuity, with exponent ζ, jointly in space and time. To see this, we note that, since ζ < α,  |t − s|ζ + |x − y|α ≤ 2 max (|t − s| + |x − y|)ζ , (|t − s| + |x − y|)α ≤ 2(1 + (T + diam(Ω))α−ζ )(|t − s| + |x − y|)ζ ,

where we have to consider the cases |t − s| + |x − y| ≤ 1 and |t − s| + |x − y| > 1 separately. 3.5. H¨ older continuity of the solution. By a similar argument to the one in the last section, we will show that uN is uniformly H¨older continuous. We only need to consider the case p ∈ [2, d], since otherwise the uniform H¨older continuity of uN follows from (3.23). As (see (3.23)) ∇2 uN ∈ L∞ (0, T ; Lp (Ω; Rm×d×d )), we have that ∇uN ∈ L∞ (0, T ; W01,s (Ω; Rd×m )) for pd pd ] when p < d, and for all s ∈ (1, ∞) when d = p. Since d−p ≥ 6, there exists an α ∈ (0, 1) all s ∈ [1, d−p N ∞ 0,α m N a such that u ∈ L (0, T ; C (Ω; R )). Furthermore, as ∇u˙ ∈ L (0, T ; L2 (Ω; Rd×m )), we deduce that u˙ N ∈ La (0, T ; Ls (Ω; Rm )), for s ∈ [1, 6] if d = 3 and s ∈ (1, ∞) for d = 2. By (3.23), m 1,a s m uN ∈ L∞ (0, T ; C0,α 0 (Ω, R )) ∩ W (0, T ; L (Ω, R ))

and the norms can be correspondingly estimated by a constant independent of N . We can therefore argue exactly as before. Indeed, we will show that uN is Lipschitz continuous with respect to the metric
ρ˜ (t, x), (s, y) := |t − s|γ + |x − y|α ,

for any α ∈ (0, 1) such that uN ∈ L∞ (0, T ; C0,α (Ω; Rm )) (our α here is different from the one in the previous section), and where d  a α(a − 1) α
= , γ= d b= +α . b 2 a − 1 + α a 2

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

17

a . By Campanato’s integral characterization of H¨older continuity, Again we use the variables a′ = a−1 we need to show that Z t+rbZ |uN − huN i(t,t+rb )×Br (x) | dx dt ≤ Cr α − − Br (x)

t

for all (t, t + r b ) × Br (x) ⊂ [0, T ] × Rd , r > 0. We assume that (t, x) = (0, 0) and estimate as before Z rbZ − − |uN − huN i(0,rb )×Br | dx dt 0

Br

Z rb Z ≤ Cr α + C − r b − |u˙ N | dx dt 0

α

≤ Cr + Cr

b a′

α

≤ Cr .

Z

Br

rb

0

a Z 2 N 2 dt − |u˙ | dx Br

Hence, we find that uN is uniformly γ-H¨older continuous. Remark 3.2. The estimates in the last two sub-sections are of general nature. Indeed, what is shown here is that, for p0 > d,   ˙ La (L1 ) , [u]C0,γ ([0,T ]×Ω) ≤ C kukL∞ (W1,p0 ) + kuk for some γ ∈ (0, 1).

3.6. Proof of Theorem 1.3. The a-priori estimate (3.22) implies that there exists a (non-relabeled) subsequence such that ∗

uN ⇀ u

in L∞ (0, T ; W2,p (Ω; Rm ))

and

uN ⇀ u

in W1,a (0, T ; W1,2 (Ω; Rm )).

By weak compactness in reflexive Banach spaces we furthermore have uN ⇀ u

in Lr (0, T ; W2,p (Ω; Rm ))

for any r ∈ (1, ∞).

We rewrite (3.6) in a time-continuous form. Observe that on (tk−1 , tk ] we have u˙ N = δkN and t − tk−1 tk − t tk − t N N ∇uN ∇uN ∇(uN k + k−1 = ∇uk + k−1 − uk ). τ τ τ

∇uN (t) = We also set kN (t) :=



k ∈ {1, . . . , N } : t ∈ (tk−1 , tk ]



for t ∈ [0, T ].

Therefore, (3.6) here reads as Z TZ N R1 (u˙ N (t)) − ∇uN ˙ N (t)) dx dt kN (t) : AkN (t) : ∇(ξ(t) − u 0

Ω

+

Z TZ 0

Ω



−DW0 (uN kN (t) )

+

fkNN (t)



N

· (ξ(t) − u˙ (t)) dx dt ≤

Z TZ 0

R1 (ξ) dx dt

(3.24)

Ω

for all ξ ∈ L1 (0, T ; W01,2 (Ω; Rm )) (first use only ξ that are piecewise constant as a function of t with N N respect to the subdivision {tN 0 , t1 , . . . , tN }, and then argue by density). Using the H¨older continuity of u, we find by the Arzel`a–Ascoli theorem a subsequence such that uN → u

in C0,γ ([0, T ] × Ω; Rm )

for 0 < γ
0 , with Xh ⊂ C00,1 (Ω; Rm ), parametrized by the discretization parameter h ∈ (0, 1]; Xh will be supposed to be a finite element space generated by m-component continuous piecewise polynomial vector functions on a quasiuniform partition T h of Ω into closed d-dimensional simplices K (with, possibly, curved faces for simplices K that have nonempty intersection with ∂Ω), with h = maxK∈T h hK , where hK := diam K, S and such that 0 0. Thus, using the Poincar´e and Young inequalities, k∇vjh kL2 ≤ C(1 + kfk kL2 )

and

kvjh kqLq ≤ C(1 + kfk k2L2 ).

Hence, as Xh has finite dimension, we deduce after selecting a subsequence of the sequence (vjh )j≥1 (not indexed) that vjh → v h in Xh , as well as pointwise, and v h =: uhk is a minimizer that satisfies kuhk kqLq + k∇uhk k2L2 ≤ C(1 + kf (tk )k2L2 ).

in

(5.3)

Next we show, that uhk satisfies the following Euler–Lagrange equation 0 ∈ ∂R1 (uhk − uhk−1 ) − Lhk uhk + DW0 (uhk ) − fk , a suitable sense. That is, for any test function ξ h ∈ Xh we have that Z

R1 (uhk − uhk−1 ) − ∇uhk : Ak : ∇(ξ h − (uhk − uhk−1 )) Ω Z   h h h h R1 (ξ h ) dx. + −DW0 (uk ) + fk · (ξ − (uk − uk−1 )) dx ≤

(5.4)

Ω

To see this, we observe that first for ξ h ∈ Xh we have
Fkh uhk + ε(ξ h + uhk−1 − uhk ) − Fkh (uhk ) 0≤ , ε > 0. (5.5) ε Next, since R1 is homogeneous of degree 1 and convex, it is subadditive, i.e. R1 (a + b) ≤ R1 (a) + R1 (b), and so

 1 R uhk + ε(ξ h + uhk−1 − uhk ) − uhk−1 − R uhk − uhk−1 ε

 1 = R εξ h + (1 − ε)(uhk − uhk−1 ) − R uhk − uhk−1 ε ≤ R1 (ξ h ) − R1 (uhk − uhk−1 ). For the regularizer we may compute using the symmetry of the coefficients in Lt , see (1.9), and setting η h := ξ h + uhk−1 − uhk , Z Ak Ak 1 [∇uhk + ε∇η h ] : : [∇uhk + ε∇η h ] − ∇uhk : : ∇uhk dx ε Ω 2 2 Z ∇uhk : Ak : ∇η h dx as ε ↓ 0. → Ω

Finally, we note that Z 1 W0 (uhk + εη h ) − fk · (uhk + εη h ) − W0 (uhk ) + fk · uhk dx ε Ω Z Z ε − DW0 (uhk + sη h ) · η ds dx + hfk , η h i = ZΩ 0 DW0 (uhk ) · η h dτ dx + hfk , η h i → Ω

by the continuity of DW0 . Thus, letting ε ↓ 0 in (5.5), we arrive at (5.4) for all ξ h ∈ Xh . 5.2. Discrete a-priori estimates in space. For k = 0, . . . , N set uhk − uhk−1 T , τ := . τ N Then, dividing (5.4) by τ and replacing ξ h /τ by ξ h , we get Z   R1 (δkτ,h ) + ∇uhk : Ak : ∇(δkτ,h − ξ h ) + −DW0 (uhk ) + fk · (ξ h − δkτ,h ) dx Ω Z R1 (ξ h ) dx. ≤ δkτ,h :=

Ω

(5.6)

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

22

Now, we may further replace ξ h − δkτ,h by ξbh and use the subadditivity of R1 at ξ = ξb + δkN to get Z Z  h  h h h b b R1 (ξbh ) dx. − ∇uk : Ak : ∇ξ + −DW0 (uk ) + fk · ξ dx ≤ Ω

Ω

Given any z h ∈ Xh , we define Lhk z h ∈ Xh as the (unique) solution of the problem Z h h bh ∇z h : Ak : ∇ξbh dx for all ξbh ∈ Xh . hLk z , ξ i = −

(5.7)

Ω

We then define ψkh := Lhk uhk . Using this test function we have that Z Z   R1 (ψhk ) dx. −DW0 (uhk ) + fk · ψkh dx ≤ hψkh , ψkh i +

(5.8)

Ω

Ω

By H¨older’s inequality we deduce from (5.8) that kψkh kL2 ≤ kDW0 (uhk )kL2 + kfk kL2 + C.

By (1.6) and Sobolev embedding, assuming additionally in the case of d = 3 that q ≤ 4, we find that q−1 Z Z Z h 2 h 2(q−1) h 2 + C. |∇uk | dx dx ≤ C 1 + |uk | |DW0 (uk )| dx ≤ C Ω

Ω

Ω

Therefore, by (5.3)

max{1,(q−1)/2}

kψkh kL2 ≤ C kfk kL2

max{1,(q−1)/2}

+ C ≤ C kf kL∞ (L2 )

+ C.

Hence, max{1,(q−1)/2}

k − Lhk uhk kL2 ≤ C kf kL∞ (L2 )

+ C.

(5.9)

Now we note that (cf. the Appendix for a proof): k∇uhk kL6 ≤ C k − Lhk uhk kL2

if d = 3, and

k∇uhk kLp ≤ C k − Lhk uhk kL2

if d = 2 for all p ∈ (1, ∞).

(5.10)

The inequality (5.9) then implies that, for all k ∈ {0, . . . , N }, max{1,(q−1)/2}

+C

if d = 3, and

(5.11)

max{1,(q−1)/2}

+C

if d = 2 for all p ∈ (1, ∞).

(5.12)

k∇uhk kL6 ≤ C kf kL∞ (L2 )

k∇uhk kLp ≤ C kf kL∞ (L2 )

Moreover, taking q = 4, by Morrey’s embedding theorem, 3/2

[uhk ]Cβ ≤ Ckf kL∞ (L2 ) + C,

(5.13)

for some β ∈ (0, 1) and uniformly in h, k. 5.3. Estimates for the time derivatives. We test the kth inequality of (3.3) with ξ = 0 and the (k − 1)st inequality with uhk − uhk−2 , and divide by τ to find that Z τ,h R1 (δkτ,h ) + R1 (δk−1 ) + ∇uhk : Ak : ∇δkτ,h − ∇uhk−1 : Ak−1 : ∇δkτ,h Ω   τ,h + DW0 (uhk ) − DW0 (uhk−1 ) · δkτ,h − (fk − fk−1 ) · δkτ,h − R1 (δkτ,h + δk−1 ) dx ≤ 0. Consequently, Z τ,h R1 (δkτ,h ) + R1 (δk−1 ) + (∇uhk − ∇uhk−1 ) : Ak : ∇δkτ,h + (DW0 (uhk ) − DW0 (uhk−1 )) · δkτ,h Ω

τ,h − (fk − fk−1 ) · δkτ,h − R1 (δkτ,h + δk−1 ) − ∇uhk−1 : (Ak − Ak−1 ) : ∇δkτ,h dx ≤ 0.

Dividing by τ > 0 and using the subadditivity of R1 together with the ellipticity gives Z Z Z |fk − fk−1 | τ,h |Ak − Ak−1 | τ,h 2 κ |∇δk | dx ≤ |δk | dx + |∇uhk−1 | |∇δkτ,h | dx τ τ Ω Ω Ω Z  1 DW0 (uhk ) − DW0 (uhk−1 ) · (uhk − uhk−1 ) dx. − 2 Ω τ

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

The first term on the right-hand side can be bounded as   Z tk Z |fk − fk−1 | τ,h |δk | dx ≤ − kf˙kL2 dt kδkτ,h kL2 τ Ω tk−1   Z tk ˙ ≤ C − kf kL2 dt k∇δkτ,h kL2 .

23

(5.14)

tk−1

For the second term we deduce from (1.9) and (5.3) that Z |Ak − Ak−1 | |∇uhk−1 | |∇δkτ,h | dx ≤ CkAkC0,1 ([0,T ]×Ω) k∇uhk−1 kL2 k∇δkτ,h kL2 τ Ω ≤ C(1 + kfk−1 kL2 ) k∇δkτ,h kL2 .

For the third term we use (1.7) and the Poincar´e inequality to get Z Z   1 − 2 DW0 (uhk ) − DW0 (uhk−1 ) · (uhk − uhk−1 ) dx ≤ µ |δkτ,h |2 dx τ Ω Ω Z 2 ≤ µCP (Ω) |∇δkτ,h |2 dx, Ω

where we recall that by CP (Ω) > 0 we denote the Poincar´e constant of Ω. Hence, combining, we get   Z tk C τ,h k∇δk kL2 ≤ (5.15) 1 + − kf˙kL2 dt + kfk−1 kL2 . κ − µCP (Ω)2 tk−1 By (1.8), the constant appearing in front of the square bracket on the right-hand side is positive. 5.4. Proof of Theorem 1.8. We define the approximation sequence uhτ (t) :=

t − tk−1 h tk − t h uk + uk−1 τ τ

for t ∈ (tk−1 , tk ], k = 1, . . . , N .

Integrating (5.15) in time, we find (also using Jensen’s inequality)  Z tk N  N X X τ,h a a a h a ˙ 1 + − kf kL2 dt + kf kL∞ (L2 ) k∇δk kL2 ≤ Cτ k∇u˙ τ kLa (L2 ) = τ k=0

≤ C(1 + kf˙kaLa (L2 ) +

tk−1 k=0 kf kaL∞ (L2 ) ).

Now, by (5.13) and since f˙ ∈ La (L2 ) for some a > 1, we have that sup [uhτ (t)]Cβ + k∇u˙ hτ kLa (L2 ) ≤ C,

t∈[0,T ]

uniformly in h, τ . Hence, Remark 3.2 yields that [uhτ ]Cγ ([0,T ]×Ω) + k∇u˙ hτ kLa (L2 ) ≤ C,

(5.16)

for some γ ∈ (0, 1). T Next we define τN = N and couple it with a sequence of space discretizations, with hN → 0, as N → ∞. Here, the hN is associated with the approximation space XhN . Let uN := uhτNN be the sequence of discrete solutions to (5.6) with respect to τN and XhN . The above estimates imply that there is a subsequence of the sequence of approximate solutions converging uniformly in Cγ0 ([0, T ] × Ω; Rm ) by the Arzel`a–Ascoli theorem, for some γ0 ∈ (0, γ), and weakly in W1,a (0, T ; W01,2 (Ω; Rm )). The proof that this subsequence, in fact, converges to the (unique) solution of the problem proceeds as in the proof of the existence of solutions in Section 3.6, except for one difference: since one cannot use a fixed function ξ ∈ L1 (0, T ; W01,2 (Ω; Rm )) as a test function on the discrete level, one has to replace it with a sequence of admissible functions (ξ N )N ∈N such that ξ N → ξ, in L1 (0, T ; W01,2 (Ω; Rm )). For example, one can N take ξ N to be continuous piecewise affine function with respect to t on the partition {tN 0 , . . . , tN } of S XhN is dense in [0, T ] such that ξ N (t) ∈ XhN for all t ∈ [0, T ], and then appeal to the fact that

W01,2 (Ω; Rm ),

N ∈N

and the set of all such piecewise affine functions defined on [0, T ] is dense in L1 (0, T ).

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

24

6. Convergence with a rate Theorem 6.1. Under the assumptions made on L, f, u0 and W0 , the sequence of approximations generated by the numerical method formulated in the previous section converges to the solution u with a rate, in the sense that there exists a positive constant C, independent of h and τ , such that Z T kuhτ − uk2W1,2 dt ≤ C (h + τ min{1,a−1} ), 0

where a ∈ (1, ∞] is the assumed integrability exponent of f˙ in time.

6.1. Key estimates that are used in the proof of Theorem 6.1. We will collect here the bounds established in earlier sections that are critical for the proof of Theorem 6.1. Thanks to the a-priori estimates leading to (5.16), we have that kuhτ kL∞ ((0,T )×Ω) + k∇u˙ hτ kLa (L2 ) ≤ C, uniformly in h and τ ; here we have also used that q ≤ 4 if d = 3. Moreover, thanks to Theorem 1.3 we have that kukL∞ ((0,T )×Ω) + k∇uk ˙ La (L2 ) ≤ C. These bounds will be used repeatedly throughout the proof without mentioning them explicitly. Without loss of generality we will assume that h, τ ∈ (0, 1). Also, as is clear from the statement of Theorem 6.1, there is no improvement in the rate of convergence with respect to τ for a > 2; we shall therefore assume in the proof of Theorem 6.1 that a ∈ (1, 2]. 6.2. A C´ ea type inequality. We recall the definition t − tk−1 h tk − t h uk + uk−1 for t ∈ (tk−1 , tk ], k = 1, . . . , N . uhτ (t) := τ τ In the following we will fix a t ∈ (tk−1 , tk ], such that k∇u(t)k ˙ L2 < ∞, which holds for almost every t ∈ (tk−1 , tk ]. Henceforth, all time-dependent functions considered on (tk−1 , tk ] will be evaluated at that point t. We start with testing the continuous system. We use the test function δkτ,h = u˙ hτ , which is admissible by (5.15). Then we find for the chosen t ∈ (tk−1 , tk ], that Z Z   h h R1 (u˙ hτ ) dx. R1 (u) ˙ + ∇u : A : ∇(u˙ − u˙ τ ) + DW0 (u) − f · (u˙ − u˙ τ ) dx ≤ (6.1) Ω

Ω

We now wish to imitate this inequality for the discrete problem. We introduce for each time step k ∈ {1, . . . , N } the Ritz projector Πhk : W01,2 (Ω; Rm ) → Xh defined as hAk ∇(g − Πhk g), ∇ϕh i = 0 for all ϕh ∈ Xh . For any g ∈ W01,2 (Ω; Rm ) we have that k∇(g − Πhk g)k2L2 ≤ ChAk ∇(g − Πhk g), ∇(g − Πhk g)i = ChAk ∇(g − Πhk g), ∇(g − Ph g)i ≤ Ck∇(g − Πhk g)kL2 k∇(g − Ph g)kL2 ,

where we have used the definition of Πhk in the transition to the second line. Hence, using the properties of the projector Ph , it follows that k∇(g − Πhk g)kL2 ≤ CkgkW1,2 k∇(g − Πhk g)kL2 ≤ ChkgkW2,2

for all g ∈ W01,2 (Ω; Rm ), for all g ∈ (W2,2 ∩

W01,2 )(Ω; Rm ),

(6.2) (6.3)

with a constant C > 0 that depends only on the ellipticity of A and the properties of P, but is independent of the discretization parameters h and τ . We also require a bound in the L2 norm on the difference between a function g ∈ (W2,2 ∩ W01,2 )(Ω; Rm ) and its Ritz projection Πhk g. The derivation of this proceeds by an Aubin–Nitsche type duality argument: for g ∈ (W2,2 ∩ W01,2 )(Ω; Rm ) we let vg ∈ W01,2 (Ω; Rm ) be the unique solution of the problem h∇vg , Ak ∇ϕi = hAk ∇vg , ∇ϕi = hg − Πhk g, ϕi

for all ϕ ∈ W01,2 (Ω; Rm ).

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

25

Hence, by the regularity theory of uniformly elliptic systems, (6.2) and (6.3) yield that kg − Πhk gk2L2 = h∇vg , Ak ∇(g − Πhk g)i = h∇(vg − Πhk vg ), Ak ∇(g − Πhk g)i ≤ Ck∇(vg − Πhk vg )kL2 · k∇(g − Πhk g)kL2 ≤ Chkvg kW2,2 · hs kgkW1+s,2 ≤ Ch1+s kg − Πhk gkL2 kgkW1+s,2 ,

s = 0, 1.

Therefore, kg − Πhk gkL2 ≤ Ch1+s kgkW1+s,2 ,

(6.4)

s = 0, 1.

Now, for the fixed t ∈ (tk−1 , tk ], we find Z ˙ dx R1 (u˙ hτ ) + ∇uhτ : Ak : ∇(u˙ hτ − u) Ω Z ˙ dx R1 (u˙ hτ ) + ∇uhτ : Ak : ∇(u˙ hτ − Πhk u) = Ω Z R1 (u˙ hτ ) + ∇uhk : Ak : ∇(u˙ hτ − Πhk u) ˙ dx = Ω Z tk − t + ˙ dx, ∇(uhk−1 − uhk ) : Ak : ∇(u˙ hτ − Πhk u) τ Ω

(6.5)

using the fact that

uhτ (t) := uhk +

tk − t h (uk−1 − uhk ) τ

for t ∈ (tk−1 , tk ], k = 1, . . . , N .

We now proceed to estimate using (6.2), recalling that a ∈ (1, 2] and using the fact that ∇uhk is bounded in L2 , uniformly in k, h, thanks to (5.11), (5.12). Hence, Z tk − t ∇(uhk−1 − uhk ) : Ak : ∇(u˙ hτ − Πhk u) ˙ dx τ Ω ˙ L2 ≤ Ck∇(uhk−1 − uhk )kL2 k∇(u˙ hτ − Πhk u)k

˙ L2 ≤ C k∇(uhk−1 − uhk )kL2 k∇u˙ hτ kL2 + k∇(uhk−1 − uhk )kL2 k∇(Πhk u)k




k∇uk ˙ aL2 ˙ hτ kaL2 + Cτ a−1 k∇(uhk−1 − uhk )k2−a ≤ Cτ a−1 k∇(uhk−1 − uhk )kL2−a 2 k∇u L2 ≤ Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk ˙ aL2 ).

(6.6) uh −uh

Here in the transition from the third line to the fourth line we have noted that u˙ hτ (t) = k τ k−1 at the point t ∈ (tk−1 , tk ], and used the stability of the Ritz projector Πhk in the W1,2 (Ω; Rm )-seminorm together with Young’s inequality, to deduce the following bound on the second term in the third line: ˙ L2 ˙ L2 ≤ C k∇(uhk−1 − uhk )kL2 k∇uk k∇(uhk−1 − uhk )kL2 k∇(Πhk u)k k∇uk ˙ L2 ˙ hτ ka−1 = Cτ a−1 k∇(uhk−1 − uhk )kL2−a 2 k∇u L2


k∇u˙ hτ kaL2 + k∇uk ˙ aL2 , ≤ Cτ a−1 k∇(uhk−1 − uhk )kL2−a 2

and the first part of this expression can be combined with the first part in the fourth line of (6.6). Next we apply (5.6) with the test function ξ h = Πhk u. ˙ This implies, noting that u˙ hτ (t) = at the point t ∈ (tk−1 , tk ], and using first (6.5) and then (6.6) that Z

h uh k −uk−1 τ

= δkτ,h

R1 (u˙ hτ ) + ∇uhτ : Ak : ∇(u˙ hτ − u) ˙ dx Ω Z   tk − t ∇(uhk−1 − uhk ) : Ak : ∇(u˙ hτ − Πhk u) ˙ dx ˙ + R1 (Πhk u) ˙ + −DW0 (uhk ) + fk · (u˙ hτ − Πhk u) ≤ τ Ω Z   ˙ dx + Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk ˙ aL2 ). (6.7) R1 (Πhk u) ˙ + −DW0 (uhk ) + fk · (u˙ hτ − Πhk u) ≤ Ω

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

26

As ∇u in bounded in L∞ (0, T ; L2 (Ω; Rd×m )), ∇uhτ is uniformly bounded (with respect to h and τ ) in L∞ (0, T ; L2 (Ω; Rd×m )), and since A is assumed to be Lipschitz continuous, we have that Z ∇(u − uhτ ) : A : ∇(u˙ − u˙ hτ ) dx (I) := ZΩ ∇u : A : ∇(u˙ − u˙ hτ ) + ∇uhτ : Ak : ∇(u˙ hτ − u) ˙ + ∇uhτ : (A − Ak ) : ∇(u˙ hτ − u) ˙ dx = Ω Z ˙ L2 ). ∇u : A : ∇(u˙ − u˙ hτ ) + ∇uhτ : Ak : ∇(u˙ hτ − u) ˙ dx + Cτ (k∇u˙ hτ kL2 + k∇uk ≤ Ω

This allows us to use the estimate derived by summing up (6.1) and (6.7), which implies that Z Z h ˙ dx DW0 (uhk ) · (Πhk u˙ − u˙ hτ ) + DW0 (u) · (u˙ hτ − u) ˙ − R1 (u) ˙ dx + R1 (Πk u) (I) ≤ Ω Ω Z ˙ aL2 ) ˙ + f · (u˙ − u˙ hτ ) dx + Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk fk · (u˙ hτ − Πhk u) + Ω

˙ aL2 ). =: (II) + (III) + (IV ) + Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk

(6.8)

6.3. Imitation of the uniqueness estimate for the error. Using the 1-homogeneity and subadditivity of R1 , we obtain Z ˙ L2 . (6.9) ˙ dx ≤ Cku˙ − Πhk (u)k |u˙ − Πhk u| (II) ≤ C Ω

Consequently, by taking s = 0, g = u˙ in (6.4), and using Poincar´e’s inequality, ˙ L2 ≤ Chk∇uk ˙ L2 . (II) ≤ Cku˙ − Πhk uk

(6.10)

Next, using the fact that |DW0 (uhτ )| is uniformly bounded and noting (6.10), we find that Z     ˙ DW0 (uhk ) − DW0 (uhτ ) · (u˙ − u˙ hτ ) + DW0 (u) − DW0 (uhτ ) · (u˙ hτ − u) (III) = Ω

≤

Z

+ DW0 (uhk ) · (Πhk u˙ − u) ˙ dx     ˙ dx + Chk∇uk ˙ L2 . DW0 (uhk ) − DW0 (uhτ ) · (u˙ − u˙ hτ ) + DW0 (u) − DW0 (uhτ ) · (u˙ hτ − u)

Ω

Moreover, using that D2 W0 (θuhk + (1 − θ)uhτ ) is uniformly bounded with respect to h and τ thanks to the uniform bounds on uhτ and the assumptions on W0 , noting that |uhτ (t) − uhk (t)| ≤ τ |u˙ hτ | for all t ∈ (tk−1 , tk−1 ], and that, by hypothesis a ∈ (1, 2], we find that, pointwise, |DW0 (uhk ) −

DW0 (uhτ )|

Z ≤

1

2

D 0

W0 (θuhk

+ (1

− θ)uhτ )

≤ C|uhk − uhτ |

˙ hτ |a−1 ≤ Cτ a−1 kuτh k2−a L∞ |u

dθ |uhk − uhτ |

≤ Cτ a−1 |u˙ hτ |a−1 . Therefore, combining the above estimates with Young’s inequality (using a1 + a−1 a = 1) and the Sobolev– Poincar´e inequality implies that Z Z   h h a−1 DW0 (u) − DW0 (uτ ) · (u˙ τ − u) ˙ dx + Cτ (III) ≤ |u˙ hτ |a−1 |u˙ − u˙ hτ | dx + Chk∇uk ˙ L2 Ω Ω Z   DW0 (u) − DW0 (uhτ ) · (u˙ hτ − u) ˙ dx + Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk ˙ aL2 ) + Chk∇uk ˙ L2 . ≤ Ω

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

27

Using the symmetry of D2 W0 yields that   DW0 (u) − DW0 (uhτ ) · (u˙ hτ − u) ˙ =

Z

0

1

Z

D2 W0 (θu + (1 − θ)uhτ ) dθ [u − uhτ , u˙ hτ − u] ˙ 1

D2 W0 (θu + (1 − θ)uhτ ) dθ [u − uhτ , u˙ − u˙ hτ ] 0   Z d 1 1 2 h h h D W0 (θu + (1 − θ)uτ ) dθ [u − uτ , u − uτ ] =− dt 2 0 Z  1 1  2 + ∂t D W0 (θu + (1 − θ)uhτ )) dθ [u − uhτ , u − uhτ ]. 2 0 =−

We estimate further using (1.11) and the a-priori bounds in the space-time L∞ norm on uhτ and u, H¨older’s inequality and the Sobolev–Poincar´e inequality, to get Z Z 1   ∂t D2 W0 (θu + (1 − θ)uhτ )) dθ [u − uhτ , u − uhτ ] dx Ω 0 Z 1 |D3 (θu + (1 − θ)uhτ )) |θ u˙ + (1 − θ)u˙ hτ | |u − uhτ |2 | dθ ≤ 0

≤ C(kuk ˙ L2 + ku˙ hτ kL2 ) k∇(u − uhτ )k2L2 .

Finally we get that (III) ≤ −

d dt

Z

Ω

1 2

Z

1

0

D2 W0 (θu + (1 − θ)uhτ ) dθ [u − uhτ , u − uhτ ] dx



˙ aL2 ) + Chk∇uk ˙ L2 . + C(kuk ˙ L2 + ku˙ hτ kL2 ) k∇(u − uhτ )k2L2 + Cτ a−1 (k∇u˙ hτ kaL2 + k∇uk The last term is estimated using that, for a ∈ (1, 2] and t ∈ (tk−1 , tk ], a−1 Z t a−1  Z tk k 2−a a−1 2−a ˙ ˙ . ≤ Cτ − |f | ds f ds |f (tk ) − f (t)| ≤ 2 kf kL∞ t

tk−1

Hence, using Young’s inequality, the Sobolev–Poincar´e inequality, (6.2), Jensen’s inequality, Fubini’s theorem and (6.10), we find Z ˙ + f · (u˙ − Πhk (u)) ˙ dx (IV ) = (fk − f ) · (u˙ hτ − Πhk (u)) Ω

≤ Cτ

a−1

a−1 Z  Z tk ˙ ˙ dx + Chkf kL∞ k∇uk ˙ L2 (|u˙ hτ | + |Πhk (u)|) − |f | ds Ω

tk−1

 a  Z  Z tk h a a a−1 ˙ L2 + k∇u˙ τ kL2 + Chk∇uk ˙ L2 ≤ Cτ − |f˙| ds dx + k∇uk Ω

tk−1

  Z tk ≤ Cτ a−1 − kf˙kaLa ds + k∇uk ˙ aL2 + k∇u˙ hτ kaL2 + Chk∇uk ˙ L2 . tk−1

We now wish to replicate the Gronwall argument from the uniqueness section. We therefore define the error function e(t) := k∇(u − uhτ )k2L2 . Using the uniform ellipticity assumption on A, we find that Z d A (I ′ ) := ∇(u − uhτ ) : : ∇(u − uhτ ) dx dt Ω 2 Z Z ∂t A h h : ∇(u − uhτ ) dx ∇(u − uhτ ) : ∇(u − uτ ) : A : ∇(u˙ − u˙ τ ) dx + = 2 Ω Ω Z |∇(u − uhτ )|2 dx ≤ (I) + C Ω

≤ (I) + Ce(t).

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

28

Collecting the bounds on the terms (II), (III) and (IV ) and substituting them into (6.8), yields a bound on (I), from which we then deduce that  Z Z d 1 1 2 ′ h h h (I ) + D W0 (θu(t) + (1 − θ)uτ (t)) dθ [u − uτ , u − uτ ] dx dt Ω 2 0   Z tk a−1 a h a a ˙ ≤ C(h + τ ) 1 + 1(tk−1 ,tk ] (t) − kf kLa ds + k∇u˙ τ (t)kL2 + k∇u(t)k ˙ L2 tk−1

+ C(1 + ku(t)k ˙ L2 +

ku˙ hτ (t)kL2 ) e(t),

t ∈ (tk−1 , tk ],

k = 1, . . . , N.

For any s ∈ (0, T ] there exists a k ∈ {1, . . . , N } such that s ∈ (tk−1 , tk ]. If k = 1 we integrate the above inequality over (0, s]; else we integrate it over each of the intervals (0, t1 ], . . . , (tk−1 , s], and sum up the resulting inequalities. Thus we deduce that Z A(s) : ∇(u(s) − uhτ (s)) ∇(u(s) − uhτ (s)) : 2 Ω Z 1 1 2 D W0 (θu(s) + (1 − θ)uhτ (s)) dθ [u(s) − uhτ (s), u(s) − uhτ (s)] dx + 2 0 Z s (1 + kuk ˙ L2 + ku˙ hτ kL2 )e(t) dt ≤C 0

˙ aLa (L2 ) ) + C(h + τ a−1 )(1 + kf˙kaLa (La ) + k∇u˙ hτ kaLa (L2 ) + k∇uk Z A0 : ∇(u0 − uh0 ) ∇(u0 − uh0 ) : + 2 Ω Z 1 1 2 + D W0 (θu0 + (1 − θ)uh0 ) dθ [u0 − uh0 , u0 − uh0 ] dx. 2 0

We use (3.5), (3.3), (5.2) and (1.6) to find that Z 1 Z h h ′′ D2 W0 (θu0 + (1 − θ)uh0 ) dθ [u0 − uh0 , u0 − uh0 ] dx ∇(u0 − u0 ) : A0 : ∇(u0 − u0 ) + (I ) := 0 Ω Z ∇u0 : A0 : ∇(u0 − uh0 ) + DW0 (u0 ) · [u0 − uh0 ] dx = Ω Z ∇uh0 : A0 : ∇(uh0 − u0 ) + DW0 (uh0 ) · [uh0 − u0 ] dx + Ω Z R1 (uh0 − u0 ) + f (0) · (u0 − uh0 ) dx ≤ Ω Z ∇uh0 : A0 : ∇(uh0 − Πh0 u0 ) + DW0 (uh0 ) · [uh0 − Πh0 u0 ] dx + Z ZΩ h h DW0 (uh0 ) · [Πh0 u0 − u0 ] dx ∇u0 : A0 : ∇(Π0 u0 − u0 ) dx + + Ω Ω Z Z q−1
h h |Πh0 u0 − u0 | dx. R1 (Π0 u0 − u0 ) dx + kf (0)kL∞ + C(1 + ku0 kL∞ ) ≤ Ω

Ω

Here in the transition from the right-hand side of the first inequality on the right-hand side of the second inequality we have used that the second integral on the right-hand side of the first inequality is equal to zero since uh0 = Πh0 u0 , and the third integral on the right-hand side of the first inequality is also equal to zero as uh0 = Πh0 u0 is the Ritz projection of u0 . Next, using that u0 , uh0 ∈ (W01,2 ∩ L∞ )(Ω; Rm ) by (5.13), we find by (6.4) that Z ′′ |Πh0 (u0 ) − u0 | dx ≤ Chk∇u0 kL2 . (I ) ≤ C Ω

Hence, by using (4.4), the bounds on ∇u, ˙ ∇u˙ hτ , the assumptions on f˙, and also the (uniform) positivedefiniteness of A(s), we have that Z s (1 + kuk ˙ L2 + ku˙ hτ kL2 ) e(t) dt + C(h + τ a−1 ). e(s) ≤ C 0

REGULARITY AND APPROXIMATION OF RATE-INDEPENDENT SYSTEMS

29

Since 1 + kuk ˙ L1 (L2 ) + ku˙ hτ kL1 (L2 ) is bounded (uniformly in h, τ ) Gronwall’s lemma implies the desired error bound. Appendix Our aim in this appendix is to prove the inequalities (5.10). Let z h ∈ Xh , and define the discrete Green operator Gkh : Xh → Xh by Z Z z h · ξbh dx for all ξbh ∈ Xh , ∇(Gkh z h ) : Ak : ∇ξbh dx = Ω

Ω

which is the finite element counterpart of the operator Gk : L2 (Ω; Rm ) → W01,2 (Ω; Rm ), defined, for any z ∈ L2 (Ω; Rm ), by Z Z z · ϕ dx for all ϕ ∈ W01,2 (Ω; Rm ). ∇(Gk z) : Ak : ∇ϕ dx = Ω

Ω

By elliptic regularity theory Gk z ∈ (W2,2 ∩ W01,2 )(Ω; Rm ) for all k ∈ {0, . . . , N }, and kGk zkW2,2 ≤ CkzkL2 ,

where C is a positive constant, independent of k. It is easily seen that Gkh Lhk z h = z h for all z h ∈ Xh . Indeed, using the definitions of Gkh and Lhk , we have that Z Z Z ∇z h : Ak : ∇ξbh dx for all ξbh ∈ Xh ; Lhk z h · ξbh dx = ∇(Gkh Lhk z h ) : Ak : ∇ξbh dx = Ω

Ω

Ω

hence, by taking ξbh = Gkh Lhk z h − z h we deduce that k∇(Gkh Lhk z h − z h )k2L2 = 0, whereby Gkh Lhk z h = z h . Let us first concentrate on the case when d = 3. By writing ξkh := Lhk z h , noting (5.1), using the Sobolev embedding W2,2 (Ω; Rm ) ֒→ W1,6 (Ω; Rm ), an inverse inequality (recall that T h has been assumed to be quasiuniform), and that by an Aubin–Nitsche duality argument kGk ξkh − Gkh ξkh kL2 ≤ Ch2 kGk ξkh kW2,2 ≤ Ch2 kξkh kL2 , we have that kz h kL6 = k∇Gkh Lhk z h kL6 = k∇Gkh ξkh kL6 ≤ k∇Gk ξkh kL6 + k∇(I − Ph )Gk ξkh kL6 + k∇(Ph Gk − Gkh )ξkh kL6 1

1

≤ CkGk ξkh kW2,2 + CkGk ξkh kW1,6 + Ch−1+3( 6 − 2 ) kPh (Gk ξkh ) − Gkh ξkh kL2 ≤ CkGk ξkh kW2,2 + Ch−2 (kPh (Gk ξkh ) − Gk ξkh kL2 + kGk ξkh − Gkh ξkh kL2 ) ≤ CkGk ξkh kW2,2 + Ch−2 (Ch2 kGk ξkh kW2,2 + Ch2 kξkh kL2 ) ≤ CkGk ξkh kW2,2 + Ckξkh kL2 ≤ Ckξkh kL2 = CkLhk z h kL2 , as required. For d = 2 the proof is analogous: the L6 norm is replaced by the Lp norm for p ∈ (1, ∞), the embedding W2,2 (Ω; Rm ) ֒→ W1,6 (Ω; Rm ) is replaced by W2,2 (Ω; Rm ) ֒→ W1,p (Ω; Rm ), p ∈ (1, ∞), and the exponent −1 + 3( 61 − 12 ) in the third line of the displayed chain of inequalities above is replaced by −1 + 2 min{( p1 − 12 ), 0}. References [ADN59] [ADN64] [Bar14] [Bro60] [Bro61]

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. , Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. S. Bartels, Quasi-optimal error estimates for implicit discretizations of rate-independent evolutions, SIAM J. Numer. Anal. 52 (2014), no. 2, 708–716. MR 3180858 F. E. Browder, A-priori estimates for solutions of elliptic boundary-value problems. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 145–169. , A priori estimates for solutions of elliptic boundary-value problems. III, Nederl. Akad. wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 404–410.

¨ FILIP RINDLER, SEBASTIAN SCHWARZACHER, AND ENDRE SULI

30

[Cam63]

S. Campanato, Propriet` a di h¨ olderianit` a di alcune classi di funzioni, Ann. Scuola Norm. Sup. Pisa 17 (1963), 175–188. [DE08] L. Diening and F. Ettwein, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Mathematicum 20 (2008), 523–556. [Eva10] L. C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 2010. [Gia83] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, 1983. [Giu03] E. Giusti, Direct methods in the calculus of variations, World Scientific, 2003. [GM12] M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, 2nd ed., Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 11, Edizioni della Normale, Pisa, 2012. [HR00] W. Han and B. D. Reddy, Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity, Numer. Math. 87 (2000), no. 2, 283–315. MR 1804659 [KRZ15] D. Knees, R. Rossi, and C. Zanini, A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains, Nonlinear Anal. Real World Appl. 24 (2015), 126–162. [LU68] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, 1968. [Mie11] A. Mielke, Differential, Energetic, and Metric Formulations for Rate-Independent Processes, Calculus of Variations and Nonlinear Partial Differential Equations, Springer, 2011, pp. 87–170. [MPPS10] A. Mielke, L. Paoli, A. Petrov, and U. Stefanelli, Error estimates for space-time discretizations of a rateindependent variational inequality, SIAM J. Numer. Anal. 48 (2010), 1625–1646. [MR09] A. Mielke and F. Rindler, Reverse approximation of energetic solutions to rate-independent processes, NoDEA Nonlinear Differential Equations Appl. 16 (2009), 17–40. [MR15] A. Mielke and T. Roub´ıˇcek, Rate-Independent Systems. Theory and Application, Applied Mathematical Sciences, vol. 193, Springer, 2015. [MRS09] A. Mielke, R. Rossi, and G. Savar´e, Modeling solutions with jumps for rate-independent systems on metric spaces, Discrete Contin. Dyn. Syst. 25 (2009), 585–615. [MRS12] , BV solutions and viscosity approximations of rate-independent systems, ESAIM Control Optim. Calc. Var. 18 (2012), 36–80. [MT04] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl. 11 (2004), 151–189. [MTL02] A. Mielke, F. Theil, and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle, Arch. Ration. Mech. Anal. 162 (2002), 137–177. [MZ14] A. Mielke and S. Zelik, On the vanishing-viscosity limit in parabolic systems with rate-independent dissipation terms, Ann. Sc. Norm. Super. Pisa Cl. Sci. 13 (2014), 67–135. [Sim87] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. 146 (1987), 65–96. [TM10] M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain—existence and regularity results, ZAMM Z. Angew. Math. Mech. 90 (2010), 88–112. Filip Rindler: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK E-mail address: [email protected] ´ 83, 186 75 Praha Sebastian Schwarzacher: Department of Analysis, Charles University Prague, Sokolovska 8, Czech Republic E-mail address: [email protected] Endre S¨ uli: Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, United Kingdom E-mail address: [email protected]