DUALITY BASED A POSTERIORI ERROR ESTIMATES FOR HIGHER ORDER VARIATIONAL INEQUALITIES WITH POWER GROWTH FUNCTIONALS

Transcription

1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 33, 2008, DUALITY BASED A POSTERIORI ERROR ESTIMATES FOR HIGHER ORDER VARIATIONAL INEQUALITIES WITH POWER GROWTH FUNCTIONALS Michael Bildhauer, Martin Fuchs and Sergey Repin Universität des Saarlandes, Fachbereich 6. Mathematik Postfach 5 50, D-6604 Saarbrücken, Germany; bibi@math.uni-sb.de Universität des Saarlandes, Fachbereich 6. Mathematik Postfach 5 50, D-6604 Saarbrücken, Germany; fuchs@math.uni-sb.de V. A. Steklov Mathematical Institute, St. Petersburg Branch Fontanka 27, 90 St. Petersburg, Russia; repin@pdmi.ras.ru Abstract. We consider variational inequalities of higher order with p-growth potentials over a domain in the plane by the way including the obstacle problem for a plate with power hardening law. Using duality methods we prove a posteriori error estimates of functional type for the difference of the exact solution and any admissible comparision function.. Introduction On a bounded Lipschitz domain R 2 we consider the minimization problem (P) J u := π p ( 2 u) dx min in K, where the class K consists of all functions v from the space W 2 p() s.t. v(x) Ψ(x) on, the potential π p is given by the formula π p (E) := p E p for symmetric (2 2)- matrices E and 2 u represents the matrix of the second generalized derivatives. It is assumed that Ψ Wp 2 () is a given function s.t. Ψ < 0 and Ψ(x 0 ) > 0 for some point x 0 and the exponent p is arbitrarily chosen in the interval < p <. For a definition of the Sobolev spaces W 2 p(), Wp 2 () and related classes we refer to Ad. We recall that by Sobolev s embedding theorem the functions Ψ and v Wp 2 () have a representative in C 0 () and that this observation can be used to show that the class K is not empty (compare FLM), which means that (P) has a unique solution u K. The second order obstacle problem (P) is of some physical relevance: consider a plate which is clamped at the boundary and whose undeformed state is represented by the region. If some outer forces are applied acting in vertical direction, then the 2000 Mathematics Subject Classification: Primary 65N5, 65K0, 74K20, 49J40, 49M29. Key words: A posteriori error estimates, higher order variational inequalities, duality methods, power growth, elastic plates with obstacles.

2 476 Michael Bildhauer, Martin Fuchs and Sergey Repin equilibrium configuration can be found as a minimizer of an energy with principal part g( 2 w) dx, where the mechanical properties of the plate are characterized by the given convex function g. In the case of linear elastic plates, we have g = π 2 and since our exponent p is arbitrary we can include any power-hardening law. In particular, for p close to we have an approximation of perfectly plastic plates. The new feature of problem (P) however is that the plate has to respect the given side condition. In various papers mainly the regularity properties of minimizers (with or without obstacle) for different functions g have been investigated, we refer to Se for the case of plastic behaviour, whereas the case of nearly linear growth is studied in FLM and BF. For the classical case p = 2 we refer, e.g., to Fr. In the present note we concentrate on a posteriori error estimates of functional type for the solution of problem (P) by combining the methods developed by the third author for first order obstacle problems with quadratic potentials (see Re and also NR, Chapter 8) with the techniques introduced in BR and BFR for unconstrained variational problems with a power growth potential. To be more precise, let u K denote the solution of problem (P) and consider any function v from the class of comparison functions. Then our goal is to prove the estimate (.) 2 u 2 v L p M (v, D), where D stands for the set of known data and where M is a non-negative functional depending on v, on the data of the problem such as p,, Ψ and on parameters which are under our disposal. M should satisfy the following requirements: i.) the value of M is easy to calculate for any choice of an admissible function v; ii.) the estimate is consistent in the sense that M (v, D) = 0 if and only if v = u, moreover M (v k, D) 0 if 2 v k 2 u L p 0; iii.) M provides a realistic upper bound for the quantity 2 u 2 v L p. Of course iii.) means that for obtaining the bound (.) one carefully tries to avoid over-estimation so that (.) can be used for a reliable verification of approximative solutions obtained by various numerical methods. As already outlined in BR and BFR the cases p 2 and < p < 2 require different techniques: in Section 3 we will prove an estimate like (.) if p 2, which without a priori estimates for the exact solution we could not verify for the subquadratic case. Therefore, in Section 4, we pass to the dual variational problem (P) and discuss a variant of (.) involving the dual solution σ. We like to remark that just for the case of technical simplicity we did not include terms like uf dx or u F dx with functions f : R, F : R 2 into our variational integral J. These quantities can be added without substantial changes provided f and F satisfy suitable integrability assumptions. In

3 Duality based a posteriori error estimates for higher order variational inequalities 477 the same spirit we could include the double obstacle problem Ψ u Φ combined with different boundary conditions being compatible with the obstacle(s). It is worth noting that if we start from any function Ψ W 2 p () and if we require that K is not empty, then all our calculations remain valid if R d, d 2, with constants partially depending on d. 2. Preliminaries Basic facts in duality theory. We recall some facts from duality theory (see, e.g. ET) valid for all < p < : if we let q := p, Y := (; R 2 2 p sym), Y := L p (; R 2 2 sym) and l(v, τ ) := τ : 2 v πp(τ ) dx, where π p is the conjugate function of π p, i.e. π p(e) = q E q, then and Jv = sup τ Y l(v, τ ) Ju = J σ, where J τ := v K l(v, τ ), τ Y, is the dual functional, σ denotes its unique maximizer, and u K is the unique solution of the problem (P). Clarkson s inequality. If p 2, then we will use a version of Clarkson s inequality Cl presented in MM for tensor-valued functions τ, τ 2 Y : τ + τ 2 (2.) p + τ τ 2 p dx τ p L + p 2 τ 2 p L p. If p < 2, then we apply this inequality to the dual variable. Basic deviation estimate. In the superquadratic case, i.e. if p 2, (2.) implies for all v K 2 (u v) p p 2 p L p 2 Jv + u + v 2 Ju J, 2 and since u+v K we deduce from the J-minimality of u that 2 (2.2) 2 (u v) p p 2 Jv p Ju. L p If p < 2, then the basic deviation estimate (2.2) takes the form (2.3) and is derived as follows: with v K chosen arbitrarily we have q 2 q τ σ q L τ q dx + σ q dx 2 τ + σ q dx q q q q 2

4 478 Michael Bildhauer, Martin Fuchs and Sergey Repin = q τ q τ : v 2 dx + q σ q σ : v 2 dx 2 τ + σ q σ + τ : 2 v dx q 2 2 = l(v, τ ) l(v, σ ) + 2l (v, τ + σ ) 2 ( ) ( ) sup l(v, τ ) + sup l(v 2, σ ) + 2l (v, τ + σ ) v K v 2 K 2 = l(v, τ ) l(v 2, σ ) + 2l (v, τ + σ ) v K v 2 K 2 = J τ J σ + 2l (v, τ + σ ), 2 and since v K is under our disposal, we may pass to the w.r.t. v K on the r.h.s. with the result σ q 2 q σ τ q L J τ J σ + 2J + τ, q 2 and the J -maximality of σ implies (2.3) q 2 q σ τ q J σ J τ. A modified functional. Following Re we introduce a relaxtion of (P): for we let (P λ ) J λ w := Jw λ Λ := {ρ () : ρ 0 a.e.} λ(w Ψ) dx min in W 2 p(). Clearly (P λ ) is well-posed with unique solution u λ. Also we note that { Jw, if w K sup J λ w = Jw λ(w Ψ) dx = λ Λ λ Λ +, if w / K. Letting L(w, τ, λ) := we define the dual functional 2 w : τ π p(τ ) λ(w Ψ) dx, w W 2 p(), τ Y, λ Λ, J λτ := with unique maximizer σλ and get L(w, τ, λ) w W 2 p () (2.4) J λ u λ = J λσ λ.

5 Duality based a posteriori error estimates for higher order variational inequalities 479 The reader should observe that Jλ τ > for τ Y implies that τ is in the class { Q λ := η Y : η : 2 w λw dx = 0 for all w W p()} 2, which means that in the distributional sense τ = (τ αβ ) satisfies In this case we have (2.5) Jλτ = We further note that (2.6) J λ J λ = W 2 p () K v K div(div τ ) := α ( β τ αβ) = λ. v K π p (τ ) + ψλ dx. π p ( 2 v) dx π p ( 2 v) dx = K J. λ(v Ψ) dx 3. Estimates for the superquadratic case Let us now state our first result: Theorem 3.. Let p 2. With the notation introduced above we have for any v K, for any η Y, for all λ Λ and for any choice of β > 0 the estimate 2 (u v) p L p p2 p {D p 2 v, η q (3 q) + q β q d(η ) q (3.) where for ρ Y, κ Y and D p ρ, κ := + p βp η q 2 η 2 v p + L p d(κ ) := πp (ρ) + π p(κ ) ρ : κ dx κ τ. τ Q λ If in addition div(div η ) (), then we have the inequality (3.2) d(η ) C p () λ div(div η ). The constant C p () is defined in formula (3.8). } λ(v Ψ) dx, Remark 3.. Note that all the quantities on the r.h.s. of (3.) are non-negative, and the r.h.s. of (3.) vanishes if and only if 2 v = η q 2 η, div(div η ) = λ, λ(v Ψ) = 0.

6 480 Michael Bildhauer, Martin Fuchs and Sergey Repin Let w K. Then the validity of the above equations gives 2 v p 2 2 v : 2 (w v) dx = η : 2 (w v) dx = λ(w v) dx = λ(w Ψ) dx + λ(ψ v) dx = λ(w Ψ) dx 0, which means that v is the unique solution of Problem (P). If d is estimated via (3.2) then all the functions on the r.h.s. of (3.) are either known or in our disposal. Thus (3.) gives a practical way to measure the accuracy. Having proved Theorem 3. we will give variants of (3.) by optimizing the function λ. Proof of Theorem 3.. We recall (2.6), i.e. so that according to (2.4) Ju J λ, W 2 p () Ju J λσ λ J λτ for all choices of λ Λ and τ Q λ. This gives in combination with (2.2) (3.3) 2 (u v) p p 2 p Jv J L λτ. p By (2.5) we find that Jv Jλτ = = π p ( 2 v) + πp(τ ) Ψλ dx π p ( 2 v) + πp(τ ) τ : 2 v dx + λ(v Ψ) dx, = D p 2 v, τ + and according to (3.3) we have shown that (3.4) 2 (u v) p L p p 2 p {D p 2 v, τ + } λ(v Ψ) dx λ(v Ψ) dx valid for all v K, τ Q λ and λ Λ. Consider any tensor η Y. Then (3.4) and the convexity of πp imply 2 (u v) p p2 {D p L p p 2 v, η + π p (τ ) πp(η ) (τ η ) : 2 v dx + } λ(v Ψ) dx

7 Duality based a posteriori error estimates for higher order variational inequalities 48 (3.5) p2 {D p p 2 v, η + τ q 2 τ 2 v : (τ η ) dx } + λ(v Ψ) dx = p 2 {D p p 2 v, η + τ q 2 τ η q 2 η : (τ η ) dx + η q 2 η 2 v } : (τ η ) dx + λ(v Ψ) dx. As demonstrated in BR we have τ q 2 τ η q 2 η : (τ η ) dx 2 2 q (3 q) τ η q, moreover, from Hölder s inequality it follows that η q 2 η 2 v : (τ η ) dx η q 2 η 2 v L p τ η ρ Q λ p βp η q 2 η 2 v p L p + q β q τ η q, where in the last line we used Young s inequality with some β > 0. Inserting these estimates into (3.5) and taking the w.r.t. τ Q λ, inequality (3.) is proved. To establish the second part of the theorem, we consider η Y with the property div(div η ) (). Then τ Q λ η τ is attained for some τ Q λ, and η τ is a measure for the distance from η to Q λ. Letting λ := λ α β ηαβ we find (with an obvious meaning of Q) λ (3.6) q κ ql, q (3.7) sup κ Q λ q ρ η q L = sup q κ Q λ q κ ql = q w W 2 p() For w W 2 p() we have by Poincaré s inequality p (3.8) w L p C p () 2 w L p t 0 2 w p λw dx. with a positive constant C p () depending on p and. Using (3.8) on the r.h.s. of (3.7), we see that the r.h.s. of (3.7) is bounded from below by p tp C p ()t λ, and this is attained at t 0 := p λ C p ().

8 482 Michael Bildhauer, Martin Fuchs and Sergey Repin From (3.6) we therefore get (3.9) ρ η =: d(η ) C p () λ div(div η ). ρ Q λ Now we discuss two variants of how to choose λ Λ in a suitable way. Variant. Given v K and η Y s.t. div(div η ) () we let (following Re) { λ = 0 on v > Ψ, λ = div(div η ) on v = Ψ, where f := max(0, f), f := min(0, f), hence f = f +f for real-valued functions f. With this choice of λ we get moreover λ div(div η ) q dx = and we arrive at v>ψ λ(v Ψ) dx = 0, div(div η ) q dx + v=ψ div(div η ) q dx, Corollary 3.. Let p 2. For any v K, for all η Y s.t. div(div η ) () and for all β > 0 we have with K p (, β) := Cp() 2 q 2 q (3 q) + q β q the estimate 2 (v u) p p 2 {D p L p p 2 v, η + K p (, β) (3.0) + v=ψ v>ψ div(div η ) q dx ( div(div η ) ) q dx + p βp η q 2 η 2 v } plp. Assume that the r.h.s. of (3.0) is zero for some triple (v, η, β). Then 2 v = η q 2 η together with div(div η ) = 0 on v > Ψ, div(div η ) = 0 on v = Ψ.

9 Duality based a posteriori error estimates for higher order variational inequalities 483 This implies for any w K 2 v p 2 2 v : 2 (w v) dx = = = η : 2 (w v) dx div(div η )(w v) dx v=ψ and therefore v coincides with the solution u of (P). div(div η ) (w Ψ) dx 0, Variant 2. As an alternative to (3.0) we again follow ideas of Re and estimate d(η ) q on the r.h.s. of (3.) by Cp() q λ div(div η ) q (recall (3.2)) and then try L to find λ Λ s.t. q λ(v Ψ) dx + K p (, β) λ div(div η ) q dx becomes minimal for a fixed triple (v, η, β). pointwise minimization of the function f(t) := t ( v(x) Ψ(x) ) + K t δ(x) q Of course this can be achieved by on 0, ). Here K := K p (, β), δ := div(div η ), and in the following we will omit the fixed argument x. Note that f is strictly convex on R and f(t) + as t ±, thus there is a unique number t 0 R s.t. f(t 0 ) = R f. From f (t 0 ) = 0 it follows that 0 = v Ψ + Kq t 0 δ q 2 (t 0 δ), and since v Ψ, we must have t 0 δ, hence (δ t 0 ) q = Kq (v Ψ), i.e. t 0 = δ qk (v Ψ) q. Case. t 0 < 0. Since f is strictly increasing on t 0, ), we get Case 2. t 0 0. Then we have min t 0 f(t) = f(0) = K δ q. min t 0 f(t) = f(t 0) For v K, η Y s.t. div(div η ) () and β > 0 we define the sets + := { x : div(div η )(x) := + qk p (, β) ( v(x) Ψ(x) ) q },

10 484 Michael Bildhauer, Martin Fuchs and Sergey Repin and set λ = 0 on and λ = (v(x) Ψ(x)) q qk p on +. We further introduce (,β) the quantity ε(v, η, β) := K p (, β) div(div η ) q dx ( { v Ψ } ) + (v Ψ) div(div η q v Ψ } p ) + K p (, β){ dx. qk p (, β) qk p (, β) + With the above choice of λ it is immediate that µ(v Ψ) dx + K p (, β) µ Λ µ div(div η ) q dx ε(v, η, β), and for p = 2 this estimate reduces to the one given in Remark 2 of Re. Summing up we arrive at Corollary 3.2. With the notation introduced above we have in case p 2 (3.) 2 (u v) p p 2 D p L p p 2 v, η + p βp η q 2 η 2 v p +ε(v, η, β) L p valid for all v K, for any η Y such that div(div η ) () and for any choice of β > 0. Note that the r.h.s. of (3.) vanishes only on the exact solution. Estimate (3.) gives an upper bound on the error related to the approximations of variational inequalities. It should be emphasized that it does not require a priori knowledge of the form of the unknown free boundaries. Elimination of the quantity div(div η ). Let us finally reconsider the quantity d(η ) := τ Q λ η τ for a tensor η Y and a function λ Λ. We have (3.2) τ Q λ q η τ q L = q ρ Y sup L (w, ρ ), where In fact, it holds L (w, ρ ) := sup w W 2 p w W 2 p() π p (η ρ ) + ρ : 2 w λw dx. +, L (w, ρ if ρ / Q λ, ) = πp(η ρ ) dx, if ρ Q λ, and this implies (3.2). Now, using standard results from duality theory, we have (3.3) sup ρ Y w W 2 p L (w, ρ ) = sup L (w, ρ Y ρ ). w W 2 p

11 Since Duality based a posteriori error estimates for higher order variational inequalities 485 we get from (3.2) and (3.3) (3.4) τ Q λ q η τ L (w, ρ Y ρ ) = L (w, κ Y η κ ), = sup w W κ Y 2 p π p (κ ) + (η κ ) : 2 w λw dx. Note that (3.4) corresponds to formula (4.) established in NR2 for linear equations related to the biharmonic operator. Proceeding as in this reference, we write π p (κ κ Y ) + (η κ ) : 2 w λw dx = sup κ : 2 w πp(κ ) dx + η : 2 w λw dx κ Y = πp ( 2 w) η : 2 w + λw dx. Inserting this into (3.4) we have shown that (3.5) τ Q λ q η τ q L = q πp ( 2 w) η : 2 w + λw dx. w W 2 p If div(div η ) (), then (3.5) reduces to (3.7), and we arrive at (3.9). Without further ormation concerning η (3.5) just states that the quantity d(η ) can be obtained by solving an auxiliary variational problem, which means to compute a lower bound for the functional w πp ( 2 w) η : 2 w + λw dx defined on the space W 2 p(). Here of course no side condition enters but for each choice of η and λ a new problem has to be considered. A rather natural assumption concerning η Y is the requirement that div η = ( ) α ηαβ is in the space (). Then πp ( 2 w) η : 2 w + λw dx = = β 2 πp ( 2 w) + div η w + λw dx πp ( 2 w) + (div η y ) w + (λ div y )w dx,

12 486 Michael Bildhauer, Martin Fuchs and Sergey Repin where y is a vector-function from () such that div y (). From (3.5) we get by Hölder s inequality (3.6) q d(η ) q w W 2 p() p 2 w p L p div η y w L p λ div y w L p. From the Poincaré inequality we have for all v W p() (3.7) v L p K p () v L p and applying (3.7) to w W 2 p() as well as to the vectorial function w W p() we see that the r. h. s. of (3.6) is bounded from above by w W 2 p() p 2 w p L p K p() div η y 2 w L p K p () 2 λ div y 2 w L p t 0 p tp K p () div η y + K p () λ div y t = q K p() q div η y + K p () λ div y q. Let us summarize our results: Theorem 3.2. Suppose that we are given η () and λ Λ. Then we have τ Q λ q η τ q L = q πp ( 2 w) η : 2 w + λw dx. w W 2 p () If in addition div η (), then τ Q λ q η τ q, q K p() q div η y + K p () λ div y where y is any vector-function in () s.t. div y () and where K p () is defined according to (3.7). From the proof it is immediate that the statements of Theorem 3.2 are also valid in the case < p < 2.

13 Duality based a posteriori error estimates for higher order variational inequalities Estimates for the subquadratic case Obviously we have for τ Q λ Jλτ = l(w, τ ) λ(w Ψ) dx w W 2 p() w K l(w, τ ) λ(w Ψ) dx w K l(w, τ ) dx = J τ, moreover J σ = Ju Jw for any w K, hence we get from (2.3) (4.) τ σ q L Jw q q 2q Jλτ. Observe that (4.) exactly corresponds to (3.3), and as outlined in Section 3 inequality (4.) can be rewritten as } (4.2) τ σ q L {D q q 2q p 2 w, τ + λ(w Ψ) dx valid for all τ Q λ, w K and λ Λ, D p 2 w, τ having the same meaning as before. If now η Y and τ Q λ, then (using (4.2)) η σ ( σ q τ + η τ q ) ( σ 2 q τ q + L η τ ) (4.3) q q L q q 4 D q p 2 w, τ + λ(w Ψ) dx + 2 q η τ q. The quantity D p 2 w, τ can be estimated as before: D p 2 w, τ D p 2 w, η + τ q 2 τ 2 w : ( τ η ) dx = D p 2 w, η + τ q 2 τ η q 2 η : (τ η ) dx (4.4) + η q 2 η 2 w : (τ η ) dx =: D p 2 w, η + I + I 2. According to the calculations after (3.5) in BFR we have I (q ) τ η 2 τ η + 2 η q 2, and for I 2 we get with Hölder s and Young s inequality I 2 β 2 η q 2 η 2 w 2 + L τ η 2, p 2β

14 488 Michael Bildhauer, Martin Fuchs and Sergey Repin where β > 0 is arbitrary. If we insert these estimates in (4.4) and return to (4.3) it is shown that η σ q q 4 q D p 2 w, η + (q ) τ η 2 ( τ η + 2 η + β 2 + η q 2 η 2 w 2 + L p 2β τ η 2 λ(w Ψ) dx + 2 q η τ q. ) q 2 Now we may pass to the imum w.r.t. τ Q λ with the result: Theorem 4.. Let < p < 2. Then for all η Y, for any w K and for all λ Λ, β > 0 the following error estimate holds: σ η q q 4 q D p 2 w, η + β η q 2 η 2 w 2 2 L p + q 4 q 2β + (q )( ) d(η ) + 2 η q 2 (4.5) d(η ) q d(η ) q + q 4 q λ(w Ψ) dx, where d(η ) := τ Q τ η λ. Remark 4.. i) With exactly the same arguments as used in the proof of Theorem 3. we obtain the estimate (3.9) for p < 2, i.e. (4.6) d(η ) C p () λ div(div η ), provided that in addition div(div η ) (). Starting from (4.5) and using (4.6) it is also immediate how to get variants of Corollaries 2. and 2.2 for the present situation: the arguments used for proving (3.0) and (3.) do not depend on the value of p. ii) If the r.h.s. of (4.5) vanishes for a triple (w, η, λ), then w = u and η = σ. 5. Concluding remarks Our estimates formally suffer from the fact that we are only allowed to insert functions v from the class K. If w W 2 p() is arbitrary, then we choose v K s.t. 2 v 2 w L p = 2 v 2 w L p, v K use v in our estimates, and then we have to find a bound for 2 v 2 w L p. In the first order case we have v w L p ( /p, w max(w, Ψ) L p = w Ψ dx) p w Ψ

15 Duality based a posteriori error estimates for higher order variational inequalities 489 but unfortunately we did not obtain a comparable result in our situation which is caused by the fact that in general max(w, Ψ) is not in W 2 p(). However, from the computational point of view the difficulties arising if v / K are not very serious because one can easily modify (post-process) an approximate solution such that it belongs to the class K. Our choice of the class K means that we consider functions v vanishing on and having in addition zero gradient (in the trace sense) on the boundary. Alternatively we could choose functions v from the class W p Wp 2 () satisfying v Ψ but the energy J is not coercive on this space, so that one would have to replace J by the functional Jv := 2 v p dx + v p dx p p in order to get a well posed problem. More general, let < p, r <, let Ψ as before and define { L := v Wp 2 () W } r() : v Ψ as well as Kv := 2 v p dx + v r dx. p r Then the problem K min on L makes sense and error estimates are available. Moreover, for this new problem in which only the trace of the function itselve is prescribed, it is possible to overcome the difficulty arising in measuring the distance of an arbitrary function to the admissible class of comparison functions. The details will be given in the forthcoming paper BF2. References Ad Adams, R. A.: Sobolev spaces. - Academic Press, New York San Francisco London 975. BF BF2 BR BFR Bildhauer, M., and M. Fuchs: Higher order variational inequalities with non-standard growth conditions in dimension two: plates with obstacles. - Ann. Acad. Sci. Fenn. Math. 26, 200, Bildhauer, M., and M. Fuchs: Error estimates for obstacle problems of higher order. - Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 348, 2007, 5 8. Bildhauer, M., and S. Repin: Estimates for the deviation from exact solutions of variational problems with power growth functionals. - Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 336, 2006, Bildhauer, M., M. Fuchs, and S. Repin: A posteriori error estimates for stationary slow flows of power-law fluids. - J. Non-Newtonian Fluid Mech. 42, 2007, Cl Clarkson, J. A.: Uniformly convex spaces. - Trans. Amer. Math. Soc. 40, 936, ET Fr Ekeland, I., and R. Temam: Convex analysis and variational problems. - North-Holland, Amsterdam, 976. Friedman, A.: Variational principles and free-boundary problems. - John Wiley and Sons, 982.

16 490 Michael Bildhauer, Martin Fuchs and Sergey Repin FLM MM NR NR2 Re Se Fuchs, M., G. Li, and O. Martio: Second order obstacle problems for vectorial functions and integrands with subquadratic growth. - Ann. Acad. Sci. Fenn. Math. 23, 998, Mosolov, P. P., and V. P. Mjasnikov: Mechanics of rigid plastic media. - Nauka, Moscow, 98 (in Russian). Neittaanmäki, P., and S. Repin: Reliable methods for computer simulation: error control and a posteriori estimates. - Elsevier, New York, Neittaanmäki, P., and S. Repin: A posteriori error estimates for boundary-value problems related to the biharmonic operator. - East-West J. Numer. Math. 9, 200, Repin, S.: Estimates of deviations from exact solutions of elliptic variational inequalities. - J. Math. Sci. 5, 2003, Seregin, G.: Differentiability properties of weak solutions of certain variational problems in the theory of perfect elastoplastic plates. - Appl. Math. Optim. 28, 993, Received 23 August 2007