Table Of Content

THE DECOMPOSITION-COORDINATION METHOD FOR THE p(x)-LAPLACIAN 4 1 LEANDRODEL PEZZO ANDSANDRAMARTÍNEZ 0 2 Abstract. In this paper we construct two algorithms to approximate the minimizer of a dis- n a crete functional which comes from using a discontinuous Galerkin method for a variational J problemrelatedtothep(x)-Laplacian. Wealsomakesomenumericalexperimentsindimension 0 two. 1 ] A N 1. Introduction . h Thisworkisdevotedtodevelopingandanalysingtwoalgorithmstoapproximatetheminimizer t a ofadiscretefunctionalwhichcomesfromusingadiscontinuousGalerkinmethodforanonlinear, m nonhomogeneousvariationalproblem. Thisvariationalproblemisrelatedtoanimageprocessing [ model of Chen, Levin and Rao [8], see also [3]. 1 More precisely, we consider the following nonlinear variational problem: v 2 Find u := v W1,p(·)(Ω): v u W1,p(·)(Ω) such that 4 (P) ∈ A n ∈ − D ∈ 0 o 4 J(u) = minJ(v), 2 v∈A . 1 where 0 4 J(v) := v p(x) + v ξ 2dx, 1 Z |∇ | | − | Ω : v Ω is a bounded connected open set in RN with Lipschitz continuous boundary, p : Ω [p ,p ] i 1 2 X is a log-Hölder continuous function with 1 < p p 2, u W1,p(·)(Ω) and ξ → L2(Ω). r 1 ≤ 2 ≤ D ∈ 0 ∈ a It is well-known that the functional J admits a unique minimizer u . For the definitions ∈ A of the log-Hölder continuous function and the variable exponent Sobolev spaces W1,p(·)(Ω) and 1,p(·) W (Ω), see Section 2. 0 Note that this functional is related to the so-called p(x) Laplacian operator, that is − ∆ = div( up(x)−2 u). p(x) |∇ | ∇ ThisoperatorextendstheclassicalLaplacian(p(x) 2)andthep Laplacian(p(x) p,1 < p < ≡ − ≡ + ). The interest in this operator was originally motivated by the model for electrorheological ∞ fluids, see [27, 28]. Key words and phrases. variable exponent spaces, decomposition coordination methods, discontinuous Galerkin. 2010 Mathematics Subject Classification. 65k10, 35J20, 65N30 and 46E35. Supportedby ANPCyTPICT 2006-290, UBA X078, UBA X117 . The authors are members of CONICET. 1 2 L.M.DELPEZZO&S.MARTÍNEZ In [12], the so-called discontinuous Galerkin method is considered to approximate the minimizer of (P). More precisely, the authors study the following discrete functional, I (v ) := v +R (v )p(x)+ v ξ q(x)dx+ v u p(x)h1−p(x)dS h h h h h h h D Z |∇ | | − | Z | − | Ω ∂Ω + [[v ]]p(x)h1−p(x)dS, h Z | | Γint where h is the local mesh size, h is the global mesh size, Γ is the union of the interior edges of int the elements, [[v ]] is the jump of the function between two edges, v denotes the elementwise h h ∇ gradient of v and R is the lifting operator, see Section 2 for a precise definition. Observe that h h the boundary condition is weakly imposed by the second term of the functional. With this setting, the discrete problem is to find a minimizer u of J over the space Sk( ) h h h T of all the functions that are polynomials of degree at most k in each element, with k 1, see ≥ Subsection 2.2 for details. In [12], the authors prove the following result. Theorem 1.1. Let Ω be a polyhedral domain, u W2,2(Ω), and u Sk( ) be the minimizer D h h ∈ ∈ T of J for any h (0,1]. If u is the minimizer of J then h ∈ I (u ) J(u), R (u ) 0, u u strongly in W1,p(·)(Ω), and h h h h h → → → u u p(x)h1−p(x)dS + [[u ]]p(x)h1−p(x)dS 0 h D h Z | − | Z | | → ∂Ω Γint Since we want to implement this method for some examples, the next step is to find a good approximation of the minimizer of the discrete functional. The methods for finding minimizers of functionals, such as the BFGS Quasi- Newton, work when the dimension of the space is small. However, we observe that these methods are to slow when we refine the mesh. We also observe, in some numerical experiments, that the decomposition– coordination–method (DCM), defined in [23, Chapter VI], is more suitable for our problem. The DCM is used to approximate the minimizers of functionals that can be written in the form J(v) = F(Bv)+G(v), where F : H R, G : V R are convex functions, B :V H is a linear operator and V and → → → H are topological vector spaces. In this context, the problem of finding minimizers of J over V is equivalent to find (q ,v ) 0 0 ∈ W := (q,v) V H : Bv q = 0 such that { ∈ × − } (1.1) F(q )+G(v )= min F(q)+G(v) . 0 0 (q,v)∈W{ } In the practical applications, under the following assumptions (H1) F : H R, G : V R are lower semicontinuous functions and → → dom(F B) dom(G) = ; ◦ ∩ 6 ∅ THE DECOMPOSITION-COORDINATION METHOD FOR THE p(x)-LAPLACIAN 3 (H2) F is a convex Gateaux–diffentiable functional and F(q) lim = ; |q|→∞ q ∞ | | (H3) The rank of B is close in H; (H4) B injective; In [23] the authors prove that there exists a sequence (un,qn) V H such that un solves a { } ⊂ × linear differential equation, qn solves a non-linear equation, and un v = u strongly in V 0 → qn q = B(u) strongly in H 0 → where u is the minimizer of J. If we write the functional I in this form, we have that h I (v) = F(Bv)+G(v) h where here V = Sk( ), H = Sl( ) Sl( ), k,l N with l k 1, Bv = R (v)+ v, h h h 0 h T T × T ∈ ≥ − ∇ F(q) = q p(x)dx and Z | | Ω G(v) = v ξ q(x)dx+ v u p(x)h1−p(x)dS + [[v]]p(x)h1−p(x)dS. D Z | − | Z | − | Z | | Ω ∂Ω Γint We can observethat, (H1), (H2), (H3) hold, but(H4) doesnot hold, that is B is notinjective. Moreover, G′ is not linear. To prove the convergence of the method, it is not necessary the assumption that G′ is linear, but it is useful for the implementation. For this reason and since we are interested in the case p(x) 2, we define a new discrete functional ≤ J (v) = F(Bv)+G(v) h where now G(v) = v ξ 2dx+ v u 2h−2/p′(x)dS + [[v ]]2h−2/p′(x)dS, h h D h Z | − | Z | − | Z | | Ω ∂Ω Γint and F and B are defined as before. In this manner, G′ is linear. Observe that, here we have to change the power over the function h. To overcome the lack of injectivity of the functional B, we will use that our functional G is Gateaux-differentiable and convex. Now, we are ready to state the main results of this paper. Since we change the discrete functional we have to prove a result similar to Theorem 1.1. More precisely, we prove the following theorem. Theorem 1.2. Let Ω be a polygonal domain in RN, p : Ω [p1,2] (N/2 < p1 2) be a log- → ≤ Hölder continuous and u W2,2(Ω). For each h (0,1], let u Sk( ) be the minimizer of D h h ∈ ∈ ∈ T 4 L.M.DELPEZZO&S.MARTÍNEZ J . If u is the minimizer of J then h u u strongly in Ls(·)(Ω) s , h → ∀ ∈K u u strongly in L2(∂Ω), h → J (u ) J(u), h h → R (u ) 0, h h → u u 2h−2/p′(x)dS + [[u ]]2h−2/p′(x)dS 0, h D h Z | − | Z | | → ∂Ω Γint u u strongly in Lp(·)(Ω), h ∇ → ∇ where = s L∞(Ω): 1 s(x)< p∗(x) ε for some ε> 0 . K { ∈ ≤ − } We define two algorithms that construct a sequence {unh}n∈N that approximates, for each h 0 the minimizers of J and finally we prove the convergence of both algorithms. h ≥ Theorem 1.3. Let h 0 and (u ,η ,λ ) V H H be a saddle-point of . If h h h r ≥ ∈ × × L (1.2) 0< α ρ α < 2r 0 n 1 ≤ ≤ and (un,ηn,λn) V H H is the solutions given by Algorithm 1 then h h h ∈ × × un u in V, h → h ηn η = Bu in H, h → h h λn+1 λn 0 in H, h − h → λn is bounded. h Moreover, [[un u ]] 0 in L2(Γ ), h − h → int un u in L2(∂Ω), h → h R(un) R(u ) in H, h → h un u in H. ∇ h → ∇ h Theorem 1.4. Let h 0 and (u ,η ,λ ) V H H be a saddle-point of . If h h h r ≥ ∈ × × L (1+√5) (1.3) 0< ρ = ρ < r n 2 and (un,ηn,λn) V H H is the solutions given by Algorithm 2 then h h h ∈ × × ηn η = Bu in H, h → h h λn+1 λn 0 in H, h − h → λn is bounded . h THE DECOMPOSITION-COORDINATION METHOD FOR THE p(x)-LAPLACIAN 5 Moreover, un u in V, h → h [[un u ]] 0 in L2(Γ ), h − h → int un u in L2(∂Ω), h → R(un) R(u ) in H, h → h un u in H. ∇ h → ∇ h Let us end the introduction with a brief comments on previous bibliography. In [4, 10], the convergenceofconformingfiniteelementmethodapproximationsfortheDirichletproblemofthe p() Laplacian is studied. Moreover, in [10], we study the convergence rate using the regularity · − results obtained in [11]. Finally, we want to mention that in [3] and [8] the authors find an approximation of the solutions by using an explicit finite difference scheme for the associated parabolic problem. Outline of the paper. In Section 2, we state several properties of the variable exponent Sobolev spaces, we give some definitions and properties related to the mesh and to the broken Sobolevspaces; Insection3,weproveTheorem1.2; Insection4, thedecomposition-coordination method is studied and the convergence of the algorithms are proved; Finally, in Section 5, we give some numerical examples. 2. Preliminaries We begin with a review of the basic results that will be needed in subsequent sections. The resultsaregenerallystatedwithoutproof,althoughweattempttoprovidegoodreferenceswhere the proofs can be found. Also, we introduce some of our notational conventions. 2.1. The spaces Lp(·)(Ω) and W1,p(·)(Ω). WefirstintroducethespacesLp(·)(Ω)andW1,p(·)(Ω) and state some of their properties. Let p: Ω [p ,p ] be a measurable bounded function, called a variable exponent on Ω where 1 2 → p := essinf p(x) and p := esssup p(x) with 1 p p < . 1 2 1 2 ≤ ≤ ∞ WedefinethevariableexponentLebesguespaceLp(·)(Ω)toconsistofallmeasurablefunctions u: Ω R for which the modular → ̺ (u) := u(x)p(x)dx p(·) Z | | Ω is finite. We define the Luxemburg norm on this space by kukLp(·)(Ω) = kukp(·) := inf{k > 0: ̺p(·)(u/k) ≤ 1}. This norm makes Lp(·)(Ω) a Banach space. The following properties can be obtained directly from the definition of the norm. For the proof see [21, Theorem 1.3 and Theorem 1.4 ]. Proposition 2.1. If u,u Lp(·)(Ω), u = λ, then n p(·) ∈ k k (1) λ < 1 (= 1,> 1) iff u(x)p(x)dx < 1 (= 1,> 1); Z | | Ω (2) If λ 1, then λp1 u(x)p(x)dx λp2; ≥ ≤ Z | | ≤ Ω 6 L.M.DELPEZZO&S.MARTÍNEZ (3) If λ 1, then λp2 u(x)p(x)dx λp1; ≤ ≤ Z | | ≤ Ω (4) u (x)p(x)dx 0 iff u 0. n n p(·) Z | | → k k → Ω For the proofs of the following two theorems we refer the reader to [26]. Theorem 2.2. Let q(x) p(x), then Lp(·)(Ω)֒ Lq(·)(Ω) continuously. ≤ → Theorem 2.3. Let p′(x) such that, 1/p(x)+1/p′(x) = 1. Then Lp′(·)(Ω) is the dual of Lp(·)(Ω). Moreover, if p > 1, Lp(·)(Ω) and W1,p(·)(Ω) are reflexive. 1 Now we give some well known inequalities. Proposition 2.4. For any x fixed we have the following inequalities η ξ p(x) C(η p(x)−2η ξ p(x)−2ξ)(η ξ) if p(x) 2, | − | ≤ | | −| | − ≥ p(x)−2 η ξ 2 η + ξ C(η p(x)−2η ξ p(x)−2ξ)(η ξ) if p(x) < 2, | − | | | | | ≤ | | −| | − (cid:16) (cid:17) η p(x) 2p(x)−1(η ξ p(x)+ ξ p(x)) if p(x) 1. | | ≤ | − | | | ≥ The following properties will be used throughout the paper. Proposition 2.5. Let F ,F Lp(·)(Ω). n ∈ (1) If F ⇀ F weakly in Lp(·)(Ω) n then F p(x)dx liminf F p(x)dx. n Z | | ≤ n→∞ Z | | Ω Ω (2) If F F strongly in Lp(·)(Ω) n → then F p(x)dx F p(x)dx. n Z | | → Z | | Ω Ω (3) If F ⇀ F weakly in Lp(·)(Ω) and F p(x)dx F p(x)dx n n Z | | → Z | | Ω Ω then F F strongly in Lp(·)(Ω). n → Proof. For the proof of (1) and (3) see [16, Theorem 3.9 and Lemma 2.4.17]. Finally (2) follows by [18, Proposition 2.3]. (cid:3) Let W1,p(·)(Ω) denote the space of measurable functions u such that, u and the distributional derivative u are in Lp(·)(Ω). The norm ∇ kukW1,p(·)(Ω) = kuk1,p(·) := kukp(·)+k|∇u|kp(·) makes W1,p(·)(Ω) a Banach space. We define the space W1,p(·)(Ω) as the closure of C∞(Ω) in W1,p(·)(Ω). 0 0 We now introduce the most important condition on the exponent in the study of variable exponent spaces, the log-Hölder continuity condition. THE DECOMPOSITION-COORDINATION METHOD FOR THE p(x)-LAPLACIAN 7 Definition 2.6. We say that a function α : Ω R is log-Hölder continuous if there exists a → constant C such that log C log α(x) α(y) x,y Ω. | − | ≤ log e+ 1 ∀ ∈ |x−y| (cid:16) (cid:17) For example, it was proved in [14, Theorem 3.7], that if one assumes that ∂Ω is Lipschitz and p : Ω [1,+ ) is log-Hölder continuous then C∞(Ω¯) is dense in W1,p(·)(Ω). See also → ∞ [13, 17, 19, 26, 29]. We now state two Sobolev embedding Theorems. Here, p∗ and p are the Sobolev critical ∗ exponents for these spaces, i.e. p(x)N p(x)(N 1) if p(x) < N, − if p(x)< N, p∗(x) :=N p(x) and p (x) :=  N p(x) ∗ − − + if p(x) N, + if p(x) N. ∞ ≥ ∞ ≥ For the proofs of the following theorems see [15] and [20,Corollary 2.4], respectively. Theorem 2.7. Let Ω be a Lipschitz domain. Let p : Ω [1, ) be a log-Ho¨lder continuous function. Then the embedding W1,p(·)(Ω) ֒ Lp∗(·)(Ω) is co→ntinu∞ous. → Theorem 2.8. Let Ω be a bounded Lipschitz domain. Suppose that p C0(Ω) with p > 1. If 1 ∈ r C0(∂Ω) satisfies the condition 1 r(x) < p (x) for all x ∂Ω, then there is a compact ∗ ∈ ≤ ∈ boundary trace embedding W1,p(·)(Ω)֒ Lr(·)(∂Ω). → 2.2. Themesh andpropertiesofW1,p(·)( ). Wenowgivesomedefinitionsandproperties h h T T related to the mesh and to the broken Sobolev space. Hypothesis 2.9. Let Ω be a bounded polygonal domain and ( ) be a family of partitions h h∈(0,1] T of Ω into polyhedral elements. We assume that there existsa finite numberof reference polyhedral κˆ ,...,κˆ such that for all κ there exists an invertible affine map F such that, κ= F (κˆ ). 1 r h κ κ i ∈ T We assume that each κ is closed and that diam(κ) h for all κ . h h ∈ T ≤ ∈T Now we give some notation, := κ κ′: dim (κ κ′)= N 1 κ ∂Ω: dim (κ ∂Ω) = N 1 , h H H E { ∩ ∩ − }∪{ ∩ ∩ − } Γ := e := dim (e ∂Ω)< N 1 , int h H { ∈ E ∩ − } [ where dim is the Hausdorff dimension. H We also assume that the mesh satisfies the following hypotheses. Hypothesis 2.10. The family of partitions ( ) satisfies the Hypothesis 2.9 and h h∈(0,1] T (a) There exist positive constants C and C ,independent of h, such that for each element 1 2 κ h ∈ T C hN κ C hN. 1 κ ≤ | | ≤ 2 κ (b) There exists a constant C > 0 such that for all h (0,1] and for all face e there 1 h ∈ ∈ E exists a point x e and a radius ρ C diam(e) such that B (x ) A e, where A e ∈ e ≥ 1 ρe e ∩ e ⊂ e is the affine hyperplane spanned by e. Moreover, there are positive constants such that chκ he Chκ, chκ′ he Chκ′ ≤ ≤ ≤ ≤ where e = κ κ′. ∩ 8 L.M.DELPEZZO&S.MARTÍNEZ Now, we introducethe finiteelement spaces associated with . We definethevariable broken h T Sobolev space as W1,p(·)( ) := u L1(Ω): u W1,p(·)(κ) for all κ , h κ h T { ∈ | ∈ ∈ T } and the subspaces Uk( ) := u C(Ω): u Pk for all κ , h κ h T { ∈ | ∈ ∈ T } Sk( ) := u L1(Ω): u Pk for all κ , h κ h T { ∈ | ∈ ∈ T } where Pk is the space of polynomials functions of degree at most k. For each face e , e Γ we denote by κ+ and κ− its neighboring elements. We write h int ∈ E ⊂ ν+,ν− todenotetheoutwardnormalunitvectorstotheboundaries∂κ±,respectively. Thejump of a function u W1,p(·)( ) and the average of a vector-valued function φ (W1,p(·)( ))N, h h ∈ T ∈ T with traces u±, φ± from k± are, respectively, defined as the vectors φ++φ− [[u]] := u+ν++u−ν− and φ := . { } 2 Let h: ∂Ω Γ R a piecewise constant function define by int ∪ → h(x) := diam(e) if x e, ∈ where e . h ∈ E We consider the following seminorm on W1,p(·)( ), h T −1 |u|W1,p(·)(Th) := k∇ukLp(·)(Ω) +k[[u]]hp′(x)kLp(·)(Γint). 2.3. The lifting operator. Finally we define, as in [6] (see also [1]), the lifting operator. Definition 2.11. Let l 0 and R : W1,p(·)( ) Sl( )N defined as, h h h ≥ T → T R (u),φ dx := [[u]], φ dS h Z h i −Z h { }i Ω Γint for all φ Sl( )N. h ∈ T ThisoperatorappearsinthefirsttermofthediscretizedfunctionalJ . Aswecanseefromthe h definition, this operator represents the contribution of the jumps to the distributional gradient. This is the reason why it is crucial to add this term in order to have the consistency of the method. Wepointoutthatthisliftingoperatorwasfirstusedin[2]inordertodescribethecontributions of the jumps across the interelements of the computed solution on the (computed) gradient of the solution in a mixed formulation of Navier-Stokes equations. It was also used in [5] where a solid mathematical background for the method introduced in [2] was proposed. Now, we state a bound of the Lp(·)(Ω)-norm of R (u) in terms of the jumps of u in Γ . For h int the proof see [12]. Lemma 2.12. Let p : Ω : [1, ) be a log- Hölder continuous in Ω. Then, there exists a → ∞ constant C such that, kRh(u)kLp(·)(Ω) ≤ Ckh−1/p′(x)[[u]]kLp(·)(Γint) ∀u∈ W1,p(·)(Th) ∀h ∈(0,1]. THE DECOMPOSITION-COORDINATION METHOD FOR THE p(x)-LAPLACIAN 9 3. Convergence of the discontinuous Galerkin FEM In this section we prove the convergence of the discontinuous Galerkin FEM. From now on, we make the following assumption: Let Ω be a bounded polygonal domain in RN and p :Ω [p ,2] (1 < p 2) be a log-Hölder continuous function. 1 1 → ≤ Our next result follows by using Lemma 2.12 and the fact that L2(Γ ) Lp(·)(Γ ). int int ⊂ Lemma 3.1. There exists a constant C such that kRh(v)kLp(·)(Ω) ≤ Ckh−1/p′(x)[[v]]kL2(Γint) for all v W1,p(·)( ) and for all h (0,1]. h ∈ T ∈ Now, we prove the coercivity of the functional. Theorem 3.2. For each h (0,1], let v W1,p(·)( ). If there exists a constant C independent h h ∈ ∈ T of h such that J (v ) C for all h (0,1], then h h ≤ ∈ sup kvhkL1(Ω)+|vh|W1,p(·)(Th) < ∞. h∈(0,1](cid:16) (cid:17) Moreover, sup v u p(x)h1−p(x)dS < . h D Z | − | ∞ h∈(0,1] ∂Ω Proof. Since J (v ) C, we have that h h ≤ [[v ]]2h−2/p′(x)dS C h Z | | ≤ Γint then, by Lemma 3.1, kRh(vh)kLp(·)(Ω) is bounded. Therefore, using Proposition 2.4, we have J (v )+C C v p(x)dx+ v u 2h−2/p′(x)dS + [[v ]]2h−2/p′(x)dS. h h h h D h ≥ Z |∇ | Z | − | Z | | Ω ∂Ω Γint By the above inequality and the fact that L2 Lp(·) we get ⊂ v p(x)dx C, h Z |∇ | ≤ Ω v u p(x)h1−p(x)dS C, h D Z | − | ≤ ∂Ω [[v ]]p(x)h1−p(x)dS C. h Z | | ≤ Γint Finally, the proof follows as in the end of the proof of Theorem 6.2 in [12]. (cid:3) The following theorem was proved in [12]. Theorem 3.3. Let u Sk( ) be such that h h ∈ T h∈su(0p,1](cid:16)kuhkL1(Ω)+|uh|W1,p(·)(Th)(cid:17)< ∞ and h∈su(0p,1]Z∂Ω|uh−uD|p(x)h1−p(x)dS < ∞. 10 L.M.DELPEZZO&S.MARTÍNEZ Then, there exist a sequence h 0 and a function u W1,p(·)(Ω) such that j → ∈ ∗ u ⇀ u weakly* in BV(Ω) hj u +R (u )⇀ u weakly in Lp(·)(Ω), ∇ hj h hj ∇ u u strongly in Lp(·)(∂Ω), hj → u u strongly in Ls(·)(Ω) s , hj → ∀ ∈ K where = s L∞(Ω): 1 s(x)< p∗(x) ε for some ε> 0 . K { ∈ ≤ − } Before proving the convergence of the sequence of mimizers, we need an auxiliary lemma. Lemma 3.4. Let h (0,1] and u W2,2(Ω). If v W2,2(Ω) , then there exists v U1( ) D h h ∈ ∈ ∈ ∩A ∈ T such that v v 0 as h 0, h H1(Ω) k − k → → and J (v ) J(v) as h 0. h h → → Proof. Given v W2,2(Ω) , by standard approximation theory (see [9, Theorem 3.1.5]), we ∈ ∩A have that there exists v U1( ) such that h h ∈ T v v 0 h H1(Ω) k − k → as h 0, and → v v Ch D2v . h L2(∂Ω) L2(Ω) k − k ≤ k k Since 1 < p p(x) 2, we have 1 ≤ ≤ Z |v−vh|2h−2/p′(x)dS ≤ Ch−2/p1Z |v−vh|2dS ≤ Ch−2/p1h2kD2vk2L2(Ω) → 0 ∂Ω ∂Ω as h 0. → Finally,sincev W1,p(·)(Ω),wehavethat[[v ]] = 0andR (v )= 0. Then,usingProposition h h h h ∈ 2.5, we have J (v )= v p(x)+ v ξ 2dx+ v u 2h−2/p′(x)dS ( v p(x)+ v ξ 2)dx h h h h h D Z |∇ | | − | Z | − | → Z |∇ | | − | Ω ∂Ω Ω as h 0. The proof is complete. (cid:3) → Now we prove Theorem 1.2. Proof of Theorem 1.2. By Lemma 3.4, there exists w U1( ) such that J (w ) J(u ). h h h h D ∈ T → Then (3.4) J (u ) J (w ) C h> 0. h h h h ≤ ≤ ∀ By Theorem 3.2 and Theorem 3.3, we obtain that there exists a subsequenceu of u such that hj h u u strongly in Ls(·)(Ω) s hj → ∀ ∈ K u u strongly in Lp(·)(∂Ω), hj → u +R (u ) ⇀ u weakly in Lp(·)(Ω). ∇ hj h hj ∇ On the other hand, by (3.4) u u 2h −2/p′(x)dS C Z | hj − D| j ≤ ∂Ω