مرکزی صفحہ
Abstract and Applied Analysis Variational inequalities for energy functionals with nonstandard growth conditions
Variational inequalities for energy functionals with nonstandard growth conditions
Fuchs, Martin, Gongbao, Liآپ کو یہ کتاب کتنی پسند ہے؟
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جلد:
3
سال:
1998
زبان:
english
رسالہ:
Abstract and Applied Analysis
DOI:
10.1155/s1085337598000438
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PDF, 1.94 MB
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VARIATIONAL INEQUALITIES FOR ENERGY FUNCTIONALS WITH NONSTANDARD GROWTH CONDITIONS MARTIN FUCHS AND LI GONGBAO∗ Abstract. We consider the obstacle problem minimize I(u) = Ω G(∇u)dx among functions u : Ω → R such that u∂Ω = 0 and u ≥ Φ a.e. for a given function Φ ∈ C 2 (Ω̄), Φ∂Ω < 0 and a bounded Lipschitz domain Ω in Rn . The growth properties of the convex integrand G are described in terms of a N function A : [0, ∞) → [0, ∞) with limt→∞ A(t)t−2 < ∞. If n ≤ 3, we prove, under certain assumptions on G, C 1,α partial regularity for the solution to the above obstacle problem. For the special case where A(t) = tln(1 + t) we obtain C 1,α partial regularity when n ≤ 4. One of the main features of the paper is that we do not require any power growth of G. 1. Introduction In this paper we discuss the obstacle problem (1.1) to minimize I(u) = Ω G(∇u)dx among functions u : Ω → R s.t. u∂Ω = 0 and u ≥ Φ a.e. for a given function Φ ∈ C 2 (Ω̄) with the property Φ∂Ω < 0, where Ω ⊂ Rn is a bounded Lipschitz domain. The integrand G : Rn → R is assumed to be of class C 2 and locally coercive in the sense that D2 G(P )(Q, Q) ≥ λ(P )Q2 , ∀P, Q ∈ Rn , 1991 Mathematics Subject Classiﬁcation. 49N60, 35J70, 46E35. Key words and phrases. Variational inequalities, nonstandard growth, OrliczSobolev spaces, regularity theory. ∗ Partially supported by NSFC and Academy of Finland. Received: September 15, 1997. c 1996 Mancorp Publishing, Inc. 41 42 MARTIN FUCHS AND LI GONGBAO holds with λ(P ) > 0. If the domain Ω ⊂ Rn is strictly convex, then the HilbertHaar theory applies showing that the unique minimizer u is of class C 1,α (Ω) for any α ∈ (0, 1) (see [KS]). For general Ω this result is only known to hold for integrands G with power growth condition (see [F2 ]). The papers [FS] and [FO] investigated the regularity of local minimizers for vectorial problems without side conditions and integrands G having nonstandard growth and proved (under certain additional assumptions on G) partial r; egularity in dimensions n ≥ 3 and full regularity if n = 2. These arguments do not immediately apply to problem (1.1) since then the Euler equation has to be replaced by a diﬀerential inequality or equivalently by a diﬀerential equation with a measurevalued r.h.s.. Using techniques outlined in [F1 ] and [F2 ] we ﬁrst show that this measure has a well behaved density w.r.t. Lebesgue’s measure so that we have a substitute for the Euler equation being valid in the unconstrained case. Unfortunately this step works only in the scalar case but nevertheless it can be combined with appropriate modiﬁcations of the quoted regularity arguments to give at least partial regularity of the minimizer up to a certain dimension n which can be calculated in terms of the integrand G. Let us now give precise statements of the results: in what follows Ω ⊂ Rn will always denote a bounded Lipschitz domain and we also assume that the obstacle Φ is in the space C 2 (Ω̄) satisfying Φ∂Ω < 0. To begin with, let us consider the logarithmic case G(Y ) := Y ln(1 + Y ), Y ∈ Rn . (1.2) Then problem (1.1) is wellposed on the class ◦ 1 LLnL (Ω) K := {u ∈ W : u ≥ Φ a.e.}, ◦ where W 1LLnL (Ω) is the OrliczSobolev space generated by the N function t ln(1 + t) (compare Section 2 for deﬁnitions of OrliczSobolev spaces), and we have Theorem 1.1. Suppose that G is given by (1.2). a) Then problem (1.1) admits a unique solution u ∈ K . b) Suppose n ≤ 4. Then there is an open subset Ω0 ⊂ Ω such that Ω − Ω0  = 0 and u ∈ C 1,α (Ω0 ) for any 0 < α < 1. Here  ·  denotes Lebesgue’s measure of the set Ω − Ω0 . Next, let A denote a N function having the ∆2 property (compare [A]) and consider the corresponding OrliczSobolev space WA1 (Ω). The class of admissible functions is now deﬁned as ◦ 1 A (Ω) K = {u ∈ W ◦ : u ≥ Φ a.e.}, where W 1A (Ω) is the closure of C0∞ (Ω) w.r.t. OrliczSobolev norm in WA1 (Ω) (see Section 2). Concerning the integrand G we require the following conditions to be satisﬁed with positive constants C1 , · · · , C5 , λ and a nonnegative VARIATIONAL INEQUALITIES 43 number µ: (1.3) G is of class C 2 ; C1 (A(E) − 1) ≤ G(E) ≤ C2 (A(E) + 1); (1.4) G(E) ≤ C3 (E2 + 1); (1.5) E2 D2 G(E) ≤ C4 (G(E) + 1); (1.6) A∗ (DG(E)) ≤ C5 (A(E) + 1); (1.7) D2 G(X)(Y, Y ) ≥ λ(1 + X)−µ Y 2 where X, Y, E ∈ Rn are arbitrary and A∗ denotes the N function conjugate to A (see [A]). Theorem 1.2. Let (1.3)(1.7) hold. a) Then problem (1.1) possesses a unique solution u ∈ K . b) Suppose that n ≥ 2 together with µ < n4 . Then partial regularity in the sense of Theorem 1.1 b) is true. The reader may wonder for what reason we state Theorem 1.1 since it seems to be a special case of Theorem 1.2 by letting A(t) := tln(1 + t), t ≥ 0, G(Y ) := A(Y ). It is easily checked that (1.3)(1.7) hold with µ = 1 so that by Theorem 1.2 we have partial regularity up to dimension 3. But Theorem 1.1 provides a slightly stronger result: partial regularity is also true in the 4dimensional case which means that for the concrete model given by (1.2) direct calculations yield better results than the general theory summarized in Theorem 1.2. Let us give some further examples of integrands G satisfying (1.3)(1.7): Example 1. A(t) = tp ln(1 + t), t ≥ 0, 1 ≤ p < 2; A(X), X ≥ 1 G(X) = , X ∈ Rn , g(X), X ≤ 1 where g(t) is the unique quadratic polynomial such that G is C 2 . In this case (1.7) holds for µ = 2 − p Example 2. A(t) = tln(1 + ln(1 + t)), t ≥ 0; G(X) := A(X), X ∈ Rn . Now for µ in (1.7) we may choose 1 + ε with any number ε > 0. Example 3. A(t) = t 1−α (arcsinhs)α ds, 0 < α ≤ 1, t ≥ 0; 0s G(X) = A(X), X ∈ Rn . This model occurs in the study of certain generalized Newtonian ﬂuids (see [BAH]), (1.7) now holds with µ = α. 44 MARTIN FUCHS AND LI GONGBAO In all cases (1.6) may be veriﬁed as follows: recall the equation A∗ (A (s)) = sA (s) − A(s) and observe sA (s) ≤ A(s) for large s together with A (Q) = DG(Q) for Q ≥ 1. Our paper is organized as follows: we only present a proof of Theorem 1.1 since the case of general integrands G requires some minor modiﬁcations which can be found in [FO]. In Section 2 we introduce a quadratic regularization of problem (1.1) whose solutions converge to the minimizer of the problem under discussion. Section 3 describes the method of linearization which transforms the variational inequality for the obstacle problem into a nonlinear equation. In Section 4 we use this information to derive a Caccioppolitype inequality which is the main tool for the regularity proof carried out in Section 5. We ﬁnally wish to remark that our results can be viewed as a ﬁrst step towards the regularity theory of obstacle problems with integrands G being not of power growth. The standard growth condition is replaced by (1.3) which means that we can control G in terms of a N function A. Of course it is of great importance to discuss if singular points actually occur and if the restriction on the dimension n is really needed. This investigation will be carried out in a subsequent paper. 2. Some Comments on Function Spaces and Discussion of the Regularity Problem We ﬁrst give the deﬁnition of OrliczSobolev spaces and state some results which we will use later. For a technical account of the OrliczSobolev spaces we refer the reader to the books [A] and [KR]. As in [A] we say that a function A : [0, ∞) → [0, ∞) is a N function if it satisﬁes the following properties (N1 ) and (N2 ): (N1 ) A is continuous, strictly increasing and convex; (N2 ) limt→0+ A(t)/t = 0, limt→∞ A(t)/t = ∞. We say that a N function A(t) satisﬁes a ∆2 condition near inﬁnity if (N3 ) there exist positive constants k and t0 such that A(2t) ≤ kA(t) for all t ≥ t0 . It is easy to see that (N3 ) implies the inequality lnλ A(λt) ≤ A(λt0 ) + (1 + k ln2 +1 )A(t) being valid for all t, λ ≥ 0. Let A(t) be a N function. Then the conjugate A∗ of A is deﬁned as A∗ (s) = max(st − A(t)). t≥0 A∗ is also a N function. It is easy to see that For a bounded domain Ω, the Orlicz space LA (Ω) is deﬁned as LA (Ω) := {u : Ω → R measurable ∃λ > 0 such that Ω A(λu)dx < +∞}. VARIATIONAL INEQUALITIES 45 LA (Ω) together with the Luxemburg norm uLA (Ω) = inf{l > 0 : Ω A( u )dx ≤ 1} l carries the structure of a Banach space. Let EA (Ω) be the closure in LA (Ω) of all bounded measurable functions. Then EA (Ω) is a separable linear subspace of LA (Ω), moreover, LA (Ω) = EA (Ω) iﬀ A satisﬁes a ∆2 condition near inﬁnity (see [A]). The OrliczSobolev space generated by a N function A is deﬁned as WA1 (Ω) = {u : Ω → R measurable  u, ∇u ∈ LA (Ω)} which together with the norm uW 1 (Ω) = uLA (Ω) + ∇uLA (Ω) A is a Banach space. We further let ◦ 1 A (Ω) W := closure of C0∞ (Ω) in WA1 (Ω) w.r.t. · W 1 (Ω) . A The following results were proved in [FO]. Lemma 2.1. (Theorem 2.1 in [FO]) Let Ω be a bounded Lipschitz domain and suppose that A(t) is a N function satisfying a ∆2 condition near inﬁnity. Then we have ◦ 1 A (Ω) W ◦1 = WA1 (Ω) ∩ W A (Ω). Lemma 2.2. (Lemma 2.4 in [FO], Poincare’s inequality) we have the inequality ◦ For u ∈ W 1 (Ω) A uLA (Ω) ≤ d∇uLA (Ω) where d is the diameter of Ω. It is easy to see that the following result is true (see [A] or [KR]). Lemma 2.3. Let Ω be a bounded domain in Rn and A be a N function satisfying a ∆2 condition near inﬁnity. Consider a sequence {um } in LA (Ω). Then the following conditions are equivalent: (a) Ω A(um )dx → 0 as m → ∞; (b) Ω A(λum )dx → 0 as m → ∞ for any λ ≥ 0 and (c) limm→∞ um LA (Ω) = 0. Let A be a N function with conjugate function A∗ . A sequence {um } in LA (Ω) is said to be EA∗ weakly convergent to u ∈ LA (Ω), if lim m→∞ Ω um vdx = Ω uvdx, ∀v ∈ EA∗ (Ω). 46 MARTIN FUCHS AND LI GONGBAO A sequence {um } in WA1 (Ω) is said to be EA∗ weakly convergent to some u ∈ WA1 (Ω) if both um − u and ∇um − ∇u are EA∗ weakly convergent to 0 in LA (Ω). The following results can be found in [KR]. Lemma 2.4. Let Ω be a bounded domain in Rn and A be a N function with conjugate function A∗ . Then the following statements hold: (a) If a sequence {um } in LA (Ω) is EA∗ weakly convergent, then um LA (Ω) ≤ C for some constant C and any m ≥ 1; (b) LA (Ω) is EA∗ weakly complete, i.e. for any EA∗ weakly convergent sequence {um } in LA (Ω), there is a unique u ∈ LA (Ω) such that lim m→+∞ Ω um (x)v(x)dx = u(x)v(x)dx, ∀v ∈ EA∗ (Ω) Ω (c) LA (Ω) is EA∗ weakly compact, i.e. for any bounded sequence {um } in LA (Ω), there is a EA∗ weakly convergent subsequence. It is easy to prove the following results. Lemma 2.5. Let A denote a N function. Then (a) the following imbeddings WA1 (Ω) %→ W11 (Ω), ◦ 1 A (Ω) W ◦ %→ W 11 (Ω) are continuous. ◦ (b) If {um } is a bounded sequence in WA1 (Ω)(W u∈ that ◦ WA1 (Ω)(W 1A (Ω)) 1 (Ω)), A then there is a and a subsequence {um } (still denoted by {um }) such ◦ 1 A (Ω)) um & u EA∗ − weakly in WA1 (Ω) (W and ◦ um & u weakly in W11 (Ω) (W 11 (Ω)). Lemma 2.6. Let Ω ⊂ Rn be a bounded Lipschitz domain and A be a N function satisfying a ∆2 condition near inﬁnity. ◦ (a) If u, v ∈ WA1 (Ω)(W 1A (Ω)), then both max(u, v) and min(u, v) are in ◦ WA1 (Ω) (W and 1 (Ω)) A with ∇u(x) ∇ max(u, v) = ∇v(x) ∇u(x) ∇ min(u, v)(x) = ∇v(x) if u(x) ≥ v(x), if v(x) ≥ u(x) if u(x) ≤ v(x), if v(x) ≤ u(x) . VARIATIONAL INEQUALITIES 47 ◦ ◦ (b) If uj , vj ∈ WA1 (Ω)(W 1A (Ω)) with uj → u, vj → v in WA1 (Ω) (W 1A (Ω)), then max(uj , vj ) → max(u, v) and min(uj , vj ) → min(u, v) in WA1 (Ω) ◦ (W 1 (Ω)). A Proof. We mention only that since A satisﬁes a ∆2 condition, Lemma 2.3 can be used. The proof will then be carried out as in the ordinary Sobolev space case. (See e.g. [HKM]). We now turn to our main problem (1.1): ◦ 1 to ﬁnd u ∈ K = {v ∈ W A (Ω)v ≥ Φ a.e} such that I(u) = inf v∈K I(v) where I(w) = Ω G(∇w)dx and G satisﬁes (1.3)(1.7) for some N − function with (N ), (N ), (N ). 1 2 3 The solvability of (1.1) is given by the following Theorem 2.7. Problem (1.1) admits a unique solution u. ◦ ◦ Proof. Since Φ∂Ω < 0 and Φ ∈ C 2 (Ω̄), v = max(0, Φ) ∈ W 22 (Ω) %→ W with v ≥ Φ a.e, so v ∈ K and K = φ. Let {um } be a minimizing sequence in K of I, then 1 (Ω) A I(um ) → inf I = γ and v∈K Ω G(∇um )dx ≤ C. By (1.3) we see that Ω A(∇um )dx ≤ C < +∞ ∀m. Since A satisﬁes a ∆2 condition, we have ∇um LA (Ω) ≤ C (see [A] or [KR]). The Poincare inequality (Lemma 2.2) implies that um W 1 (Ω) ≤ C. A ◦ Using Lemma 2.5 we ﬁnd a function û ∈ W such that um & u 1 (Ω) A and a subsequence {um } ◦ in W 11 (Ω). Sobolev’s imbedding implies that um → u a.e. in Ω, hence u ≥ Φ and u ∈ K . According to Mazur’s Lemma we can arrange N (m) vm = j=m 1 cm j uj → u in W1 (Ω) 48 MARTIN FUCHS AND LI GONGBAO for suitable sequences N (m) ∈ N, cm j ≥ 0, N (m) quence we may also assume ∇vm → ∇u j=m cm j = 1, and for some subse a.e. The convexity of G(X) implies that I(u) ≤ γ, and the strict convexity of G gives the uniqueness of the minimizer. In what follows we let G(E) = Eln(1 + E), A(t) = tln(1 + t) for t ≥ 0. To study the regularity problem we deﬁne δ Gδ (E) = E2 + G(E) 2 for E ∈ Rn , δ > 0. We further let ◦ K∗ = {w ∈ W 12 (Ω)w ≥ Φ a.e. in Ω}, Iδ (w) = Ω Gδ (∇w)dx. We have the following density result. Lemma 2.8. K∗ is dense in K w.r.t. the norm · W 1 (Ω) . A Proof. For any u ∈ K, since Φ ∈ C 2 (Ω̄) and Φ∂Ω < 0, we have max(0, Φ) ∈ ◦ W 12 (Ω), hence ◦ By the deﬁnition of W ◦ 1 A (Ω). u − max(0, Φ) ∈ W 1 (Ω), A there is a sequence vi ∈ C0∞ (Ω) such that ◦ 1 A (Ω). vi → u − max(0, Φ) in W Since Φ − max(0, Φ) < 0 in a neighborhood N of ∂Ω, we see max(vi , Φ − max(0, Φ)) = 0 in (Ω \ spt(vi )) ∩ N. In fact we have ◦ ◦ 1 A (Ω). max(vi , Φ − max(0, Φ)) ∈ W 12 (Ω) %→ W Let ui = max(vi , Φ − max(0, Φ)) + max(0, Φ). Then ui ∈ K∗ and by Lemma 2.6 ui → max(u − max(0, Φ), Φ − max(0, Φ)) + max(0, Φ) = u − max(0, Φ) + max(0, Φ) = u in WA1 (Ω). This proves the Lemma. We have the following result concerning the functional Iδ (w). VARIATIONAL INEQUALITIES 49 Theorem 2.9. a) The problem Iδ → min in K∗ has a unique solution uδ . b) We have uδ  u in W11 (Ω), moreover, Iδ (uδ ) → I(u) as δ ↓ 0, and δ ∇uδ 2 dx → 0 2 Ω where u is the minimizer of I(v) in K. Proof. Since K∗ = φ, we may apply the direct method in order to verify part a). Let w ∈ K∗ be ﬁxed. Then for δ < 1 Iδ (uδ ) ≤ Iδ (w) ≤ I1 (w) = C1 which implies I(uδ ) ≤ C, and as in the proof of Theorem 2.7 we see ◦ uδ & ũ weakly in W 11 (Ω) ◦ for some ũ ∈ W 1A (Ω) which belongs to the class K. Then, for any w ∈ K∗ , we have Iδ (uδ ) ≤ Iδ (w) −→ I(w) δ→0+ and I(ũ) ≤ lim I(uδ ) ≤ lim Iδ (uδ ) =: α δ→0+ δ→0+ ≤ lim Iδ (uδ ) =: β, so that δ→0+ I(ũ) ≤ α ≤ β ≤ I(w), ∀w ∈ K∗ . By Lemma 2.8, K∗ is dense in K, thus we have ũ = u. Remark 2.10. We mention that the proof of Lemma 2.9 also applies to general integrands G with (1.3)(1.7) and A satisfying a ∆2 condition near inﬁnity. We now state a higher integrability result. Theorem 2.11. For the minimizer u from Theorem 2.7 we have 1 1 + ∇u ∈ W2,loc (Ω). Corollary 2.12. ∇u is in the space Lploc (Ω, Rn ) for <∞ p ≤ n n−2 if n = 2 if n ≥ 3. 50 MARTIN FUCHS AND LI GONGBAO Corollary 2.13. If n = 2, then u ∈ C 0,α (Ω) for any 0 < α < 1; if n ≤ 4, then ∇u ∈ L2loc (Ω). Remark 2.14. In the general case we have instead of Theorem 2.11 that 1 (Ω) (compare [FO] for the unconstrained case). (1 + ∇u)1−µ/2 ∈ W2,loc Proof of Theorem 2.11. We ﬁx a coordinate direction eγ ∈ Rn , γ = 1, · · · , n, and deﬁne for h = 0 and functions f 1 ∆h f (x) = (f (x + heγ ) − f (x)). h Let {uδ } denote the sequence obtained in Theorem 2.9. With δ ﬁxed we consider ε > 0 satisfying εh−2 < 12 and deﬁne vε := uδ + ε∆−h (η 2 ∆h [uδ − Φ]) with η ∈ C02 (Ω) such that 0 ≤ η ≤ 1. Then vε ∈ K∗ , hence Iδ (uδ ) ≤ Iδ (vε ), and we deduce 1 [Gδ (∇uδ + ε∇(∆−h (η 2 ∆h (uδ − Φ)))) − Gδ (∇uδ )]dx ≥ 0. Ω ε Letting ε → 0 we infer Ω DGδ (∇uδ ) · ∇(∆−h [η 2 ∆h [uδ − Φ]])dx ≥ 0 where DGδ denotes the gradient of Gδ . Using “integration by parts” for ∆−h we end up with the result (2.15) Ω ∆h {DGδ (∇uδ )} · ∇(η 2 ∆h [uδ − Φ])dx ≤ 0. Introducing ξt = ∇uδ + th∆h (∇uδ ) we may write ∆h {DGδ (∇uδ )} · ∇(η 2 ∆h [uδ − Φ]) = 1 0 D2 Gδ (ξt )(∆h ∇uδ , ∇(η 2 ∆h [uδ − Φ])dt. Let us further deﬁne the bilinear form Bx (X, Y ) = 1 0 D2 Gδ (ξt (x))(Z, Y )dt for x ∈ Ω and X, Y ∈ Rn . Then (2.15) takes the form (2.16) Ω We have Bx (∆h ∇uδ , ∇(η 2 ∆h [uδ − Φ])dx ≤ 0. ∇(η 2 ∆h uδ ) = η 2 ∇∆h uδ + 2η∇η∆h uδ and (2.16) implies ≤ Ω Ω η 2 Bx (∆h ∇uδ , ∆h ∇uδ )dx Bx (∆h ∇uδ , ∇(η 2 ∆h Φ) − 2η∇η∆h uδ )dx. VARIATIONAL INEQUALITIES 51 Using the CauchySchwarz’s inequality in the form 1 1 Bx (X, Y ) ≤ Bx (X, X) 2 Bx (Y, Y ) 2 together with Young’s inequality we arrive at (2.17) η 2 Bx (∆h ∇uδ , ∆h ∇uδ )dx Ω ≤ C1 (η) Bx (∆h Φ2 + ∇∆h Φ2 + ∆h uδ 2 )dx sptη for some constant C1 depending also on η. It is easy to check (see [FS]) that the following bounds hold for the parameter dependent bilinear form D2 Gδ (Z)(X, Y ) (2.18) (2.19) D2 Gδ (Z) = sup D2 Gδ (Z)(X, X) ≤ δ + X =1 2ln(1 + Z) Z D2 Gδ (Z)(X, X) ≥ δX2 + (1 + Z)−1 X2 ≥ δX2 . Inserting this into (2.17), we ﬁnd that (2.20) Ω η 2 ∆h ∇uδ 2 dx ≤ C3 (δ, η){uδ 2W 1 (Ω) + Φ2W 2 (Ω) } 2 2 2 W2,loc(Ω) . and therefore uδ ∈ For this reason we can replace ∆h in (2.16) by the partial derivative ∂γ . Then, following the calculation after (2.16), we see that (2.17), (2.20) have to be replaced by (summation over γ) Ω (2.21) η 2 D2 Gδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ )dx ≤ C4 (η) sptη ≤ C5 (η)[δ Ω + C6 (η)( D2 Gδ (∇uδ )(∇Φ2 + ∇2 Φ2 + ∇uδ 2 )dx (∇uδ 2 + ∇Φ2 + ∇2 Φ2 )dx Ω ∇uδ  ln(1 + ∇uδ )dx + Φ2W 2 (Ω) )] 2 where C6 is independent of δ. We know that δ Ω ∇uδ 2 dx → 0 as δ → 0+ (cf. Theorem 2.9) and supδ>0 Ω ∇uδ  ln(1 + ∇uδ )dx < ∞. Hence (2.21) implies (2.22) Ω D2 Gδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ )dx ≤ C(Ω ) for any subdomain Ω ⊂⊂ Ω. Combining (2.19) and (2.22) we ﬁnd that 2 Ω ∇ 1 + ∇uδ  dx ≤ C(Ω ). 1 (Ω) such that Thus there is a function ω ∈ W2,loc (2.23) 1 1 + ∇uδ  & ω in W2,loc (Ω). 52 MARTIN FUCHS AND LI GONGBAO We claim (2.24) ω= 1 + ∇u. To show (2.24), we note that Iδ (uδ ) − I(u) = = δ 2 + Ω δ 2 ∇uδ 2 dx + 1 Ω 0 ∇uδ 2 dx + I(uδ ) − I(u) Ω Ω DG(∇u)(∇uδ − ∇u)dx D2 G((1 − t)∇u + t∇uδ )(∇uδ − ∇u, ∇uδ − ∇u)(1 − t)dtdx. From Theorem 2.9 we have Iδ (uδ ) −→ I(u), δ (2.25) lim { δ→0+ + δ→0+ Ω 2 Ω ∇uδ  dx −→δ→0+ 0, hence DG(∇u) · (∇uδ − ∇u)dx 1 Ω 0 D2 G((1 − t)∇u + t∇uδ )(∇uδ − ∇u, ∇uδ − ∇u)(1 − t)dtdx} = 0. On the other hand, minimality of u implies (2.26) Ω DG(∇u)(∇uδ − ∇u)dx ≥ 0. Next we observe the estimate 1 (2.27) Ω 0 ≥ D2 G((1 − t)∇u + t∇uδ )(∇uδ − ∇u, ∇uδ − ∇u)(1 − t)dtdx 1 Ω 0 ∇uδ − ∇u2 (1 − t) dt dx. 1 + (1 − t)∇u + t∇uδ  This implies that ∇uδ → ∇u a.e in Ω (possibly for some subsequence). Hence we get from (2.23) that 1 + ∇uδ  & 1 1 + ∇u in W2,loc and the theorem is proved. ◦ Remark 2.15. We mention that in [FS] W 1LLnL has a diﬀerent deﬁnition but one of the equivalent characterizations of a function u belonging to the ◦ space W 1LLnL is that u belongs to the OrliczSobolev space generated by A(t) = tln(1 + t), t ≥ 0. 3. Linearization To study the obstacle problem, it is convenient to consider the variational inequality as an equation with a measure valued right–hand side and then VARIATIONAL INEQUALITIES 53 to apply suitable methods in order to identify this measure. To this end, following [F2 ], we deﬁne wtε := uδ + tηhε ◦ (uδ − Φ) where δ ≥ 0, uδ is given in Section 2, η ∈ C01 (Ω), 0 ≤ η ≤ 1, t > 0, ε > 0, hε ∈ C 1 (R1 ), 0 ≤ hε ≤ 1, hε = 1 on (0, ε), hε = 0 on (2ε, ∞), hε ≤ 0. In case δ = 0 we have u0 = u, u denoting the solution of (1.1). Then wtε ∈ K∗ , if δ > 0, and wtε ∈ K for δ = 0, hence 1 t Gδ (∇wtε )dx Ω − Ω Gδ (∇uδ )dx ≥ 0 ⇒ (as t → 0) DGδ (∇uδ ) · ∇(ηhε ◦ (uδ − Φ))dx ≥ 0, Ω and there exists a Radon measure λ (independent of ε!) such that (3.1) Ω DGδ (∇uδ ) · ∇(ηhε ◦ (uδ − Φ))dx = Ω ηdλ The fact that λ does not depend on ε can be seen by using w̃ = uδ + ηt{hε ◦ (uδ − Φ) − hε ◦ (uδ − Φ)}(ε < ε ) as test function provided t is small enough. Note that (3.1) is valid for all small ε > 0 and any η ∈ C01 (Ω). For estimating λ we may therefore ﬁx η ≥ 0 and let ε → 0, in order to get ηdλ = Ω DGδ (∇uδ ) · ∇ηhε ◦ (uδ − Φ)dx Ω + DGδ (∇uδ ) · ∇(uδ − Φ)hε ◦ (uδ − Φ)ηdx Ω =: (α) + (β), where (β) = Ω + ≤ {DGδ (∇uδ ) − DGδ (∇Φ)} · ∇(uδ − Φ)hε ◦ (uδ − Φ)ηdx Ω Ω DGδ (∇Φ) · ∇(uδ − Φ)hε ◦ (uδ − Φ)ηdx DGδ (∇Φ) · ∇(uδ − Φ)hε ◦ (uδ − Φ)ηdx =: (γ), and the estimate holds since Gδ is convex and hε ≤ 0. We have (γ) = Ω − =− − DGδ (∇Φ) · ∇(hε ◦ (uδ − Φ)η)dx Ω Ω Ω DGδ (∇Φ)hε ◦ (uδ − Φ) · ∇ηdx div{DGδ (∇Φ)}ηhε ◦ (uδ − Φ)dx DGδ (∇Φ)hε ◦ (uδ − Φ) · ∇ηdx, 54 MARTIN FUCHS AND LI GONGBAO which implies Ω ηdλ ≤ − −→ Ω Ω {DGδ (∇uδ ) − DGδ (∇Φ)} · ∇ηhε ◦ (uδ − Φ)dx div{DGδ (∇Φ)}ηhε ◦ (uδ − Φ)dx ε→0 [uδ =Φ] {DGδ (∇uδ ) − DGδ (∇Φ)} · ∇ηdx − [uδ =Φ] div{DGδ (∇Φ)}ηdx. Since ∇uδ = ∇Φ a.e. on [uδ = Φ], we arrive at Ω ηdλ ≤ Ω χ[uδ =Φ] (−div{DGδ (∇Φ)})ηdx. In particular, χ[uδ =Φ] (div{DGδ (∇Φ)}) ≥ 0 a.e. and λ takes the form λ = λδ = Θδ (−div{DGδ (∇Φ)}) × Lebesgue measure for some density 0 ≤ Θδ ≤ 1 supported on [uδ = Φ]. Returning to (3.1) and observing that Ω DGδ (∇uδ ) · ∇(η(1 − hε ◦ (uδ − Φ)))dx = 0 we get Ω DGδ (∇uδ ) · ∇ηdx = Ω ηdλδ . Thus we have proved Theorem 3.1. For any δ ≥ 0, there exists fδ ∈ L∞ (Ω) such that fδ ∞ ≤ div{DGδ (∇Φ)}∞ and Ω DGδ (∇uδ ) · ∇ϕdx = Ω fδ ϕdx, ∀ϕ ∈ C01 (Ω). In particular, fδ ∞ is bounded independently of δ. Remark 3.2. The proof of Theorem 3.1 given before does not use the fact 2 (Ω). The higher diﬀerentiability of u allows to that uδ is of class W2,loc δ perform an integration by parts in formula (3.1), and afetr passing to the limit ; ↓ 0 we immediately deduce the representation of the measure λ. 4. A Caccioppolitype inequality In this section we prove the following Caccioppolitype inequality. VARIATIONAL INEQUALITIES 55 Theorem 4.1. Let u be the minimizer from Theorem 1.1. Then, for arbitrary balls Br (x0 ) ⊂ BR (x0 ) ⊂ Ω, we have the estimate Br (x0 ) ∇ 1 + ∇u2 dx ≤C 1 (R − r)2 BR (x0 ) ln(1 + ∇u) ∇u − Xdx ∇u 1 + (1 + ∇u)dx + R−r BR (x0 ) BR (x0 ) ∇u − Xdx , where X is any vector in Rn , C = C(n, Φ). 2 (Ω) for δ > 0 Proof. Let uδ be as in the previous sections. Using uδ ∈ W2,loc (which will be assumed from now on), we get from Theorem 3.1 that Ω 2 D Gδ (∇uδ )(∂γ ∇uδ , ∇ϕ)dx = − fδ ∂γ ϕdx. Ω Let ϕ = η 2 (∂γ uδ − Xγ ) for η ∈ C01 (Ω), 0 ≤ η ≤ 1, X ∈ Rn . Using summation over γ we deduce − Ω Ω η 2 D2 Gδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ )dx = 2D2 Gδ (∇uδ )(∂γ ∇uδ , ∇η[∂γ uδ − Xγ ])ηdx − Ω fδ ∂γ (η 2 [∂γ uδ − Xγ ])dx, and as in the proof of Lemma 3.1 in [FS] we get the estimate Ω η 2 [δ∇2 uδ 2 + ∇ 1 + ∇uδ 2 ]dx ≤ C(n) ∇η∞ (4.2) + Ω stpη [δ∇uδ − X2 + 2 ln(1 + ∇uδ ) ∇uδ − X2 ]dx ∇uδ  fδ ∇(η [∇uδ − X])dx . We recall fδ ∞ ≤ const. independent of δ and observe Ω and 2 ∇(η [∇uδ − X])dx ≤ C{ Ω η 2 ∇2 uδ dx ≤ ε Ω η2 Ω 2 2 η ∇ uδ dx + sptη 1 1 ∇2 uδ 2 dx + 1 + ∇uδ  ε ∇η∞ ∇uδ − Xdx} Ω η 2 (1 + ∇uδ )dx. Of course, inequality (4.2) remains valid if the lefthand side is replaced by Ω η 2 D2 Gδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ )dx. On the other hand, 1 D2 uδ 2 ≤ D2 Gδ (∇uδ )(∂γ ∇uδ , ∂γ ∇uδ ), 1 + ∇u 56 MARTIN FUCHS AND LI GONGBAO and by choosing ε > 0 small enough, we may absorb ε into the lefthand side. This ﬁnally implies Ω 1 2 2 2 Ω η 1+∇u D uδ  dx η 2 ∇ 1 + ∇uδ 2 dx ≤ C{∇η2∞ + sptη sptη [δ∇uδ − X2 + (1 + ∇uδ )dx + ∇η∞ ln[1 + ∇uδ ) ∇uδ − X2 ]dx ∇uδ  sptη ∇uδ − Xdx}. Using the imbedding theorem as in [FS], we may now pass to the limit δ → 0 in the above inequality which ﬁnishes the proof of the theorem. 5. Blowup: proof of partial regularity We ﬁx some 0 < µ < 1 and denote by u the solution to the obstacle problem from Theorem 1.1. Let us further assume that n ≤ 4. We have the following Lemma 5.1. Fix some L > 0 and calculate C0 = C0 (n, L) as indicated in the proof. Then, for all τ ∈ (0, 1), we ﬁnd a number ε = ε(n, τ, L) such that (∇u)x0 ,R < L and (5.2) − ∇u − (∇u)x0 ,R 2 dx + R2µ < ε2 BR (x0 ) imply (5.3) − 2 2 ∇u − (∇u)x0 ,τ R  dx ≤ C0 τ { − Bτ R (x0 ) ∇u − (∇u)x0 ,R 2 + R2µ } BR (x0 ) for any ball BR (x0 ) ⊂ Ω. Here (g)y,ρ denotes the mean value −Bρ (y) gdx. In the formulation of the Lemma we replaced the “standard assumption” (∇u)x0 ,R  < L by a slightly stronger one. Lemma 5.1 will be proved at the end of this section. We ﬁrst show our main result Theorem 1.1. Let us introduce the set Ω0 = {x ∈ Ω : x is a Lebesgue point for ∇u and ∇u and −Br (x0 ) ∇u − (∇u)x0 ,r 2 dx −→r↓0 0}. Clearly Ω−Ω0  = 0. In order to prove the theorem with the help of Lemma 5.1, we need only to show that any point x0 from Ω0 has some neighborhood in Ω on which ∇u is Hölder continuous. Let x0 ∈ Ω and let L := max{2∇u(x0 ), 1}, ∇u(x0 ) = lim r→0 − ∇udx. Br (x0 ) VARIATIONAL INEQUALITIES 57 This determines the constant C0 = C0 (L). Fix τ such that C0 τ 2 = 1 2 and calculate ε w.r.t. this data. Note the inequality (5.4) n (∇u)x0 ,τ k+1 R ≤ τ − 2 k i=0 1 E(x0 , τ i R) 2 + (∇u)x0 ,R being valid for any R such that BR (x0 ) ⊂ Ω and any k ∈ N. Here E(x0 , R) := −BR (x0 ) ∇u − (∇u)x0 ,R 2 dx. Let us further set θ = τ 2µ (w.l.o.g. θ < 12 ). A number ε̄ is chosen according to ∞ i 1 τ − n2 2− 2 √1−2θ ε̄ < L/3 . i=0 ε̄2 ≤ min{ 1 , 1−2θ }ε2 (5.5) 4 2 Finally, we ﬁx R > 0 such that E(x0 , R) + R2µ < ε̄2 , (5.6) 2 (∇u)x0 ,R < L. 3 Proposition 5.2. For any k ∈ N we have (5.7)k k −k E(x0 , τ R) ≤ 2 E(x0 , R) + k 2−j θk−j R2µ . j=1 Proof. We prove the proposition by induction. Let k = 1. Then (5.6) implies (5.2). Hence (5.3) holds. Thus, by the choice of τ, 1 E(x0 , τ R) ≤ (E(x0 , R) + R2µ ), 2 and (5.7)1 holds. 58 MARTIN FUCHS AND LI GONGBAO Assume now (5.7)k . Then = 2−k E(x0 , R) + k 1 j 2µ ( 2θ ) R j=1 1 k+1 1−( 2θ ) θk − 1 1− 2θ E(x0 , τ k R) ≤ 2−k E(x0 , R) + θk 1 = 2−k E(x0 , R) + θk 2θ (5.8) 1 k+1 −( 2θ ) R2µ 1 1− 2θ = 2−k E(x0 , R) + 1 k−1 1 −k−1 θ −θ2 2 1 1− 2θ = 2−k E(x0 , R) + θk −2−k 2µ 2θ−1 R = 2−k E(x0 , R) + 2−k −θk 2µ 1−2θ R ≤ 2−k {E(x0 , R) + ≤ 1 R2µ R2µ 1 2µ 1−2θ R } 2−k 2 1−2θ ε̄ , hence E(x0 , τ k R) + (τ k R)2µ ≤ 2−k 2 2 ε̄ + τ k2µ R2µ ≤ ε̄2 ≤ ε2 , i.e. 1 − 2θ 1 − 2θ E(x0 , τ k R) + (τ k R)2µ < ε2 . (5.9) Next, we have n (∇u)x0 ,τ k R ≤ τ − 2 k−1 i=1 1 E(x0 , τ i R) 2 + (∇u)x0 ,R , and since we may assume that (5.7)j holds for any j ≤ k, we get n (∇u)x0 ,τ k R ≤ τ − 2 k−1 2−i {E(x0 , R) + i=1 1 R2µ } 1 − 2θ 1 2 + (∇u)x0 ,R which is a consequence of (5.8). Therefore n (∇u)x0 ,τ k R ≤ τ − 2 ≤τ < −n 2 L 3 ∞ i=0 i 1 1 2− 2 √1−2θ (E(x0 , R) + R2µ ) 2 + (∇u)x0 ,R 1 ε̄ √1−2θ ∞ i=0 i 2− 2 + (∇u)x0 ,R + (∇u)x0 ,R ≤ L. VARIATIONAL INEQUALITIES 59 From this inequality and (5.9) we see that Lemma 5.1 can be applied. Thus, E(x0 , τ k+1 R) ≤ 12 (E(x0 , τ k R) + (τ k R)2µ ) ≤ 12 (2−k E(x0 , R) + = 2−k−1 E(x0 , R) + = 2−k−1 E(x0 , R) + k 2−j θk−j R2µ ) + 12 θk R2µ j=1 k 2−(j+1) θ(k+1)−(j+1) j=1 k+1 2−j θk+1−j R2µ + 1 k 2θ R2µ j=1 and the proof of the proposition is complete. As shown in (5.8), we get from (5.7)k 1 {E(x0 , R) + R2µ }, ∀k ∈ N, (5.10) E(x0 , τ k R) ≤ 2−k 1 − 2θ which turns into the inequality r (5.11) E(x0 , r) ≤ const.( )α {E(x0 , R) + R2µ }, ∀r < R R for some exponent α > 0. Clearly, (5.6) implies 2 E(y, R) + R2µ < ε̄2 , (∇u)y,R < L 3 for any y near x0 with R ﬁxed, hence (5.10), (5.11) also hold at y which means that ∇u is Hölder continuous for example in BR/2 (x0 ) with some exponent α. Hölder continuity near x0 with any exponent < µ can be seen by choosing τ in a diﬀerent way. Proof of Lemma 5.1. We follow the proof of Lemma 4.1 in [FS] and argue by contradiction assuming (5.12) (∇u)xk ,Rk ≤ L Ak := (∇u)x0 ,Rk 2µ E(u, BRk (xk )) + Rk = ε2k → 0 2 2 E(u, B τ Rk (xk )) > C0 τ εk for a sequence BRk (xk ). Let vk (z) := 1 (u(xk + Rk z) − ak − Rk Ak z), z ∈ B1 , εk Rk ak := − udx. BRk (x0 ) Then, for suitable A ∈ Rn and v ∈ W21 (B1 ), we get Ak → A, vk → v strongly in L2 (B1 ), ∇vk & ∇v weakly in L2 (B1 , Rn ), 60 MARTIN FUCHS AND LI GONGBAO εk ∇vk → 0 strongly in L2 (B1 , Rn ) and a.e. which is true at least for some subsequence. From Theorem 3.1, we know Ω C01 (B1 ) Let ϕ ∈ equation reads Rkn−1 B1 DG(∇u) · ∇ψdx = Ω f ψdx, ∀ψ ∈ C01 (Ω). k and deﬁne ψ(x) = ϕ( x−x Rk ), x ∈ BRk (xk ). Then the above DG(εk ∇vk + Ak ) · ∇ϕdz = Rkn B1 ϕ(z)f (Rk z + xk )dz which implies B1 Rk 1 {DG(εk ∇vk + Ak ) − DG(Ak )} · ∇ϕdz = εk εk B1 ϕ(z)f (Rk z + xk )dz Using Rk /εk → 0 and the boundedness of f we get lim k→∞ B1 1 {DG(εk ∇vk + Ak ) − DG(Ak )} · ∇ϕdz = 0 εk hence B1 D2 G(A)(∇v, ∇ϕ)dz = 0 which is an elliptic equation with constant coeﬃcients. Thus there exists C1 = C1 (L) such that 2 2 − ∇v − (∇v)τ  dz ≤ C1 τ − ∇v2 dz Bτ B1 Letting C0 = 2C1 , the foregoing inequality will contradict our assumption (5.12) as soon as we can establish (5.13) ∇vk → ∇v strongly in L2loc (B1 , Rn ). To this end, we argue in two steps. Step 1: Let ϕk (z) = ε1k ( 1 + εk ∇vk + Ak  − 1 + Ak ), z ∈ B1 . Then ϕk  ≤ 12 ∇vk  and since {∇vk } is uniformly bounded in L2 (B1 , Rn ) we see that sup B1 ϕ2k dz < ∞. Next we observe 1 1 ∇ϕk (z) = ∇ 1 + εk ∇vk + Ak  = ∇ 1 + ∇u(xk + Rk z) εk εk Rk = (∇ 1 + ∇u)(xk + Rk z) εk VARIATIONAL INEQUALITIES so that B1 ∇ϕk 2 dz = Rk2 ε2k 61 2 ∇ 1 + ∇u (xk + Rk z)dz B1 2−n = ε−2 k Rk 2 BRk (xk ) ∇ 1 + ∇u dx and in the same manner for 0 < t < 1 Bt 2−n ∇ϕk 2 dz ≤ ε−2 k Rk 2 BtRk (xk ) ∇ 1 + ∇u dx. Theorem 4.1 gives Bt ∇ϕk 2 dz 2−n ≤ C ε−2 (1 − t)−2 Rk−2 k Rk 2−n + ε−2 k Rk BRk (xk ) 2−n (1 +ε−2 k Rk −1 − t) −2 − t) Rk−1 Rk−2 BRk (xk ) ≤ C(1 − t)−2 ε−2 k BRk (xk ) ln(1 + ∇u) ∇u − Ak 2 dx ∇u (1 + ∇u)dx Clearly, 2−n ε−2 (1 k Rk BRk (xk ) ∇u − Ak dx . ln(1 + ∇u) ∇u − Ak 2 dx ∇u − ∇u − Ak 2 dx ≤ C(1 − t)−2 , Bk (xk ) and 2−n ε−2 k Rk BRk (xk ) (1 + ∇u)dx = cn Rk2 Rk2 + ε2k ε2k ≤ (cn + L) − ∇udx BRk (xk ) Rk2 → 0 as k → ∞, ε2k where we have used our assumption −BR (xk ) ∇udx < L. k On the other hand, 2−n (1 − t)−1 Rk−1 ε−2 k Rk ≤ 2−n ε−2 (1 k Rk −1 − t) BRk (xk ) ∇u − Ak dx Rk−1 ( 1 BRk (xk ) n n 2−n ≤ ε−2 (1 − t)−1 Rk−1 εk Rk2 Rk2 k Rk n ∇u − Ak 2 dx) 2 Rk2 = (1 − t)−1 ε−1 k Rk → 0 as k → ∞. 62 MARTIN FUCHS AND LI GONGBAO Hence we have established sup ϕk W 1,2 (Bt ) < ∞, (5.14) k ∀0 < t < 1. For some large number M (depending on L) we have ϕk ≥ so that 1 1 2 εk εk ∇vk  on Bt ∩ [εk ∇vk  > M ] Bt ∩[εk ∇vk >M ] ∇vk 2 dz ≤ 24 ε2k Bt ϕk 4 dz. By (5.14) and the embedding theorem we deduce (n ≤ 4!) Bt ∩[εk ∇vk >M ] and v ∈ C ∞ (B1 ) (5.15) ∇vk 2 dz → 0 as k → ∞ obviously implies ∇vk − ∇v2 dz → 0 as k → ∞. Bt ∩[εk ∇vk >M ] Step 2: Discussion of Bt ∩[εk ∇vk <M ] ∇vk 2 dz. Here we follow again [FS]. Consider ϕ ∈ C01 (B1 ), ϕ ≥ 0, and observe ε2k 1 B1 0 ϕD2 G(Ak + εk ∇v + sεk (∇vk − ∇v)) · (∇vk − ∇v, ∇vk − ∇v)(1 − s)dsdz (5.16) = B1 − ϕ[G(Ak + εk + εk ∇vk ) − G(Ak + εk ∇v)]dz B1 εk ϕDG(Ak + εk ∇v) · (∇vk − ∇v)dz. The right hand side of (5.16) equals G(Ak + εk ∇vk )dz − B1 − ≤ B1 B1 − B1 εk ϕDG(Ak + εk ∇v) · (∇vk − ∇v)dz G(Ak + εk ∇vk )dz − B1 {(1 − ϕ)G(Ak + εk ∇vk ) + ϕG(Ak + εk ∇v)}dz B1 G(Ak + εk [ϕ∇v + (1 − ϕ)∇vk ])dz εk DG(Ak + εk ∇v) · (∇vk − ∇v)dz. ◦ By Theorem 3.1 u is the minimizer of w −→ Ω [G(∇w)−f w]dx in W 1A (Ω). After transformation and after dropping constants from the functional we see that vk minimizes h −→ B1 {G(εk ∇h + Ak ) − εk Rk f h}dz VARIATIONAL INEQUALITIES 63 on B1 w.r.t. its boundary values, i.e., B1 G(εk ∇vk + Ak )dz ≤ B1 + G(εk ∇[vk + ϕ(v − vk )] + Ak )dz B1 εk Rk f ϕ(vk − v)dz. The last integral is the only new term which occurs compared to the calculations in [FS]. We therefore get, as in [FS], from (5.16) 1 B1 0 ϕD2 G(Ak + εk ∇v + sεk (∇vk − ∇v)) · (∇vk − ∇v, ∇vk − ∇v)(1 − s)dsdz Rk ≤ o(1) + f ϕ(vk − v)dz → 0 as k → +∞. εk B1 Using now the bounds of D2 G we ﬁnd lim k→∞ B1 ϕ∇vk − ∇v2 (1 + Ak  + εk ∇v + εk ∇vk − ∇v)−1 dz = 0. In particular, choosing ϕ = 1 on Bt , lim k→+∞ Bt ∩[εk ∇vk <M ] ∇vk − ∇v2 dz = 0 This together with (5.15) implies (5.13), the proof of the Lemma is complete. Thus Theorem 1.1 is proved. For proving Theorem 1.2, we modify the foregoing argument following the lines of [FO] with obvious modiﬁcations. The details are left to the reader. Acknowledgment. Part of this paper was written during the ﬁrst author’s stay in China during March 1997. The ﬁrst author would like to thank the Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, for showing kind hospitality. The second author was partially supported by NSFC and the Academy of Finland. References [BAH] [F1 ] [F2 ] [FO] [FS] R. Bird, R. Armstrong and O. Hassager, Dynamics of polymeric liquids, Vol. 1, Fluid mechanics, John Wiley, 1987. M. Fuchs, Topics in the Calculus of Variations, Vieweg,1994. M. Fuchs, Hölder continuity of the gradient for degenerate variational inequalities, Nonl. Analysis, 15 (1990), 85–100. M. Fuchs and V. Osmolovski, Variational integrals on OrliczSobolev spaces. J. Anal. Appl. (Z.A.A.), 17 (1998), No. 2, 393–415. M. Fuchs and G. Seregin, A regularity theory for variational integrals with LLnLgrowth, Preprint No. 471, SFB 256, Univ. Bonn, Calc. Var. Partial Diﬀerential Equations. 6 (1998) 2, 171–187. 64 [HKM] [KR] [KS] MARTIN FUCHS AND LI GONGBAO J. Heinonen, T. Kilpeläinen and O.Martio, Nonlinear potential theory of degenerate elliptic equations, Clarendon Press,1993. M. A. Krasnoselskii and J. B. Rutickii, Convex functions and Orlicz spaces, Noordhof, Groningen, 1961. D. Kinderlehrer and G.Stampacchia, Variational inequalities and their applications, Acad. Press, San Diego, 1980. Martin Fuchs Universität des Saarlandes Fachbereich 9 Mathematik Postfach 151150 D66041 Saarbrücken, GERMANY Email address: fuchs@math.unisb.de Li Gongbao Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences P.O. 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