Fachrichtung 6.1 Mathematik

Transcription

1 Universität des Saarlandes U N I V E R S I T A S S A R A V I E N I S S Fachrichtung 6.1 athematik Preprint Nr. 41 New regularity theorems for non-autonomous anisotropic variational integrals Dominic reit Saarbrücken 009

2

3 Fachrichtung 6.1 athematik Preprint No. 41 Universität des Saarlandes submitted: September, 009 New regularity theorems for non-autonomous anisotropic variational integrals Dominic reit Saarland University Department of athematics P.O. ox Saarbrücken Germany

4 Edited by FR 6.1 athematik Universität des Saarlandes Postfach Saarbrücken Germany Fax: e-ail: preprint@math.uni-sb.de WWW:

5 Abstract In the calculus of variations one prominent problem is minimizing anisotropic integrals with a (p, q)-elliptic density F : R n Ω R N. The best known sufficient bound for regularity of solutions is q < p (n + )/n. On the other hand, if we allow an additional x-dependence of the density we have the much weaker result q < p (n + 1)/n. If one additionally imposes the local boundedness of the minimizer, then these bounds can be improved to q < p+ and q < p + 1. In this paper we give natural assumptions for F closing the gap between the autonomous and non-autonomous situation. 1 Introduction In these paper we investigate regularity results for local minimizers of functionals J[w] := F (, w) dx (1.1) Ω with a function F : Ω R nn [0, ) and a domain Ω R n. efore Esposito, Leonetti und ingione found rather surprising counterexamples (see [EL]), it was a wide-spread meaning, that the theorems valid for the autonomous situation extend to the case of x-dependence are portable and therefore most authors ignored x-dependence for technical simplification of their proofs. We are interested in anisotropic growth conditions. In the following we assume λ(1 + Z ) p Q D P F (x, Z)(Q, Q) Λ(1 + Z ) q Q (1.) for all Z, Q R nn and all x Ω with positive constants λ, Λ and exponents 1 < p q <. From (1.) we can deduce D P F (x, Z) Λ 1 (1 + Z ) q 1 and c 1 Z p c F (x, Z) c 3 Z q + c 4 with further constants Λ 1, c 1, c, c 3 and c 4. Furthermore we assume for all Z R nn and all x Ω γ D P F (x, Z) Λ (1 + Z ) q 1 (1.3) with Λ > 0 and γ {1,..., n}. In [EL] Esposito, Leonetti and ingione examine the Lavrentiev gap functional which is defined as L := inf J inf J u 0 +W 1,q 0 (,R N ) u 0 +W 1,p 0 (,R N ) 1

6 on a ball Ω with boundary data u 0 W 1,p (, R N ). The results of the studies from [EL] provide the sharpness of the bound q < p n + α n for higher integrability of solutions (assuming that D P F (x, Z) is α-hölder continuous with respect to x) and without this condition they have examples for the Lavrentiev-phenomenon. If (1.) and (1.3) (with at least continuous derivatives) hold together with q < p n + 1 n, (1.4) ildhauer and Fuchs proved full C 1,α -regularity for N = 1 or n = and partial regularity in the general vector case. This statement is in accordance with the results of [EL]. Under several structure conditions ildhauer and Fuchs can improve the last result to full regularity (see [F1]). Without x-dependence we know from [F3] that the better bound q < p n + n (A1) is sufficient for regularity. Having a look at the proof in [F1], two main differences to the autonomous case become obvious. The first obstacle is that the standard-regularization u does not converge against the minimum u without (1.4). Thus we work with a regularization from below which is based on a construction from [CG] (see [F1], section 3, for its presentation). This regularization F ( 1) has the properties F (x, P ) F (x, P ) for all P R nn ; for P is F (x, P ) = F (x, P ); F (x, P ) growth isotropic: i.e. a P p b F (x, P ) A P p + (1.5) for all P R nn with uniform constants a > 0, b R and constants A and depending on. F (x, P ) is uniform (p, q)-elliptic, which means we have for Z, Q R nn and γ {1,..., n} λ(1 + Z ) p Q D P F (x, Z)(Q, Q) Λ 3 (1 + Z ) q Q, γ D P F (x, Z) Λ 3 (1 + Z ) q 1 (1.6) with constants λ, Λ 3 > 0.

7 In order to introduce this regularization we have to assume F (x, P ) = g(x, P ). (A) If (A) holds, then (1.1) reads as λ(1 + t ) p g (x, t) t Λ(1 + t ) q, λ(1 + t ) p g (x, t) Λ(1 + t ) q The second obstacle in the proof of [F1] is estimating the term γ D P F (, u) : γ u dx.. (A3) otivated from [i] (section 4...) we now require γ g (x, t) Λ 4 [ g (x, t)(1 + t ) ɛ + (1 + t ) p+q 4 1 ] (A4) for all (x, t) Ω [0, ) and γ {1,..., n} with 0 ɛ 1 in order to handle this integral. All of our assumptions have to be satisfied by the regularization function F uniformly in and this can be achieved by requiring γg (x, t) Λ 5 (1 + t ) q. (A5) Additionally to (1.5) and (1.6) we now get for the sequence F, constructed in [F1] (section 3), the following estimates: Lemma 1.1 Under the conditions for g above ((A)-(A5) and continuity of the derivatives) we have for all γ {1,..., n} and uniform in x Ω F (x, P ) is p-elliptic, i.e. for Z, Q R nn is λ(1 + Z ) p Q D P F (x, Z)(Q, Q) Λ (1 + Z ) p Q, γ D P F (x, Z) Λ (1 + Z ) p 1 with a uniform constant λ and a constant Λ depending on. For all P, Z R nn it holds γ D P F (x, Z) Λ6 (1 + Z ) q 1, γ DP F (x, Z)(P, Z) Λ6 D P F (x, Z)(P, Z) (1 + Z ) ɛ uniform in with Λ Λ 6 (1 + Z ) p+q 4 P 3

8 The proof of Lemma 1.1 is presented in section. In the isotropic situation, full regularity in the general vector case under modulus-dependence is only possible, if we know D F (x, P ) D F (x, Q) c(1 + P + Q ) q α P Q α (A6) for all P, Q R nn, all x Ω with α (0, 1) (a first theorem in this context can be found in [Uh]). Therefore we need such a condition in the anisotropic case, too. With these preparations we define the regularization u of the problem (1.1) as the unique minimizer of J [w] = F (, w) dx in u + W 1,p 0 (, R N ) with a ball = R Ω. This is the solution of a isotropic problem and so we get Lemma 1. u belongs to the space L t loc (, RnN ) for t = pn/(n ) if n 3 and t arbitrary if n =. u belongs to the space W, loc W 1, loc (, RN ) if n = or N = 1. Γ p 4 u belongs to the space L loc () and u L t loc (, RnN ) for t := min {p, }. u is in W 1,p (, R N ) uniformly bounded. If we have u L loc (Ω, RN ) then u L p+ loc (, RnN ) and sup u <. The first and the third part can be found for example in [F1] (Lemma.3 and.5 for p = q). The second statement follows from [F1] Thm. 1.1 (which is a classical result for N = 1, see [LU]). For the uniform boundedness of u in W 1,p (, R N ) we see by (1.5) [ ] [ ] u p dx c F (, u ) dx + 1 c F (, u) dx + 1 [ ] c F (, u) dx + 1 c. The Poincaré-inequality supplies u L p () c [ (u u) L p () + u L p () 4 ]

9 [ ] c u L p () + u W 1,p () c. sup u < is received by quoting the maximum-principle of [DL], since we assume (A). For the last statement we quote [F1] (proof of Lemma.8 for α = 0). Note that it is not necessary to have Lipschitz-regularity for the regularization in [F1] to proof this. Hence, one can choose there q p+ (therefore we do not need (A6) at this point). In the following, we formulate the new results valid for local minimizers of (1.1): THEORE 1.1 Under the assumptions (A1)-(A5), where all involved derivatives are assumed to be continuous, we have: (a) for a local minimizer u W 1,p loc (Ω, RN ) of (1.1) there is an open subset Ω 0 with full Lebesgue-measure such that u belongs to the space C 1,α (Ω 0, R N ) for any α (0, 1). (b) If we have additionally for n 5 one of the following conditions (i) q < p n 1 n (ii) g (x, t) cg (x, t)(1 + t ) ω (A7) ( ) pn for ω < n q + 1 (A8) for all t 0 as well as (A6) if n 3 and N, then any local minimizer u W 1,p loc (Ω, RN ) of (1.1) belongs to the space C 1,α (Ω, R N ) for all α < 1. Remark 1.3 Considering the situation F (x, Z) = f(x)g( Z ) with two C -functions f and g, where f is strictly positive and g shares suitable anisotropic growth conditions, it is easy to see, that F satisfies all assumptions necessary for Theorem 1.1. ut even in this case there is no possibility to argue any simpler than in our proof of higher integrability (see Lemma.1). The critical point is the integral η γ fg ( u ) u u : γ u dx. Since the estimate g ( u ) c(1 + u ) (q 1)/ is not sharp enough, one must integrate by parts to handle this term. 5

10 The motivation for allowing small ɛ > 0 in (A4) is the minimization of ( ) f(x) 1 + w dx Ω for f C. The density satisfies the growth condition (A4) for every ɛ > 0. Remark 1.4 The condition (A7) is necessary to ensure the uniform boundedness of the integral (1 + u ) q p ln(1 + u ) dx. The higher integrability of Theorem.1 provides (A7), which follows from (A1) for n 4. Alternatively, if we assume (A8) we have another possibility to argue at the critic point. In the proof of full regularity we return our problem to an integral with isotropic growth conditions, so that we get the claim from (A) and (A6). Under assumption (A7) or (A8) we get Theorem 1.1 a) directly from the proof of part b). There we show local boundedness of u without (A6), which is the reason why we can quote another time the results of the isotropic situation to get partial regularity (and full regularity for n = or N = 1). Fuchs [Fu] shows regularity for minimizers of variational integrals which depend on the modulus of the gradient, too. He gets an arbitrary range of anisotropy. His results are dedicated to the autonomous situation, but with a little modification and adaption of the hypotheses they extend to the case of x-dependence, too. In contrast to our assumption they are very restrictive, which can easily be seen by consideration of the functions g 1 and g 3 from of 6. In both cases the functions must be strictly increasing, convex and must satisfy lim t 0 g(x, t)/t = 0. Indeed, our functions do not necessarily perform a -condition. Considering the hypothesis (a 0) g (x, t) t g (x, t) a(1 + t ) ω g (x, t), t with an arbitrary ω 0, claimed for in [Fu], the second inequality is surely always fulfilled. ut the first inequality is very restrictive. Even 6

11 if we allow constants smaller than 1 this is in general false in our context. Fuchs does not only need this condition for having superquadratic growth, which is consistent with p in our case. He requires the monotonicity of t g (, t)/t, which he needs for (.5) and (.6) of [Fu]. Another possibility for this argumentation is a -condition for g (x, t). In case of g 3 this is false again. Obviously we can choose ω > 1 in (A8), where the difference ω 1 is the margin between q and the maximal integrability which we get from Lemma.1. For locally bounded minimizers we get dimensionless conditions between p and q: Without x-dependence ildhauer and Fuchs proved full regularity for N = 1 and in the general vector case with structure conditions under the assumption (see [F3]) q < p +, (A9) whereas the non-autonomous situation requires the much more restrictive bound (see [F1]) Our new statement is q < p + 1. THEORE 1. We assume (A)-(A5), where all involved derivatives are assumed to be continuous and (A6) if n 3 and N. Then any local minimizer u W 1,p loc L loc (Ω, RN ) of (1.1) belongs to the space C 1,α (Ω, R N ) for all α < 1, if we have (A9) and g (x, t) cg (x, t)(1 + t ) ω for ω < (p + q) + 1. (A10) Remark 1.5 Analogous to (A8) we can allow ω > 1 in (A10), where the difference ω 1 is again the range between q and the maximal integrability p +. 7

12 Regularization & higher integrability Following [F1] we consider for a fixed number 1 a cut-off function η C 1 ([0, )) with η 1 on [0, 3/], η 0 on [, ) and η c/. We define h(x, t) := η(t) + (1 η(t))λ (1 + t ) p g (x, t) Then h is a continuous function with the following properties: h(x, ) = η() = 1 for all x Ω; for all (x, t) Ω [0, ) follow 0 h(x, t) 1; g (x, t)h(x, t) λ(1 + t ) p and g (x, t)h(x, t) c()(1 + t ) p. After these preparations we define g(x, t), for 0 t g (x, t) := g(x, )+ g t ρ (x, )(t ) + g (x, τ)h(x, τ)dτdρ, for t > and finally F (x, Z) := g (x, Z ). Proof of Lemma 1.1: Obviously only the case Z > is of interest for the growth estimates. Assuming this we get D P F (x, Z) = g (x, ) Z Z + γ D P F (x, Z) = γ g (x, ) Z Z + γd P F (x, Z) = γg (x, ) Z Z + y (A4) and the definition of h we see γ D P F (x, Z) c() + c() + Z Z Z Z c()(1 + Z ) p 1 g (x, τ)h(x, τ)dτ Z Z,. γ [g (x, τ)h(x, τ)] dτ Z Z, γ [g (x, τ)h(x, τ)] dτ Z Z. γ g (x, τ)η(τ)dτ γ g (x, τ) dτ c(). 8

13 For the uniform bound of γd P F we obtain in the same way from (A4) and (A5) γ D P F (x, Z) c(1 + ) q 1 + { c (1 + Z ) q 1 c ecause of Z > we finally receive Z γg (x, τ)η(τ) dτ + 1 {(1 + Z ) q 1 + (1 + ) q 1 γ D P F (x, Z) c(1 + Z ) q 1. (1 + τ ) q 1 dτ } The uniform growth conditions of γ D P F and the uniform (p, q)-ellipticity are already established in [F1]. We only have to show that DP F is p-elliptic. We have [ ] DP F (x, Z)(P, P ) = g (x, ) P Z : P Z Z [ ] Z + g (x, τ)h(x, τ)dτ 1 P Z : P Z Z + g Z : P (x, Z )h(x, Z ) Z := T 1 + T + T 3. For the first term we see by (A3) in case p and for p < }. T 1 g (x, ) P c() P c()(1 + Z ) p P T 1 c() (1 + Z ) p Z c()(1 + Z ) p P. (1 + Z ) p P For T we conclude from the properties of g h (remember 1) T c()(1 + Z ) p P 9

14 and in an analogous way T 3 c()(1 + Z ) p P which shows the p-ellipticity. Considering the derivatives γ DP F we get γ DP F (x, Z)(P, Q) = [ ] γg (x, ) (Z : P )(Z : Q) P : Q Z Z Z [ + P : Q γ [g (x, τ)h(x, τ)] dτ 1 Z + γ [g (x, Z )h(x, Z )] (Z : P )(Z : Q) Z for arbitrary P, Q, Z R nn. In case Q = Z we see by (A4) γ D P F (x, Z)(P, Z) ] (Z : P )(Z : Q) Z = γ [g (x, Z )h(x, Z )] Z : P = η( Z ) γ g (x, Z ) Z : P ] c [ η( Z )g (x, Z ) (1 + Z ) ɛ Z : P + (1 + Z ) p+q 4 1 Z P ] c [ g (x, Z )h(x, Z )Z : P (1 + Z ) ɛ + (1 + Z ) p+q 4 P [ D = c P F (x, Z)(P, Z) ] (1 + Z ) ɛ + (1 + Z ) p+q 4 P which finally proves Lemma 1.1. Lemma.1 Under the assumptions of Theorem 1.1 we get for local minimizers u W 1,p loc (Ω, RN ) of (1.1) { u L pn n loc (Ω, RnN ) if n 3 L s loc (Ω, RnN ), for all s <, if n =. Also u belongs to the space W,t loc (Ω, RN ) for t := min {p, }. For the proof of Lemma.1 we need a Caccioppoli-type inequality: Lemma. There is a constant c > 0 independent from such that η Γ p u dx c η Γ q dx + c Γ q dx for all η C 1 0(). 10 spt η spt η

15 Proof: The growth of DP implies (sum over γ) η Γ p u dx c η DP F (, u )( γ u, γ u )dx c ηd P F (, u )( γ u, γ u η)dx η γ D P F (, u ) : γ u ηdx η γ D P F (, u ) : γ u dx := I 1 + I + I 3, where we used the method of difference quotients for the second inequlity (see [F], Lemma 3.1, for details) and Lemma 1. (part 3). Using Younginequality and Lemma 1.1 we can calculate all terms except for I 3. For this one we write { I 3 = γ η γ D P F (, u ) } : u dx = + + η γd P F (, u ) : u dx η γ D P F (, u )( γ u, u )dx γ D P F (, u ) : u γ η dx := I I 3 + I 3 3. Lemma 1.1 (part ) delivers I 1 3 c Γ q dx spt η and from (1.6) we deduce I3 3 c η Γ q dx c η Γ q dx + c Γ q dx. spt η spt η spt η 11

16 For I3 we conclude from Lemma (1.), part, I3 c η D P F (, u )( γ u, u ) (1 + u ) ɛ dx + c η Γ p+q 4 u dx. We can bound the first integral by τ η DP F (, u )( γ u, γ u )dx + c(τ) η D P F (, u )( u, u )(1 + u ) ɛ dx. If we know ɛ < 1 ( p n + ) q, n we can increase q to q + ɛ w.l.o.g. Now we can absorb the first term and bound the second one by c Γ q dx. spt η For arbitrary τ > 0 we obtain by Young s inequality η Γ p+q 4 u dx τ η Γ p u dx + c(τ) spt η Γ q dx which we handle conventionally. Proof of Lemma.1: We follow the lines of [F]. We chose η C0 ( r+ρ ) with η 1 on r and η c/ρ (asking for 0 < r R < R and 0 < ρ R r). For α := pn/(n ) (n 3) and h := Γ α(n )/n we clearly get by Lemma. r Γ α 1 dx c ρ r+ρ r q Γ dx + r+ρ dx q Γ n n. (.1) 1

17 Since p < q < α (compare (A1)), there is a θ (0, 1) such that 1 q = θ p + 1 θ α. (.) Using interpolation inequality (compare [GT], 7.9, p. 146) we obtain u q u θ p u 1 θ α, where the norms are taken over r+ρ r. It follows 1 ρ r+ρ r q Γ dx c ρ + c ρ r+ρ r dx p Γ p Γ dx. θq p r+ρ r Γ α dx (1 θ)q α (.3) R Now (A1) shows (1 θ) q p = n ( ) q p 1 < 1 and so we can find positive exponents β 1 and β such that 1 ρ R q Γ dx c ρ β 1 R p Γ dx β + r+ρ r Γ α dx n n. (.4) Apply this into (.1), for suitable β 3, β 4 > 0 we see r Γ α dx c ρ β 3 R β4 dx + c p Γ r+ρ dx q Γ n n + c r+ρ r Γ α dx and the hole filling technique provides for a ϑ (0, 1) r Γ α dx c ρ β 3 R β4 dx + c p Γ r+ρ dx q Γ n n + ϑ r+ρ Γ α dx. (.5) 13

18 Exactly as in (.3) we can calculate the second term and can follow Γ α c p dx Γ β4 ρ β 3 dx + ϑ Γ α dx (.6) r+ρ r R for a ϑ (0, 1). From a well-known Lemma by Giaquinta (see [Gi], Lemma 5.1, p. 81) we deduce Γ α c p dx Γ β4 (R r) β 3 dx, r and therefore the uniform boundedness of u in L α loc (, RnN ). In case n = we argue similarly (see [i] or [F]) for an arbitrary α > 1 p. To p q transfer the integrability to the solution u we have to show the convergence u u. For p we get the uniform boundedness of u in L loc by a combination of Lemma. and the uniform W 1,q loc (, RN )-bound of u. In case p < it appears by Young-inequality, the uniform W 1,p loc (, RN )-bound of u and Lemma. for an arbitrary r < R u p dx = Γ p 4 u p Γ p 4 dx r r Γ p u dx + dx c(r). r R r Γ p In both cases u is uniformly bounded in W,t loc (, RN ) for t := min {, p}, so we can follow after passing to a subsequence u : v in W,t loc (, RN ) and u v almost everywhere on for a function v W,t loc (, RN ). So we achieve F (, u ) F (, v) almost everywhere on and by the Lemma of Fatou and the minimality of u we see F (, v)dx lim inf F (, u )dx lim inf F (, u)dx F (, u)dx on account of F F. y uniqueness of local minimizers of (1.1) this implies u = v and so the result of Lemma.1. 14

19 3 Partial regularity A first preparation for proving Theorem 1.1 a) is Lemma 3.1 Let H := Γ p 4, Γ := 1 + u and H := Γ p 4. Than we have H W 1, loc (), H H in W 1, loc () for and u u almost everywhere for. Proof: The second statement we get from Lemma. and.1 under observation of H Γ p 4 u. This shows the first part directly, whereas the third point was shown at the end of the last section. y lower semicontinuity of the map v v for v W 1, and Lemma. we obtain η H dx lim inf η H dx lim inf c η Γ q dx + Γ q dx. spt η Since q < pn/(n ) we see by the higher integrability of Γ, the convergence Γ q Γ q almost everywhere on (see Lemma 3.1) and the Vitali convergence theorem Γ q Γ q in L 1 (spt η) and thus spt η Lemma 3. For η C0 () and arbitrary balls Ω we have η H dx c η Γ q dx + c Γ q dx. spt η spt η In case n = an analogous argumentation is possible. We define if q E + (x, r) := u ( u) x,r q dy + u ( u) x,r dy r(x) r(x) 15

20 and in the other situation E (x, r) := V ( u) V (( u) x,r ) dy, r(x) V (ξ) := (1 + ξ ) q 4 ξ for ξ R nn. So we get LEA 3.3 Fix L > 0. Then there exists a constant C (L) such that for every τ (0, 1/4) there is an ɛ = ɛ(τ, L) > 0 satisfying: if r R and we have ( u) x,r L, E(x, r) + r γ ɛ then E(x, τr) C τ [E(x, r) + r γ ]. Here γ (0, ) is an arbitrary number. We follow the lines of [F] and so the only part which needs a comment is the uniform bound of ρ ψ m dx for ρ < 1 (the function ψ m is defined in [F]). For Θ(Z) := (1 + Z ) p 4 (Z R nn ) we see ρ ψ m (z) dz = DΘ(A m + λ m u m (z)) : u m (z) dz ρ = rm n rm λ m ρrm (x m) c(ρ) r mλ m rm (x m) H dz Γ q dz, (3.1) where λ m := E(x m, r m ) + rm. Furthermore we receive for q Γ q dz c 1 + u q dz rm (x m) rm (x m) c 1 + rm (x m) u ( u) xm,rm q dz + 16 rm (x m) ( u) xm,rm q dz

21 ce(x m, r m ) + c(l). If q < we define for t (1, ) and ξ R nn V t (ξ) := (1 + ξ ) t 4 ξ and Ht (ξ) := (1 + ξ ) t. y Lemma.3 of [Ha] then we have H t (ξ 1 ) H t (ξ ) c V t (ξ 1 ) V t (ξ ). Now we can calculate [ Γ q dz H q ( u) H q (( u) xm,rm ) H + q (( u) xm,rm ) ] dz rm (x m) rm (x m) c rm (x m) V q ( u) V q (( u) xm,r m ) dz + c(l) = ce(x m, r m ) + c(l) In both cases we obtain [ ψ m (z) dz c(ρ) r m + rmλ m c(l) ]. ρ Recalling the choice of γ we have rmλ m 0 for m and the boundedness of ρ ψ m dx follows. Now the proof can be completed as in [F]. 4 Full regularity Here we consider the standard regularization: u is defined as the unique minimizer of (F (Z) = F (Z) + (1 + Z ) eq ) I [w, ] := F ( w)dx (4.1) in (u) ɛ + W 1,eq 0 () for Ω and q > q. Thereby (u) ɛ is the mollification of u with parameter ɛ and = (ɛ) := ɛ 1 + (u) ɛ eq L eq() For u we obtain, since u is a W 1,q loc -minimizer by Lemma.1 (compare [F1], Lemma.1 and.7): 17

22 Lemma 4.1 As ɛ 0 we have: u u in W 1,p (, R N ), ( 1 + u ) eq dx 0; F ( u )dx F ( u)dx; u W 1, loc L loc (Ω, RN ). If we have u L loc (Ω, RN ) then we get sup u <. For the last statement we need (A) and quote [DL]. Exactly as in the proof of Lemma.1 we can show u L pn n loc (, RnN ) uniformly. (4.) Note, that we can chose q arbitrary near to q. Therefore we can replace in all of our estimations q by q and especially (A1) stays true. For showing Theorem 1.1 b) i) we define τ(k, r) := Γ ν (ω k) dx A(k,r) where A(k, r) := r [ω > k] for balls r with the choice ν := q p < 1 pn n. Lemma.1 guarantees together with (A7) the uniform boundedness of τ. Following the argumentation of [i] (p. 65 ff.) we see { τ(h, r) X ηγ p 4 (ω h)} dx, X := A(h,r) Γ χ χ 1 β dx for β := (ν p)/ and χ := n/(n ) if n 3. To estimate the second term on the r.h.s. we have to handle the integrals γ D P F (, u ) : { η γ u [ω h] } dx, χ 1 χ 18

23 γ D P F (, u ) : { η γ u [ω h] } dx additionally to the terms in [i], p. 6. So we get for the first integral the three integrals η γ D P F (, u ) η u (ω h) dx + η γ D P F (, u ) u (ω h) dx + η γ D P F (, u ) u ω dx :=S 1 + S + S 3. Considering S we obtain for an arbitrary τ > 0 S c η Γ q 1 u (ω h) dx τ +c(τ) η Γ p u (ω h) dx Γ ν dx where the first term can be observed in the same way as in [i]. Furthermore we have S 1 c η Γ q η (ω h) dx c Γ ν η (ω h) dx + c Γ ν dx by making use of Young s inequality. For S 3 one estimates S 3 c η Γ q ω dx 19

24 τ η Γ p ω dx + c(τ) Γ ν dx for all τ > 0, which allows an absorption for another time. After estimating γ D P F (, u ) : { η γ u [ω h] } dx in the same way, we finally have showed τ(h, r) cx η Γ ν (ω h) dx + cτ(h, r) + Γ ν dx. Assuming w.l.o.g. h k 1 now we have τ(h, r) The choice of ν and (A7) deliver or equivalently c ( r r) (h k) X τ(k, r). (4.3) ν < pn n, χ χ 1 β < ν. In case n = we can achieve this, if we choose χ big enough. Now we receive X Combining (4.3) and (4.4) we see τ(h, r) Γ ν dx n c τ(k, r) 1+ ( r r) (h k) + 4 n, n c τ(k, r) n. (4.4) (h k) 4 n and u W 1, loc (Ω, RN ) follows by [St], Lemma 5.1, as in [i]. So we locally have a variational problem with isotropic growth conditions and by (A6) the 0

25 claim follows (compare [F1], Lemma.7). For showing Theorem 1.1 b) ii) we similarly use DeGiorgi-arguments but with other estimates at the critical point. For a technical simplification we assume ɛ = 0 in (A4). So we define (with an obvious meaning of Γ and ω ) τ(h, r) := Γ ν (ω k) dx A(h,r) for ν := q + ω 1. y (A8) and (4.) we get the uniform boundedness of τ for another time. We obtain by decomposition ν = ν 1 + ν for ν 1 := (n )ν/n, ν := ν/n and χ := n/(n ) τ(h, r) = Γ ν 1 (ω h) Γ ν dx A(h,r) X A(h,r) Γ ν 1 χ 1 χ (ω h) χ dx { ηγ p 4 (ω h) A(h,r) 1 χ } χ dx. Γ χ ν χ 1 dx Note, that ν 1 < p since ν < pn/(n ). Here X stands for the second integral in the second line of the estimation and η C0 ( br ) is a suitable cut-off function. y Sobolev s inequality we have τ(k, r) X y the choice of ν we have X = A(h,r) Γ ν dx n χ 1 χ { ηγ p 4 (ω h)} dx. (4.5) 1 τ(k, r) (h k) 4 n. n For the remaining integral we can argue as before in the proof of part b) i). During the calculation of this integral we have to estimate the terms S and S 3 as in the proof of part b) i). We receive from (A4) γ g (x, t) c [g (x, t) + (1 + t ) p+q 4 ]. (4.6) 1

26 Assuming w.l.o.g. ɛ = 0 we get γ g (x, t) γ g (x, st) tds + γ g (x, 0) ] [g (x, st)t + (1 + (st) ) p+q 4 1 t c 0 c [g (x, t) + (1 + t ) p+q 4 ], ds + c here we quote [AF], Lemma.1, if p + q < 4. y (4.6) we have S c η g (, u ) u (ω h) dx +c η Γ p+q 4 u (ω h) dx. For the second term we obtain the upper bound τ η Γ p u (ω h) dx + c(τ) η Γ q (ω h) dx for all τ > 0 and argue is in part i). Finally we receive for the first integral S 1 by Young s inequality S 1 τ η g (, u ) u Γ ω (ω h) dx +c(τ) η g (, u )Γ ω (ω h) dx. We want to absorb the first one. First we estimate D F (x, P )(X, X) = g (x, P ) ( X (P : X) P P { min g (, P ), g (, P ) P for all P, X R nn. Finally we get by (A8) g (, u ) u c min Γ ω ) (P : + g X) (x, P ) P } X { g (, u ), g (, u ) u } u

27 cd P F (, u )( γ u, γ u ) cd P F (, u )( γ u, γ u ), (4.7) where the sum is taken over γ {1,..., n}. Considering S, 1 the remaining term is S := Γ q 1+ω (ω h) dx τ(h, r) + Γ ν dx (4.8) by (A8). For the examination of S 3 we argue similar, using (4.6) S 3 c η g (, u )Γ 1 ω dx +c =:S S 3. η Γ p+q 4 ω dx So we can follow S 3 τ η Γ p ω dx + c(τ) η Γ q dx and absorb the first term. For S3 1 we get by Young s inequality S3 1 τ η g (, u ) Γ Γ ω ω dx +c(τ) η g (, u )Γ ω dx. The second integral is bounded by S from (4.8), whereas we obtain by (4.7) c η DP F (, u )(e i ω, e i ω )Γ dx as a upper bound for the first one (compare [i], (3), together with (A)). Now it is possible to end up the proof like before (compare [i] for more details). 3

28 5 Locally bounded minimizers In this section we prove Theorem 1.. In a first step we show Lemma 5.1 Under the assumptions of Theorem 1. we have u L p+ loc (Ω, RnN ). Proof: We show the uniform boundedness of u in W 1,t loc Lp+ loc (, RnN ) (t := min {, p}). Then the claim of Lemma 5.1 follows as at the end of section. We fix η C0 ( r ) with 0 η 1, η 1 on ρ r and η c/(r ρ) 1 and get by integrating by parts, following the lines of [i], p. 155, (let h 1, k = 1 and α = 0) Γ p+ ρ dx c η Γ p u dx + 1 r 4 η Γ p+ r To calculate the only critical integral we use Lemma. and see for a suitable β > 0 c η Γ p u dx c(r ρ) r r Γ q c(r ρ) β dx Γ p+ r dx dx (5.1) because q < p +, using Young s inequality. Finally we obtain Γ p+ dx c(r ρ) β + 1 Γ p+ dx. (5.) ρ r y [Gi], Lemma 5.1, p. 81, one can follow from (5.) the uniform boundedness of u L p+ loc (, RnN ). Now we can deduce from (5.1) the uniform boundedness of u in W 1,t loc (, RnN ). So the existence of a W 1,q loc -minimizer is proved and therefore we can work with the -regularization from the proof of Theorem 1.1 b) ii) and Lemma 4.1. Analogous to Lemma 5.1, we can show u L p+ loc (, RnN ) uniformly, (5.3) since the additional term Γ eq dx is not critical by Lemma 4.1, part 1. Now we can show Lemma 5. Under the assumptions of Theorem 1. we have u L t loc (Ω, RnN ) for all t < uniformly in. 4

29 Proof: We quote for another time [i] and get for α 0 and k 1 η k Γ p+α+ dx c(η) + c η k Γ p +α u dx. (5.4) For calculating the u -integral we need an inequality of Caccioppoli-type: Lemma 5.3 Under the assumptions of Theorem 1. for all α 0 and all η C0 () there is a constant c = c(α, η) independent of with [ ] η k Γ p +α u dx c 1 + η k Γ p+α dx. Proof: Inserting φ := γ u Γ α η k and summing up over γ {1,..., n}, we only have one term of interest (compare [F5], p. 36): { } γ D P F (, u ) : γ u Γ α η k dx. (5.5) Using (1.3) and (4.6) one obtains the three integrals R 1 := η Γ q+α η k 1 dx, R := η k g (, u ) u Γ α dx, R 3 := η k Γ p +q+α 4 u dx. For k 1 we get by Young s inequality R 1 τ η k Γ p+α+ dx + c(η, τ), since q < p +. y (5.4) we can handle the r.h.s. for τ small enough. Considering R 3 we see R 3 τ η k Γ p +α u dx + c(τ) η k Γ q+α dx. We absorb the first integral by (5.4) and see η k Γ q+α dx τ η k Γ p++α dx + c(τ ) 5

30 for the second one by Young s inequality and q < p +. Obviously, the critical term is R : For this integral we get R τ η k g (, u ) u Γ ω Γ α dx + c(τ) η k g (, u )Γ ω+α dx. The first term on the r.h.s. can be absorbed in the l.h.s. of (5.5) for τ 1 if we note (4.7) and (A10). For the remaining integral we have the estimate τ Γ q+ω 1+α Γ p+α+ dx + c(τ ) if we use Young s inequality and ω < (p + q) + 1. This ends up the proof of Lemma 5.3 and we have showed [ ] η k Γ p+α+ dx c(η) 1 + η k Γ p+α dx (5.6) by (5.4). Iteratively we can follow the claim of Lemma 5.. As a starting point one can choose α = by Lemma 5.3. In the next step we show Lemma 5.4 Under the assumptions of Theorem 1. u is uniformly bounded in L loc (Ω, RnN ). The claim of Theorem 1. follows by reducing our problem to a variational integral with isotropic growth. Proof of Lemma 5.4: We consider the function τ(h, r) := A(h,r) where A(h, r) := r [Γ > h] and proof τ(h, r) ( r r) n 1 n 1 c Γ q p +1 (Γ h) dx s (h k) n 1 1 n 1 s t τ(k, r) 1 n 1 n 1 s[1+ 1 t ] for 0 < k < h and 0 < r < r < R with s, t > 1 and c independent of h, k, r, r,. For an arbitrary s > 1 we calculate (compare [i], p. 157, for details) [ τ(h, r) c(s) I n 1 n 1 s 1 n 1 s 1 + I n 6 ]. (5.7)

31 y a later choice of t > 1 we get I n 1 n 1 s 1 ( r r) n 1 n 1 c s (h k) n 1 1 n 1 s t τ(k, r) 1 n 1 n 1 s[1+ 1 t ]. (5.8) For I one obtains with sum over j {1,..., N) c I n n 1 c (h k) n 1 n 1 t η D F (, u )(e j Γ, e j Γ )dx τ(k, r) 1 For the term in brackets we show n n 1 1 n n 1 1 t. (5.9) Lemma 5.5 Suppose the assumptions of Theorem 1.. Then there is a constant c > 0 for which we have the estimate η D c F ( u ) (e j Γ, e j Γ ) dx τ(k, r) ( r r) (h k) for all η C 0 ( br (x 0 )). Proof We choose the test function ϕ := η γ u [Γ h] + and get Q 1 := ηγ q η (Γ h) dx Q := Q 3 := η Γ q 1 γ u (Γ h) dx η Γ q Γ dx. as the interesting terms (see [F1]). It follows c Q 1 Γ q p +1 (Γ h) dx ( r r)(h k) c τ(k, r), ( r r) (h k) 7

32 since we can assume w.l.o.g. h k 1. The Young s inequality delivers Q τ η Γ p γ u (Γ h) dx + c(τ) η Γ q p (Γ h) dx. The growth of DP F allows us to absorb the first term (compare [F1], (.41)). The other term can be bounded by c τ(k, r). ( r r) (h k) For Q 3 we obtain by Young s inequality Q 3 τ η Γ p Γ dx + c(τ) The inequality η Γ q p +1 dx η Γ q p +1 dx. c (h k) τ(k, r) c τ(k, r) ( r r) (h k) ends up the proof of Lemma 5.5. y recapitulating (5.7)-(5.9) and Lemma 5.5 we have τ(h, r) ( r r) n 1 n 1 c s (h k) n 1 1 n 1 s t τ(k, r) 1 n 1 n 1 Since s, t are arbitrary numbers in (1, ), we can achieve [ 1 n ] > 1 n 1 s t s[1+ 1 t ]. and the claim follows by [St], Lemma Examples Let η C ([0, ), [0, 1]) and θ > 0 such that η(t) = { 0, on [4kθ, (4k + 1)θ] 1, on [(4k + )θ, (4k + 3)θ] 8

33 for all k N 0. We define for 1 < p q < and t 0 g θ (t) := Then we have t ρ 0 0 ] [η(τ)(1 + τ ) p + (1 η(τ))(1 + τ ) q dτdρ Lemma 6.1 The function g θ satisfies the condition (A3). Whereas the estimates for g θ are trivial, we need for handling g θ the inequality (1 + t ) p 1 t t 0 (1 + τ ) p dτ c (6.1) for a c > 0, which delivers directly the first of the two requested estimates. We see lim(1 + t ) p 1 t 0 t and lim (1 + t ) p t 1 t t 0 t 0 (1 + t ) p = 1 which proves (6.1). y similar arguments we get (1 + t ) q 1 t t and thereby the estimate from above. dependence is 0 (1 + τ ) p = 1 p 1, (1 + τ ) q dτ c g 1 (x, t) := α(x)g θ (t) A trivial way to receive a spatial for a C -function α which is bigger than ɛ 0 > 0. Now it is easy to see, that g 1 satisfies the assumptions (A3), (A4) for ɛ = 0 and (A5). A possibility which is not as trivial is g (x, t) := (1 + t ) f(x) for a C -function f satisfying f(x) > 1 for all x Ω. Than we obtain Lemma 6. The function g satisfies (A3), (A4) for any ɛ > 0, (A5) and if f(x) on Ω we have (A6) for F (x, P ) := g (x, P ). 9

34 Proof: W.l.o.g. let f(x) >. Note g 8(x, t) = f(x)(1 + t ) f(x) 1 t and g 8(x, t) = f(x)(f(x) )(1 + t ) f(x) t + f(x)(1 + t ) f(x) 1. So we see (A3) by p := inf f and q := sup f. Furthermore we get γ g 8(x, t) = γ f(x) [f(x) 1] (1 + t ) f(x) t We can bound γ g 8 by Using the estimate + 1 f(x)(f(x) ) γf(x) ln(1 + t )(1 + t ) f(x) t + γ f(x)(1 + t ) f(x) f(x) γf(x) ln(1 + t )(1 + t ) f(x) 1. ln(1 + t )(1 + t ) f(x) 1. ln(1 + t ) c(ɛ)(1 + t ) ɛ takes (A4). If we derive γ g 8(x, t) once again we receive (A5) for q := sup f + ɛ. Now we prove (A6): DF 8 (x, P ) = g (x, P ) P P, D F 8 (x, P ) = g (x, P ) P ( I P P P ) + g (x, P ) P P P = f(x)(1 + P ) f(x) 1 I + f(x)(f(x) )(1 + P ) f(x) P P. Thereby we have for P, Q R nn D F 8 (x, P ) D F 8 (x, Q) f(x) n (1 + P ) f(x) 1 (1 + Q ) f(x) 1 + f(x) f(x) (1 + P ) f(x) P P (1 + Q ) f(x) Q Q =: α 1 + α. Estimating α 1 and α seperatly and using [AF], Lemma.1, if < f(x) < 3 we obtain (i {1, }) α i c(1 + P + Q ) f(x) γ P Q γ 30

35 for an arbitrary γ (0, 1). This shows for q := sup f the requested Höldercondition for D F. Due to a local argumentation it is possible to choose p and q arbitrarily near and to quote older regularity results for minimizers of g Ω (, w) dx. This motivates the following construction: let t ρ ] g 3 (x, t) := [η(τ)(1 + τ ) p+f(x) + (1 η(τ))(1 + τ ) q+f(x) dτdρ 0 0 for an η as in the definition of g θ and a function f C (Ω, R + 0 ). Here we can produce an arbitrarily wide range of anisotropy with a non-trivial x- dependence. Following the arguments of the two examples before, it is easy to see, that all of our assumptions are satisfied. References [Ad] [AF] [i] R. A. Adams (1975): Sobolev spaces. Academic Press, New York- San Francisco-London. E. Acerbi, N. Fusco (1989): Partial regularity of minimizers of nonquadratic functionals: the case 1 < p <. J. ath. Anal. Appl. 140, ildhauer (003): Convex variational problems: linear, nearly linear and anisotropic growth conditions. Lecture Notes in athematics 1818, Springer, erlin-heidelberg-new York. [r] D. reit (009): Regularitätssätze für Variationsprobleme mit anisotropen Wachstumsbedingungen. PhD thesis, Saarland University. [F1] [F] [F3]. ildhauer,. Fuchs (005): C 1,α -solutions to non-autonomous anisotropic variational problems. Calculus of Variations 4(3), ildhauer,. Fuchs (001): Partial regularity for variational integrals with (s, µ, q)-growth. Calculus of Variations 13, ildhauer,. Fuchs (00): Elliptic variational problems with nonstandard growth. International athematical Series, Vol. I, Nonlinear problems in mathematical physics and related topcs I, in honor of Prof. O.A. Ladyzhenskaya. y Tamara Rozhkovska, Novosibirsk, Russia (in Russian), y Kluwer/Plenum Publishers (in English),

36 [CG] [DL] [EL] [Fu] [Gi] [G] [GT] [Ha] [LU] [Uh] G. Cupini,. Guidorzi, E. ascolo (003): Regularity of minimizers of vectorial integrals with p q growth. Nonlinear Analysis 54 (4), A. D Ottavio, F. Leonetti, C. ussciano (1998): aximum prinziple for vector valued mappings minimizing variational integrals. Atti Sem. at. Fis. Uni. odena XLVI, L. Esposito, F. Leonetti, G. ingione (001): Sharp regularity for functionals with (p, q)-growth. Journal of Differential Equations 04, Fuchs (008): Local Lipschitz regularity of vector valued local minimizers of variational integrals with densities depending on the modulus of the gradient. ath. Nachrichten (in press).. Giaquinta (1993) Introduction to regularity theory for nonlinear elliptic systems. irkhäuser Verlag, asel-oston-erlin.. Giaquinta, G. odica (1986): Remarks on the regularity of the minimizers of certain degenerate funcionals. anus. ath. 57, D. Gilbarg und N. S. Trudinger (001): Elliptic partial differential equations of second order. Springer, erlin-heidelberg-new York. C. Hamburger (199): Regularity of differential forms minimizing degenerate elliptic functionals. J. reine angew. ath. 431, O. A. Ladyzhenskaya, N. N. Ural tseva (1964/1968): Linear and quasilinear elliptic equations. Nauka, oskow/ English Translation: Academic Press, New York. K. Uhlenbeck (1977): Regularity for a class of nonlinear elliptic systems. Acta. ath. 138,