Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth

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1 Partial regularity of strong local minimizers of quasiconvex integrals with p, q)-growth Sabine Schemm Mathematisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg, ismarckstr. 1 1/, Erlangen, Germany schemm@mi.uni-erlangen.de) Thomas Schmidt Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 405 Düsseldorf, Germany schmidt.th@uni-duesseldorf.de) We consider strictly quasiconvex integrals F[u] := fdu) Ω for u :R n Ω R N in the multi-dimensional calculus of variations. For the C -integrand f :R Nn R we impose p, q)-growth conditions γ ξ p fξ) Γ1+ ξ q ) for all ξ R Nn with γ, Γ > 0 and 1 < p q < min { p+ 1 n, n 1 n p}. Under these assumptions we prove partial C 1,α loc -regularity for strong local minimizers of F and the associated relaxed functional F. 1. Introduction In this paper we investigate the regularity of strong local minimizers of autonomous variational integrals F[u] := fdu) 1.1) Ω defined on vector-valuedmaps u : Ω R N, N 1. Here Ω denotes a bounded open set in R n, n, and f : R Nn R is a C -function satisfying suitable assumptions described below. The existence and the partial regularity of minimizers of F are classical issues in the modern calculus of variations, and they have been extensively studied, especially over the last twenty years. Specifically, in the present paper we will focus on gradient regularity under the basic assumption that f is quasiconvex; that is fξ +Dϕ) fξ) 1

2 S. Schemm and T. Schmidt holds for all ξ R Nn and all ϕ C c ; R N ), where denotes the unit ball in R n. Quasiconvexity,introduced by Morreyin his seminal paper [39], generalizesthe classical convexity assumption in the calculus of variations and has turned out to be a key concept for both, the existence and the partial regularity of minimizers. In addition, the central role of quasiconvexity in nonlinear elasticity has been pointed out in the fundamental work of all [5]. efore presenting our theorems let us briefly describe some previous existence and regularity results. Primarily, imposing the standard growth conditions γ ξ p fξ) Γ1+ ξ p ) 1.) for some p > 1, Morrey [39] proved that quasiconvexity is a necessary and sufficient condition for lower semicontinuity of F with respect to weak W 1,p -convergence see also [37, 1, 33, 30, 38]). This semicontinuity property is in turn, via the direct method of the calculus of variations, intimately linked to the existence of minimizers of F. As for regularity, classical examples of minimizers with singularities of the gradient can be constructed [15, 41, 49, 50], even for smooth convex functionals and n = 3, showing that in the vectorial case everywhere regularity of minimizers in the interior of Ω does not hold. Therefore, one is led to consider partial regularity, i.e. regularity outside a negligible closed subset of Ω, called the singular set. For quasiconvex functionals and p partial C 1,α loc -regularity of minimizers has first been shown by Evans [1]; see [3, 5,, 4, 17, 31] for extensions and variants. Partial regularity in the subquadratic case 1 < p < has been eventually proved by Carozza & Fusco & Mingione [13]; see also [48, 16]. Contrary to convex functionals, quasiconvex functionals may in general admit non-trivial critical points, i.e. weak solutions of the Euler equation which are not absolutely) minimizing. Actually, Müller & Sverák [40] have even constructed examples of critical points which are nowhere C 1. This result is in sharp contrast to the partial regularity of minimizers and leads to the investigation of an intermediate notion, namely local minimizers of F in the sense of the following definition. Definition 1.1 W 1, q local minimizer, [3]). Let 1 p, 1 q. A map ū W 1,p Ω, R) with F[ū] < is called a W 1, q local minimizer of F if there exists some δ > 0 such that F[ū] F[ū+ϕ] holds for all ϕ W 1, q 0 Ω, R N ) with Dϕ L q Ω,R Nn ) δ. In particular, for 1 q < we call ū a strong local minimizer and for q = a weak local minimizer. Having introduced this notion it is natural to ask whether non-trivial local minimizers of F exist and if they are still regular or not. Actually, the investigation of the existence and non-existence) of local minimizers has followed previous developments [6, 7] for critical points, and has, up to now, focussed on the case of L 1 local minimizers with affine boundary data. For instance, if the underlying domain Ω is an annulus in R there exist non-trivial L 1 local minimizers [44, 3], while for a starshaped Ω every L 1 local minimizer is already absolutely minimizing [51]. In

3 Strong local minimizers of quasiconvex integrals with p, q)-growth 3 fact, generalizing these ideas, Taheri [5, 53] has provided multiplicity bounds for local minimizers in terms of topological invariants of Ω. Clearly, the examples of L 1 local minimizers are also W 1, q local minimizers for every 1 q. However, in the light of [3, Section ] it would be interesting to discuss whether non-trivial examples of W 1, q local minimizers still exist in the simple case that Ω is a ball. Moreover, in view of Remark.5 below they should ideally possess the additional feature that they are not W 1,p local minimizers. Indeed, for q > p we are not aware of any theoretical obstruction but no such examples seem to be present in the literature. Assuming standard growth 1.), the regularity theory for W 1, q local minimizers has been started in a recent interesting work of Kristensen & Taheri [3]. Let us restate their result: Theorem 1. [3]). Let p <, 1 q <. Assume that f C R Nn ) is uniformly strictlyquasiconvex with 1.) and that ū W 1, q loc Ω, RN ) W 1,p Ω, R N ) is a W 1, q local minimizer of F. Then there exists an open set Ω 0 Ω with Ω\Ω 0 = 0 such that ū C 1,α loc Ω 0, R N ) for every 0 < α < 1. A similar result for weak local minimizers is contained in [3] and an analogous statement in the subquadraticcase 1 < p < has been established in [1]. We stress that in the light of the counterexamples from [40] these theorems treat a borderline case of regularity. Finally, we mention that it would be desirable to remove the technical assumption ū W 1, q loc Ω, RN ) in Theorem 1.. However, at present it seems quite hard to achieve this. Next we turn to a generalization of 1.). Actually, starting with a series of papers by Marcellini see e. g.[35, 36]) an increasing interest in more flexible growth conditions than 1.) has emerged. In this paper we concentrate on the so-called p, q)-growth conditions γ ξ p fξ) Γ1+ ξ q ) 1.3) with two growth exponents 1 < p q <. In the general vectorial setting the regularity theory for 1.3) was started in [4]. Assuming that f is strictly convex pn and q < min{p+1, n 1 } the authors showed partial C1,α loc -regularity of minimizers of F. Subsequently, considering less restrictive conditions on the growth exponents, higher integrability results for the gradient of minimizers have been given in [18, 19], and partial regularity has been established in [9]. Here, the most general condition on the exponents, appearing in [18, 9], reads q < n+ n p. For results concerning non-autonomous functionals we refer to [0, 10, 14, 8]. The quasiconvex case is more recent. In [7, 34,, 8, 9] the semicontinuity properties in W 1,p Ω, R N ) of quasiconvex functionals satisfying 1.3) have been investigated. For our approach the following notion from [7] has turned out to be crucial: Definition 1.3 W 1,p -Quasiconvexity, [7]). We say that f is W 1,p -quasiconvex iff fξ +Dϕ) fξ) holds for all ξ R Nn and all ϕ W 1,p 0 ; R N )

4 4 S. Schemm and T. Schmidt Relying on [7, ] it has been shown in [46] that strict) W 1,p -quasiconvexity together with 1.3) and some restrictions on the exponents p and q allows to establish both, existence and partial regularity, of absolute minimizers of F. Precisely, the restriction on the exponents reads q < np n 1 for the existence and 1 < p q < p+ min{,p} n 1.4) for the regularity. We refer to the forthcoming paper [45] for similar results in the higher order case. For n = N an important class of examples is given by the polyconvex integrands fξ) = 1+ ξ ) p +hdet ξ), 1.5) where h is a convexfunction of growthrate q n. These integrands areof some interest in non-linear elasticity as pointed out in [5, 6, 7, 34]. Moreover, we recall from [7] that f from 1.5) is W 1,p -quasiconvex if and only if p n holds. Thus, in this case the above existence and regularity results apply. Let us mention, at this stage, that polyconvex integrands with a structure related to the one in 1.5) and p > n 1, but with a completely different growth behaviour, have previously been treated in [4] by means of more specific methods taking into account the peculiar nature of the functional. In the case p < n the integrands 1.5) are not W 1,p -quasiconvex and the above mentioned results do not apply. However, this case, in which F can potentially admit discontinuous minimizers, is of particular physical interest. To extend the existence and regularity results, a relaxation method, which is closely related to the classical idea of the Lebesgue-Serrin extension, has been introduced in [34,, 11, 47]. Precisely, one considers the relaxed functional { } F[u] := inf lim inf F[u k] : W 1,q k loc Ω, RN ) u k u weakly in W 1,p Ω, R N ) 1.6) for u W 1,p Ω, R N ). It is not hard to see from this definition that the minimum of F is attained on every Dirichlet class; see[47]. Furthermore, invoking representation results from [, 11], the approach of [46] has been carried over to minimizers of F. Precisely, assuming that f is strictly quasiconvex with 1.3) and 1.4) partial regularity of minimizers of F has been established in [47]. The aim of the present paper is now to examine the regularity properties of local minimizers of quasiconvex functionals satisfying p, q)-growth conditions. Clearly, our main interest remains in the model case 1.5), where examples of L 1 local minimizers have been provided in [53, Section 4]. Precisely, our results are the following: If f is strictly W 1, q -quasiconvex, we prove partial C 1,α loc -regularity for W 1, q local minimizers ū of F; see Theorem.1. In addition, if f is only strictly quasiconvex, following the approach of[47] we prove a similar regularity theorem for W 1, q local minimizers ū of the relaxed functional F; see Theorem.. In both cases we are mainly interested in the case q > p where, as in [3], we need to impose the technicalintegrabilityassumptionū W 1, q loc Ω, RN ). Notethat,asabyproduct,this assumptionallowstoworkwithw 1, q -quasiconvexityinsteadofw 1,p -quasiconvexity in Theorem.1. Our results generalize those obtained in [3, 1, 46, 47].

5 Strong local minimizers of quasiconvex integrals with p, q)-growth 5 Finally, let us briefly comment on some technical issues: In the subquadratic case 1 < p < we improve the condition 1.4) replacing it by { n 1 1 < p q < min n p,p+ 1 }. 1.7) n Note that in the model case 1.5) with q = n = N = the bound 1.7) allows to replace the condition p > 8 5 from 1.4) by p > 3. However, the reason for this improvement is mainly a technical one. Moreover, we mention that following[3, 1] we use a blow-up argument based on the excess Ex,r) = 1+ Dū Dū) x,r ) p Dū Dū)x,r 1.8) rx) to prove the partial regularity. In particular, even in the case of absolute minimizers, we provide an alternative proof of the results in [46, 47], where the A-harmonic approximation method has been used.. Statement of the results In this section we state our main results concerning partial regularity of strong local minimizers. Starting with a growth and a coercivity condition we will now supply precise statements of our assumptions. H1) q-growth: There exists a bound Γ > 0 such that we have 0 fξ) Γ1+ ξ q ) for every ξ R Nn. H) p-coercivity: There is a coercivity constant γ > 0 such that we have fξ) γ ξ p for every ξ R Nn. Next we state two quasiconvexity conditions, which will be imposed in Theorem.1 and Theorem., respectively. H3) Strict W 1, q -Quasiconvexity: For each L > 0 there is a convexity constant ν L > 0 such that we have ) fξ +Dϕ) fξ) νl 1+ Dϕ ) p Dϕ for all balls r x 0 ) R n, for all ξ R Nn with ξ L + 1 and for all ϕ W 1, q 0 r x 0 ), R N ). H4) Strict Quasiconvexity: For each L > 0 there is a convexity constant ν L > 0 such that we have ) fξ +Dϕ) fξ) νl 1+ Dϕ ) p Dϕ for all balls r x 0 ) R n, for all ξ R Nn with ξ L + 1 and for all ϕ C c rx 0 ), R N ).

6 6 S. Schemm and T. Schmidt Now we present our first main result, a regularity result for strong local minimizers of the functional F defined in 1.1). Theorem.1. Let q [1, ) and 1 < p q < min { p+ 1 n, n 1 } n p..1) Assume that f C R Nn ) satisfies H1) and H3) and that ū W 1, q loc Ω, RN ) W 1,p Ω, R N ) is a W 1, q local minimizer of F in the sense of Definition 1.1. Then there exists an open set Ω 0 Ω with Ω\Ω 0 = 0 such that ū C 1,α loc Ω 0, R N ) for every 0 < α < 1. Our second main result concerns strong local minimizers of F from 1.6), where we define local minimizers of F along the lines of Definition 1.1 with F replaced by F. efore stating the result we recall some properties of the functional F: Assuming H1) with 1 < p q < np n 1 one has from [, 47] F[u] QfDu) for u W 1,p Ω, R N ),.) F[u] = Ω Ω QfDu) for u W 1,q Ω, R N ),.3) where Qf denotes the quasiconvex envelope of f. Furthermore, it has been shown in [, 11] that F depends on the domain Ω like a Radon measure, whose absolutely continuous part has density QfDu). These facts will be crucial in the proof of the following result. In particular, we mention that by an argument of [47] they can be used to prove the validity of Euler s equation for minimizers of F. This is an important observation for the proof of the following theorem. Theorem.. Let q [1, ) and 1 < p q < min { p+ 1 n, n 1 } n p..4) Assumethat f C R Nn ) satisfies H1),H) and H4) and that ū W 1, q loc Ω, RN ) W 1,p Ω, R N ) is a W 1, q local minimizer of F on Ω. Then there exists an open set Ω 0 Ω with Ω\Ω 0 = 0 such that ū C 1,α loc Ω 0, R N ) for every 0 < α < 1. We highlight some features of the previous theorems. Remark.3. In some sense W 1, q -quasiconvexity is necessary for the existence of W 1, q local minimizers. Precisely, adapting the proof of [53, Proposition 4.1] one finds: If ū W 1,p Ω, R N ) is a W 1, q local minimizer of F then for every ϕ W 1, q, R N ) there holds fdūy)+dϕx)) fdūy)) for a.e. y Ω.

7 Strong local minimizers of quasiconvex integrals with p, q)-growth 7 Remark.4. If q q then strict) W 1, q -quasiconvexity is equivalent to strict) quasiconvexity. Combining this with.3), Theorem.1 and Theorem. turn out to be equivalent in this case. Remark.5. In the case q p it follows from [3, Section ] that Theorem.1 can be reduced to the case of absolute minimizers. Remark.6. Under the additional assumption lim sup r 0 + Du Du) x,r L rx),r Nn ) < δ Theorem.1 and Theorem. hold also in the case of weak local minimizers, i e. for q =. This generalization is straightforward along the lines of [3, 1]. Remark.7. The proofs of the theorems will show that we can choose Ω 0 such that Ω\Ω 0 { x Ω : liminf Ex,r) > 0 or limsup Du) x,r = } r 0+ r 0+ holds, where Ex,r) is defined in 1.8). Remark.8. Under the assumptions of Theorem.1 or Theorem., if f C R Nn ) then we have u C Ω 0, R N ). Once C 1,α loc -regularity is proved, this higher regularity result follows from the application of linear theory to the Euler equation. 3. Preliminaries Throughout this paper we denote by a c a positive constant possibly varying from line to line. The dependences of such constants will only occasionally be highlighted. We write r x) for the open ball with center x and radius r in R n and abbreviate r := r 0) and := 1. In addition, we will use the common abbreviations u x,r := 1 u := u and u r := u 0,r for mean values, where rx) rx) rx) denotes the n-dimensional Lebesgue measure. Moreover, for β > 0 we define the functions V β : R k R k and W β : R k R k by V β ξ) = 1+ ξ ) β 1 ξ, W β ξ) = 1+ ξ ) β 1 ξ 3.1) for ξ R k, k N. Since we are mostly dealing with β = p, where p is a fixed exponent, we abbreviate V = V p and W =. Next, we will collect some useful Wp properties of V and W: Clearly, we have V β ξ) = V β ξ ), W β ξ) = W β ξ ) and c 1 W β ξ) V β ξ) c W β ξ), 3.) where c depends only on β. Furthermore, the functions V β t) and W β t) are both non-decreasing in t 0 and some elementary calculations show that W is convex for 1 p < and W p is convex for 1 p in contrast to V and V p ). Some additional properties are summarized in the following lemma.

8 8 S. Schemm and T. Schmidt Lemma 3.1. Let β > 0, 1 < p <, and M > 0. Then, for all ξ,η R k and t > 0 we have i) Vtξ) max { t,t p } Vξ) ; ii) V β ξ +η) c V β ξ) + V β η) ); iii) 1+ ξ + η ) p c1+ Vξ) + Vη) ); iv) V p 1 ξ) η Vξ) + Vη). Here, c depends only on β and p, respectively. Proof. The assertions i) and iii) are easy to check. ii) has been proved for 1 < β < 1 in [13, Lemma.1] and is easily seen to hold for all β > 0. iv) follows from the fact that V p 1 is non-decreasing. Next we restate an integral inequality for V; see for instance [46]. Lemma 3.. Let 1 < p < and u W 1,p Ω, R N ). Then we have VDu Du) Ω ) c VDu) with a constant c depending only on p. Ω Furthermore, we recall a Poincaré type inequality and a Sobolev-Poincaré type inequality for V. Lemma 3.3. We consider 1 < p <, a ball r x 0 ) in R n and a function u W 1,p r x), R N ). Then, we have u ux0,r ) V c VDu). 3.3) r Ω In addition, setting p # := n n p > for 1 < p < we have V u ux0,r r ) ) p # 1 p # c The constant c depends only on n, N and p in both inequalities. )1 VDu). 3.4) Here, 3.4) has been proved in [16, Theorem ] and 3.3) follows easily from the standard Poincaré inequality for p and from 3.4) for 1 < p <. The reader should note that a weaker version of 3.4) has previously been established in [13]. Next we restate some estimates for smoothing operators, which will be crucial for our approach. These estimates, introduced first in [, Lemma.], have already been used in the regularity theory of integrals with p, q)-growth; see [4, 46, 47]. We state them in the form of [46, Lemma 6.3].

9 Strong local minimizers of quasiconvex integrals with p, q)-growth 9 Lemma 3.4. Let 0 < r < s and s Ω. We define a linear smoothing operator T r,s : W 1,1 Ω; R N ) W 1,1 Ω; R N ) for u W 1,1 Ω; R N ) and x Ω by T r,s ux) := ux+ϑx)y)dy, where ϑx) := 1 1 { } max min{ x r,s x },0. 3.5) With this definition, for all 1 p q < np n 1 and all u W1,p Ω; R N ) the following assertions are true: T r,s u W 1,p Ω; R N ), u = T r,s u almost everywhere on Ω\ s ) r, 3.6) T r,s u u+w 1,p 0 s \ r ; R N ), 3.7) DT r,s u cn)t r,s Du almost everywhere on Ω, 3.8) T r,s u q;s\ r cn,p,q)s r) n q n 1 p DT r,s u q;s\ r cn,p,q)s r) n q n 1 p T r,s u p;s\ r cn,p) u p;s\ r, 3.9) DT r,s u p;s\ r cn,p) Du p;s\ r, 3.10) [ Ξt) sup Ξr) Ξs) + sup Ξt) ] 1 p, t ]r,s[ t r t ]r,s[ s t 3.11) [ ] 1 p Ξt) Ξr) Ξs) Ξt) sup + sup. t ]r,s[ t r t ]r,s[ s t 3.1) Here we have used the abbreviations Ξt) := u p p; t and Ξt) := Du p p; t. Further estimates of the terms on the right-hand side of 3.11) and 3.1) can be obtained by means of the following simple lemma. Lemma 3.5. Let < r < s < and a continuous non-decreasing function Ξ : [r,s] R be given. Then, there are r ]r, r+s 3 [ and s ] r+s 3,s[ for which hold: In particular we have Ξt) Ξ r) t r Ξ s) Ξt) s t 4 Ξs) Ξr) s r 4 Ξs) Ξr) s r s r 3 Proof. An elementary proof is given in []. for every t ] r, s[. 3.13) s r s r. 3.14)

10 10 S. Schemm and T. Schmidt Remark 3.6. Assume that Ξ is absolutely continuous and non-decreasing and a set N R of Lebesgue measure zero is given. Then, we can choose r and s as in Lemma 3.5 even with the additional property r, s / N; see [47, Lemma 4.6]. Finally, we state another useful lemma concerning the smoothing operator T r,s : Lemma 3.7. Let 1 < p <, 0 < r < s and s x 0 ) Ω. Then, for u W 1,p Ω, R N ) we have VDTr,s u) ct r,s [ VDu) ] where c depends only on n and p. a. e. on Ω, 3.15) Proof. Due to 3.) it suffices to show the claim with V replaced by W. Since W is a non-decreasing and convex function we obtain using 3.5), 3.8) and Jensen s inequality WDTr,s u) c WTr,s Du ) [ ct r,s WDu) ] a. e. on Ω. This proves the claim. 4. Proof of Theorem.1 First we note that the definition 1.8) of the excess reads Ex,r) = VDū Dū) x,r ) rx) in the terminology of Section 3. We will establish a decay estimate for this excess in the following proposition, which we prove by an indirect blow-up argument. Proposition 4.1. Under the assumptions of Theorem.1 for every L > 0 there is a constant C > 0 with the following property: For each 0 < τ 1 there exists a number ε > 0 such that the conditions for a ball r x) Ω imply Dū) x,r L, r < ε and Ex,r) < ε Ex,τr) Cτ Ex,r). Proof. We argue by contradiction. Assuming the proposition to be false there are L > 0 and 0 < τ 1, corresponding to a constant C which will be chosen later, such that the following holds: There is a sequence of balls r x ) Ω with r 0 such that Dū) x,r L and 0 < := Ex,r ) 0 as, 4.1) but Ex,τr ) > Cτ Ex,r ). 4.)

11 Strong local minimizers of quasiconvex integrals with p, q)-growth 11 Step 1: low-up. We define ξ := Dū) x,r and Then we have and u y) := 1 r [ūx +r y) ū) x,r ξ r y ] for y. Du y) = Dūx +r y) ξ for y, u ) 0,1 = 0, Du ) 0,1 = 0 VDu ) = ) Since Du ) 0,τ = Dū) x,τr ξ we get from 4.) λ V Du Du ) 0,τ ) > Cτ. 4.4) τ Further we define f ξ) := fξ + ξ) fξ ) Dfξ ) ξ λ for ξ R Nn. Noting ξ L we get from [, Lemma II.3] that there is a positive constant kl) with f ξ) kλ V q ξ), Df ξ) kλ 1 V q 1 ξ) 4.5) for all ξ R Nn. Moreover, we rewrite the quasiconvexity hypothesis H3) in the following form: ν L Vλ Dϕ) f ξ +Dϕ) f ξ) ) 4.6) for all ξ R Nn with ξ 1 and for all ϕ W 1, q 0, R N ). In addition, setting F [u] = f Du) the minimizing property of ū can be rephrased as follows: For all ϕ W 1, q 0, R N ) with Dϕ L q,r Nn ) δ 4.7) r n q we have F [u ] F [u +ϕ]. 4.8)

12 1 S. Schemm and T. Schmidt Next we claim Du min{,p} c. 4.9) Actually, 4.9) follows immediately from 4.3) for p. In contrast, for p we first deduce VDu ) c from 4.3) by virtue of Lemma 3.1 i) and then get 4.9) by Lemma 3.1 iii). Thus, passing to subsequences we may assume that for some u W 1,min{,p}, R N ) and some ξ R Nn we have u u weakly in W 1,min{,p}, R N ); u u strongly in L min{,p}, R N ); Du 0 a.e. on ; ξ ξ in R Nn. 4.10) Step : Linearization. In this step we will show that u is a weak solution of a linear system. Precisely, we claim D fξ )Du,Dϕ) = 0 for all ϕ Cc 1, RN ). 4.11) Actually, the derivation of the limit equation 4.11) is well-known see for instance [1,, 4, 9, 3]) and we will only sketch it. From the minimality property of u in 4.8) we get the following Euler-Lagrange equation: Df Du )Dϕ = 0 for all ϕ Cc 1, RN ). We will show that the preceding equation converges to 4.11) as. Setting + := {x : Du x) > 1} and using 4.5) and q p+1 we obtain Df Du )Dϕ c V q 1 Du ) sup Dϕ + + csup Dϕ V Du ). y 4.3) we infer that this term vanishes as and it remains to treat the integral over := {x : Du x) 1}. Here, noting + c 0 as in [3] we have Df Du )Dϕ = 1 Dfξ + Du ) Dfξ ) ) Dϕ λ λ = 1 0 D fξ +t Du )dtdu,dϕ) D fξ )Du,Dϕ)

13 Strong local minimizers of quasiconvex integrals with p, q)-growth 13 and 4.11) follows. The condition H3) implies that D fξ ) is elliptic in the sense of Legendre-Hadamard with ellipticity constant ν L and upper bound K L := sup ξ L D fξ). Thus, we can apply linear theory to deduce that u is C 1 on and Du Du) 0,τ cτ 4.1) τ is valid. Here, c depends only on n, N, ν L and K L. The remainder of the proof is now mostly devoted to showing λ V Du Du)) 0 as. 4.13) τ Once we have proved 4.13) we will see that 4.1) contradicts 4.4). Step 3: Construction of test functions and preliminary estimates. We fix τ < σ < 1, 0 < α < 1 and consider r x 0 ) σ. We define affine functions a x) := u ) x0,r +Du ) x0,rx x 0 ) and set v x) := u x) a x). Moreover, we introduce the abbreviation Ξ t) := λ V λ v ) + Vλ Dv ) ) 1 α)r tx 0) and choose for this function αr r < s r as in Lemma 3.5. In particular we have α)r s r 1 α)r. 4.14) Now we consider smooth cut-off functions η : R n [0,1] which satisfy η 1 in a neighborhood of r x 0 ), η = 0 in a neighborhood of R n \ s x 0 ) and η s r on r x 0 ). We define χ = [ 1 η )v ], ψ := T r, s χ and ϕ := v ψ, where the smoothing operator T is defined in Lemma 3.4. According to 3.6) and 3.7) of Lemma 3.4 we have ϕ W 1,p 0 s x 0 ), R N ), ϕ = v, ψ = 0 on r x 0 ) and Du Da = Dv = Dϕ +Dψ on. 4.15) In addition, the product rule and 4.14) give Dχ Dv + v 1 α)r. 4.16) Next, we will derive two preparatory estimates for χ, namely 4.17) and 4.18): Setting Y := λ V λ v ) + Vλ Dv ) ). 1 α)r \ αrx 0)

14 14 S. Schemm and T. Schmidt we apply in turn 3.8), Lemma 3.7, 3.9) with p = 1), 4.16) and Lemma 3.1 ii) to get the estimate λ s x 0)\ r x 0) V Dψ ) cλ cλ s x 0)\ r x 0) s x 0)\ r x 0) T r, s [ V Dχ ) ] V Dχ ) cy.4.17) Arguing in a similar way, but using 3.11) with p = 1, q = κ) instead of 3.9) we find for 1 κ < n λ s x 0)\ r x 0) n 1 : V Dψ ) κ cλ cλ s r ) n n 1)κ cλ κ s r ) n1 κ)+κ s x 0)\ r x 0) sup t ] r, s [ 1 t r + sup t ] r, s [ sup t ] r, s [ T r, s [ V Dχ ) ] κ tx 0)\ r x 0) 1 s t V Dχ ) s x 0)\ tx 0) Ξ t) Ξ r ) t r ) κ V Dχ ) + sup t ] r, s [ Combining the last inequality with the estimates of Lemma 3.5 we obtain λ s x 0)\ r x 0) V Dψ ) κ ) κ Ξ s ) Ξ t). s t λ c Y ) κ 1 ) n Y. 4.18) 1 α)r Step 4: The main estimate. In this step we will combine ideas of [3] and [46] to establish a key estimate. This estimate will lead to 4.13) later in the proof. Here, our first aim is to verify 4.7) for ϕ with large, which will enable us to use 4.8). We start with the following computation and use for this purpose 4.14), formula 3.9) from Lemma 3.4 and the Poincaré inequality: Dϕ q Dv q + Dψ q r x 0) Dv q +c s x 0)\ r x 0) Dχ q s x 0 )\ r x 0) Dv q +c s x 0)\ r x 0) Dv q + s x 0)\ r x 0) ) v ) q 1 α)r

15 Strong local minimizers of quasiconvex integrals with p, q)-growth 15 c 1+ 1 ) q Dv q. 1 α Changing coordinates in view of rr x +r x 0 ) r x ) we obtain Dϕ L q,r Nn ) c r n q Therefore the condition 4.7) is fulfilled if 1+ 1 ) ) 1 Dū q q. 1 α r x ) c 1+ 1 ) ) 1 Dū q q δ 1 α r x ) holds and this is satisfied for sufficiently large, say for 1 α). Furthermore, from the definition of a and 4.9) we see Da cr n. Hence, there exists a r) such that Da 1 holds for all r). We define 0 α,r) := max{ 1 α), r)}. Then, for 0 α,r) we use 4.6), 4.8) and 4.15) to get ν L r x 0) = s x 0) s x 0) VDv ) νl s x 0) f Da +Dϕ ) f Da ) ) VDϕ ) f Du Dψ ) f Du ) ) + f Du ) f Du Dϕ ) ) + f Da +Dψ ) f Da ) ) s x 0) f Du Dψ ) f Du ) ) + f Da +Dψ ) f Da ) ). s x 0) s x 0) 4.19) Recalling ψ = 0 on r x 0 ) we estimate the right-hand side by inequality 4.5) and Lemma 3.1 ii): ν L VDv ) r x 0) c s x 0)\ r x 0) =: ci +II +III) 1 s x 0) 0 Df Da +tdψ ) Df Du tdψ ) ) Dψ dt V q 1 Da ) Dψ + V q 1 Dv ) Dψ + Dψ Vq ) )

16 16 S. Schemm and T. Schmidt with the obvious labeling. Estimation of III. We estimate the last integral by Lemma 3.1 iii), 4.17) and 4.18) with κ = q p < n n 1 = 1+ 1 n 1 : III = s x 0 )\ r x 0) λ s x 0 )\ r x 0) cλ cλ c Y + VqDψ ) s x 0)\ r x 0) s x 0)\ r x 0) 1+ λ Dψ ) p q p p V Dψ ) 1+ Vλ Dψ ) )q p p Vλ Dψ ) V Dψ ) + V Dψ ) q p ) λ Y ) q ) p ) 1 n Y c 1 α)r Y + λ Y ) 1 n 1 ) n Y ). 1 α)r Estimationof II.WeestimatetheintegralII distinguishingthecasesp > n 1 n and p n 1 n. Case p > n 1 n : In this case.1) reads q < p+ 1 n and we have p +1 < p+ 1 n. Hence, enlarging q if necessary we may assume q p +1 without destroying q < p+ 1 n ). Next we give a pointwise estimation of the integrand in II. For Dv 1 we obtain with Young s inequality 1+ λ Dv ) q Dv Dψ = λ 1+ λ Dv ) p 4 +q p 4 Dv Dψ cλ 1+ λ Dv ) p 4 Dv Dψ V Dv ) c +λ Dψ ), while for Dv > 1 a similar computation yields 1+ λ Dv ) q p Dv Dψ = λ Thus, using p+1 q, q < p+ 1 n argue essentially as for III: II c VDv ) s x 0)\ r x 0) 1+ λ Dv ) p q 1 p +q p p Dv Dψ cλ 1+ λ Dv ) p q 1 V Dv ) c +λ Dψ p Dv q 1 p p p+1 q ). ) Dψ, 4.17) and 4.18) with κ = 1 p+1 q < n n 1 ) we +cλ s x 0)\ r x 0) λ Dψ + Dψ p p+1 q)

17 Strong local minimizers of quasiconvex integrals with p, q)-growth 17 V Dv ) V Dψ ) c + +λ Vλ Dψ ) ) p+1 q s x 0)\ r x 0) c Y + λ Y ) 1 ) p+1 q ) 1 n Y c 1 α)r Y + λ Y ) 1 n 1 ) n Y ). 1 α)r Case p n 1 n 1 : In this case.1) reads q < n n p and we have, in particular, q p + 1. Again we will give an estimate for the integrand in II. In the case Dv Dψ we find since V q 1 is non-decreasing V q 1 Dv ) Dψ Dψ Vq ) and in the case Dv > Dψ we get with Young s inequality and q p +1 V q 1 Dv ) Dψ = λ 1+ λ Dv ) p 4 +q p 4 Dv Dψ λ Vq p Dψ ) Vλ Dv ) λ Vq p Dψ ) + Vλ Dv ) ). Arguing essentially as supplied above and using q p q p p following estimate for II in this case: < n n 1 λ II c Y + Y ) q p p ) 1 λ n Y + Y ) q ) p ) 1 n Y 1 α)r 1 α)r λ c Y + Y ) 1 n 1 ) n Y ). 1 α)r we derive the Estimation of I. It remains to control I. Here, employing Da 1 for 0 α,r), Lemma 3.1 iv) and 4.17) we get I c s x 0)\ r x 0) c V p 1 Da ) Dψ c \ αrx 0) VDa ) +Y s x 0)\ r x 0) Collecting the estimates for I, II and III we have proved αrx 0) ). VDv ) λ c Y + Y ) 1 n 1 ) n Y )+ c 1 α)r V Da ) V Dψ ) ) + \ αrx 0) VDa ).

18 18 S. Schemm and T. Schmidt y the Poincaré-type inequality 3.3), Lemma 3.1 i), Lemma 3. and 4.3) we have Vλ Y λ Dv ) λ v ) ) + V 1 α)r c ) 1+ 1 α) max{,p} Combining the last two inequalities we find VDv ) c Y + αrx 0) VDv ) c 1 α) max{,p}. \ αrx 0) VDa ) ) +c α,r λ n 1, where c α,r > 0 is a fixed constant depending, in particular, on α and r. The reader should note that, contrarily, the constants c in the preceding estimates do not depend on α or r. Applying Lemma 3.1 ii) we deduce VDu Du) αrx 0) [ c αrx 0) [ c VDv ) + \ αrx 0) + \ αrx 0) VDu Du) + VDa ) +λ VDu Da ) ] V ywidman s holefillingtrick,that isaddingc αrx 0) sides, we finally arrive at the main estimate VDu Du) αrx 0) θ + VDu Du) + \ αrx 0) VDa ) +λ VDu Da ) λ v 1 α)r ) ]+c α,r λ n 1. V λdu Du)) VDu Da ) onboth λ v ) V +cα,r λ n 1 1 α)r for 0 α,r) with θ = c 1+c < 1. We stress that θ does not depend on α or r. 4.0) Step 5: Strong convergence. Recalling that u is C 1 on it follows from 4.3) that λ σ V Du Du)) remains bounded as. Thus, there exists

19 Strong local minimizers of quasiconvex integrals with p, q)-growth 19 a non-negative Radon measure µ on σ such that, passing to subsequences again, we have λ V Du Du)) L n µ weakly in the sense of measures on σ. Introducing the affine function ax) := u) x0,r+du) x0,rx x 0 ) we obviously have a a and λ 1 V Da ) Da. Furthermore, setting v := u a we claim λ V λ v v) ) ) 1 α)r For 1 < p < we will now prove 4.1) following an argument of [13]: We choose t 0,1) such that 1 = t + 1 t holds recall p # = n p # n p > ) and apply the interpolation inequality and the Sobolev type inequality 3.4) to get Vv v)) t 1) ) t v v ) t v v ) 1 t) V v v)) p# p # VDv Dv)) ) 1 t). y 4.3) and 4.10) the right-hand side converges to 0 for and 4.1) is verified for p <. For p 4.1) follows from 4.10) and 4.3) by a simpler argument and we omit further details. Returning to the general case, for every measurable subset A of σ we have µinta) lim inf limsup A A λ V Du Du)) λ V Du Du)) µa). Keeping this in mind and passing to the limit in 4.0) we obtain µ αr x 0 )) θµ r x 0 )) + Dv +1 α n )r n Da +c v. 1 α)r Here, for the treatment of the fourth term on the right-hand side we have used Lemma 3.1 ii) and 4.1). Since 0 < α < 1 is arbitrary we can, by virtue of a continuity argument, replace µ αr x 0 )) by µ αr x 0 )) on the left-hand side of the previous inequality. Hence, dividing by r n we have established α nµ αrx 0 )) α n r n θ µ rx 0 )) r n +ε 1 r)+ Da 1 α n )+ ε r) 1 α), 4.)

20 0 S. Schemm and T. Schmidt where we have abbreviated ε 1 r) = 1 r n Dv and ε r) = c r n+ v. Since u is C 1, we have ε 1 r) +ε r) 0 and Da Dux 0 ) as r 0 +. Next we claim µ r x 0 )) lim inf r 0 + r n = ) µ) To prove 4.3), following [3], we first suppose limsup r 0 + r > 0. Then, n by an argument of [3, p ] we can pass r 0 + in 4.) arriving at α n θ + Dux 0 ) 1 α n )limsup r 0 + r n µ r x 0 )). for all 0 < α < 1. Thus, passing α 1 recall that θ < 1 is independent of α) we get 4.3) in any case and for all x 0 σ. Hence, following [3] again, by Vitali s covering theorem we deduce µ τ ) = 0, which, in turn, implies the strong convergence stated in 4.13). Step 6: Conclusion. Noting Du ) 0,τ Du) 0,τ we deduce from Lemma 3.1 ii), 4.1) and 4.13) 1 lim λ V Du Du ) 0,τ )) τ c [ lim Vλ Du Du)) + V Du Du) 0,τ )) = c λ τ τ Du Du) 0,τ C τ + V Du) 0,τ Du ) 0,τ )) ] for some constant C > 0. Finally, the last inequality contradicts 4.4) if we choose C = C +1 and the proof is finished. The readershould note that C and C depend only on n, N, p, ν L and K L. Once Proposition 4.1 is established, Theorem.1 follows by a well-known iteration argument and Campanato s integral characterization of Hölder continuity. For further details see for instance [1, 13]. 5. Proof of Theorem. In this sectionwe presentthe proofoftheorem.,modifying the proofoftheorem.1 along the lines of [47].

21 Strong local minimizers of quasiconvex integrals with p, q)-growth 1 First we recall some simple estimates for the non-degenerate p-energy see e. g. [47] for a proof. e p ξ) := 1+ ξ ) p ; 5.1) Lemma 5.1. For 1 < p <, L > 0, ξ R Nn with ξ L+1, a ball r x 0 ) in R n and ϕ W 1,p 0 ρ x 0 ), R N ) we have VDϕ) [ ep ξ +Dϕ) e p ξ) ] C 1 VDϕ) C 1 1 for some constant C 1 > 0 depending only on p and L. In the following lemmas we collect several properties of the relaxed functional F. These lemmas have been proposed in [47] and rely heavily on.),.3) and the measure and integral representation results obtained in [, 11]. We mention that later in this section we will also apply.3) and the measure representation result [, Theorem 3.1] explicitly. Next we give a reformulation of [47, Lemma 7.1]. Note that the growth condition imposed on Df in [47] follows from H1) and the quasiconvexity of f. Lemma 5.. We suppose that f C 1 is quasiconvex with H1) and 1 < p np q < min{p + 1, n 1 }. Then, for u W1,p Ω, R N ) with F[u] < and ψ W 1, p p+1 q Ω, R N ) we have F[u+ψ] F[u] = F[u+ψ] F[u]. As in [47, Lemma 7.3] we see that Lemma 5. implies the validity of Euler s equation for W 1, q local minimizers of F: Lemma 5.3 Euler s equation). Let 1 q. We suppose that f C 1 is np quasiconvex with H1) and 1 < p q < min{p+1, n 1 }. Then, every W1, q local minimizer ū W 1,p Ω, R N ) of F is a weak solution of the Euler equation of F, i. e. DfDū)Dϕ = 0 for all ϕ Cc Ω, RN ). Ω Now we introduce the additional notation { } F[u;O]:=inf lim inf fdu k ) : W 1,q loc O, RN ) u k u weakly in W 1,p O, R N ) k O for open subsets O of Ω. We will need the next two lemmas, which can also be found in [47]. Lemma 5.4 W 1,p -quasiconvexity). Assume H1) and H) with 1 < p q < np n 1. Then, the following W1,p -quasiconvexity condition holds for F: For every ball r x 0 ) in R n, every ξ R Nn and every ϕ W 1,p r x 0 ), R N ) with compact support in r x 0 ) we have where we have set l ξ x) := ξx. F[l ξ +ϕ; r x 0 )] F[l ξ ; r x 0 )], 5.)

22 S. Schemm and T. Schmidt Lemma 5.5 Additivity property.). Assume H1) and H) with 1 < p q < np n 1. We consider a ball s x 0 ) Ω and u W 1,p Ω, R N ) such that the boundary regularity condition 1 lim sup Du p < 5.3) εց0 ε holds. Then we have s+εx 0)\ s εx 0) F[u;Ω] = F[u; s x 0 )]+F[u;Ω\ s x 0 )]. After these preparations we turn to the proof of Theorem.: As for Theorem.1 it suffices to establish the following proposition, whose statement is completely analogous to Proposition 4.1. Proposition 5.6. Under the assumptions of Theorem. for every L > 0 there is a constant C > 0 with the following property: For each 0 < τ 1 there exists a number ε > 0 such that the conditions for a ball r x) Ω imply Dū) x,r L, r < ε and Ex,r) < ε Ex,τr) < Cτ Ex,r). Sketch of proof. We argue by contradiction. Assuming the proposition to be wrong we proceed by blow-up as for Proposition 4.1 and we will highlight only the necessarymodifications in the proof.first we note that by Lemma 5.3the Eulerequation used in Step is available. Thus, the remaining modifications, which will be outlined now, concern only the handling of the quasiconvexity hypothesis and the minimizing property. We use the nomenclature of the proof of Proposition 4.1 up to the following difference: We choose r, s as in Remark 3.6 avoiding the set { } N := t ]αr,r[: t Du p is not differentiable at t. tx 0) Thus, u satisfies the condition 5.3) near s x 0 ). As explained in the proof of [47, Lemma 7.13] it is easy to see that the same condition holds also for a + ϕ and u ϕ. We will use this fact later when applying Lemma 5.5. Next we will rewrite the quasiconvexity hypothesis H4) in an adequate form for our purposes. To this aim we introduce the auxiliary integrand gξ) := fξ) ν L C 1 e p ξ) for ξ R Nn. where e p is defined in 5.1) and C 1, γ and ν L denote the constants from Lemma 5.1, H) and H4). Moreover, for W 1,p -functions w we set G[w] := gdw), Ω { G[w] := inf F [w] := inf } lim inf G[w k] : W 1,q k loc Ω, RN ) w k w weakly in W 1,p Ω, R N ), { lim inf k F [w k ] : W 1,q loc, RN ) w k w weakly in W 1,p, R N ) },

23 Strong local minimizers of quasiconvex integrals with p, q)-growth 3 and we will also use the obvious modifications of this notations for open subsets O of Ω and, respectively. Following an argument from the proof of [47, Lemma 7.13] it is not hard to see from the definitions of F and G that we have G[w;O] F[w;O] ν L C 1 O e p Dw) for all open subsets O of Ω and all w W 1,p O, R N ). In addition, from H1) and H) we see that g fulfills the growth conditions γ p ν ) L ξ p ν p L gξ) Γ1+ ξ q ) C 1 C 1 forallξ R Nn.Imposingthecondition p ν L < C 1 γ,whichisclearlynotrestrictive, we infer that g satisfies H1) and H) up to an additive constant. Obviously, this is sufficient to allow the application of Lemma 5.4 to G. Furthermore, we deduce from H4) and Lemma 5.1 that g is quasiconvex at all ξ with ξ L+1. In particular, recalling.3) this gives G[l ξ ; r x )] = r x ) Qgξ) = r x ) gξ) for these ξ, where we have used the notation l ξ from Lemma 5.4. Consequently, applying Lemma 5.4 to G we get 0 G[l ξ +ϕ; r x )] G[l ξ ; r x )] F[l ξ +ϕ; r x )] r x ) fξ) ν L C 1 r x ) [ ep ξ +Dϕ) e p ξ) ] for ξ L +1 and all ϕ W 1,p r x ), R N ) with compact support in r x ). y Lemma 5.1 we conclude VDϕ) c F[l ξ +ϕ; r x )] r x ) fξ) ) r x ) for all ξ R Nn with ξ L + 1 and all ϕ W 1,p r x ), R N ) with compact support in r x ). Rescaling gives us 1 Vλ Dϕ) cf [l ξ +ϕ;] f ξ)), for all ξ R Nn with ξ 1 and for all ϕ W 1,p, R N ) with compact support in. Next, we recall ϕ W 1,p 0 s x 0 ), R N ) and F [a ;O] = O f Da ) for all open subsets O of Ω; see Step 3 in the proof of Proposition 4.1 and.3). Using these facts, Lemma 5.5 and the choice of s we deduce from the previous inequality VDϕ ) c F [a +ϕ ; s x 0 )] s x 0 ) f Da ) ) 5.4) s x 0) 1 Actually, this can be verified by a straightforward computation using the definitions of F, F, f F, F and the fact that the integral of the linear term in the definition of f is weakly continuous.

24 4 S. Schemm and T. Schmidt for r). In the following we will use the quasiconvexity hypothesis in the form 5.4). Next we turn to a reformulation of the minimizing property: We have assumed that ū is a W 1, q local minimizer of F on Ω. y the measure representation theorem [, Theorem 3.1] this is easily seen to imply F[ū; r x )] F[ū ϕ; r x )] for all functions ϕ W 1, q r x ), R N ) with compact support in r x ) and Dϕ L q r x ),R Nn ) δ. Rescaling as before we get F [u ;] F [u ϕ;] for all ϕ W 1, q, R N ) with compact support in and Dϕ L q,r Nn ) δ Recalling Dϕ L q,r Nn ) δ n q r for 1 α) see Step 3 in the proof of Proposition 4.1) we get from Lemma 5.5 and the choice of s n q r F [u ; s x 0 )] F [u ϕ ; s x 0 )]. 5.5) Finally, recalling 4.15) and putting together 5.4) and 5.5) we find αrx 0) VDϕ ) c F [u ψ ; s x 0 )] F [u ; s x 0 )] ) +F [a +ψ ; s x 0 )] s x 0 ) f Da ). Since q p p+1 q < np n 1 we see ψ W 1, p p+1 q s x 0 ), R N ) W 1,q s x 0 ), R N ) from the estimates of Lemma 3.4. Thus, we can apply Lemma 5. and.3) to simplify the right-hand side of the preceding formula deriving αrx 0) VDϕ ) c s x 0) + f Du Dψ ) f Du ) ) s x 0) f Da +Dψ ) f Da ) ) ). for 0 α,r). Since the last inequality coincides with the estimate in 4.19) we can now argue exactly as in the proof of Proposition 4.1. Acknowledgment The second author would like to express his gratitude for the kind hospitality of the Mathematical Institute in Erlangen-Nürnberg, where parts of this paper were written in summer 007.

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