A Pointwise Differential Inequality and Second-Order Regularity For

Home , Vladimir Mazya

arXiv:2102.09423v1 [math.AP] 18 Feb 2021 (1.1) ∇ iegneoeao n( in operator divergence (1.2) operator Laplace ic h qain ntelna ytm( exp system the linear involving the integral in boundary equations a the plus since set, same the over ned dniy( identity Indeed, in ntergthn side right-hand the on tions in n ytm sotnms etydsrbdi em ft of terms in described neatly most often is systems and tions hr sa pnstin set open an is Ω where iiypoete fteexpression the of properties bility h onayo [ of boundary the basic ftergthn iei loasmd[ assumed also is side right-hand the of [ of mates [ in considered earlier was equation single a of case The second-order the in instance, for applications, (1.3) where oan,Lrnzspaces. Lorentz domains, Keywords: Classifications: Subject Mathematics u u ONWS IFRNILIEULT N EODODRREGULARITY SECOND-ORDER AND INEQUALITY DIFFERENTIAL POINTWISE A 1 2 u , . . . , lsia dniy hc ik h alca ∆ Laplacian the links which identity, classical A ieetaiiypoete of properties Differentiability r the contributions, recent more several in demonstrated As equations nonlinear of theory regularity second-order The dniy( Identity Ω : u el sthat us tells , p > p → Lpaeeuto rsystem or equation -Laplace u ocuin rae h ag fteamsil auso values admissible the of range the broaden conclusions our supin ntebudr ftedmi r eurdfrth for required are domain the of boundary the on assumptions hebc tutr,i ffrd sacneune pia s optimal consequence, in domains a in As systems elliptic offered. nonlinear is structure, Uhlenbeck Abstract. uslna litcsses eododrderivatives, second-order systems, elliptic Quasilinear N 41 R n “ and 1 N of 1.1 ,rarneetieulte f[ of inequalities rearrangement ], srgre stemti in matrix the as regarded is u a efuda al smr hnoecnuyaoi [ in ago century one than more as early as found be can ) oevr h ux“ suffix the Moreover, . NAK.AC,ADE INH,LR INN,ADVAII M VLADIMIR AND DIENING, LARS CIANCHI, ANDREA KH.BALCI, ANNA 12 1.1 div hr onws ieeta nqaiyfrvcoilsec vectorial for inequality differential pointwise sharp A .Frhrrslsi hscneto a efudeg [ e.g. found be can connection this in results Further ]. nbe n obudteitga of integral the bound to one enables ) eoe the denotes ” 1.3 .Ti ufcs o ntne rmBOadHodrbud o Hölder bounds and BMO from instance, for surfaces, This ). R | n ∆ f O OLNA LITCSYSTEMS ELLIPTIC NONLINEAR FOR ee n nwa follows, what in and Here, . hsi h aeo [ of case the is This . u 52,3J0 35B65. 35J60, 35J25, | |∇ 2 |∇ div = u R u | p N | − p − vle iegneoeao.Sadr eut ocr wea concern results Standard operator. divergence -valued T 2 − 2 1.2 30 ∇ tnsfrtranspose. for stands ” 2 div R (∆ ∇ u n ,weerslsfrbudr au rbescnb on und found be can problems value boundary for results where ], r nope,ister srdcdt hto t igeeq single its of that to reduced is theory its uncoupled, are ) u r eie.Bt oa n lbletmtsaeestablishe are estimates global and local Both derived. are 25 R u aecsoaiybe eetdudrsrn euaiyass regularity strong under detected been customarily have ( hytaebc o[ to back trace They . − |∇ 1. ) N L ,pitieoclainetmtso [ of estimates oscillation pointwise ], T × ∆ 2 u ∇ Introduction rglrt hoyfrsltost h oso ytmfrt for system Poisson the to solutions for theory -regularity n | u u p u − hs osaetegainsin gradients the are rows whose = − 44 2 favco-audfunction vector-valued a of ∇ f ,weelclsltosaecniee.Hg regularity High considered. are solutions local where ], 1 1 2 p u 56 ∇|∇ Lpain iihe rbes oa ouin,capacit solutions, local problems, Dirichlet -Laplacian, = ) .Frhrdvlpet r n[ in are developments Further ]. h exponent the f n |∇ atr nteseilcs fthe of case special the In latter. e nΩ. in u cn-re euaiypoete fsltosto solutions of properties regularity econd-order f eso ne h iegneoeao.O course, Of operator. divergence the under ression ≥ | 2 n ytm smc esdvlpd e o the for yet developed, less much is systems and 2 glrt fsltosto solutions of egularity u eexpression he 2, | 2 + n-re ata ieeta prtr,with operators, differential partial ond-order nΩ, in vrsm e nΩb h nerlof integral the by Ω in set some over N 55 |∇ for ] ≥ 2 u p 10 ,adtegradient the and 1, | 2 rvosyknown. previously > p for ] 3 , nΩ, in 26 |∇ ,adt [ to and 2, n , u 42 u e lo[ also see – 2 = | p 11 − ∈ ]. 2 R C ,rglrt eut pto up results regularity ], ∇ 1 AZ’YA 3 p u p × (Ω Lpaintp equa- type -Laplacian 1 Lpaesystem, -Laplace n pern ne the under appearing 8 , [ f , , 20 ftecomponents the of R 34 18 ∇ N .Minimal d. o every for ] ,ptnilesti- potential ], , u oisHessian its to ) 29 52 fafunction a of differentia- k ]. , 37 uations. ,convex y, .I has It ]. > p | ump- ∆ u he er 1. | 2 2 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA smoothness assumptions on ∂Ω. Both papers [44] and [30] deal with scalar problems, i.e. with the case when N = 1. Fractional-order regularity of the gradient of solutions to quasilinear equations of p-Laplacian type has been studied in [51], and in the more recent contributions [3, 16, 19, 49, 50]. The question of fractional-order p 2 regularity of the quantity u − u, when N = 1 and the right-hand side of equation (1.3) is in divergence form, is addressed in [5], where,|∇ | in∇ particular, sharp results are obtained for n = 2. Optimal second-order L2-estimates for solutions to a class of problems, including (1.3) for every p> 1, in the p 2 scalar case, have recently been established in [28]. Loosely speaking, these estimates tell us that u − u W 1,2 if and only if f L2. Such a property is shown to hold both locally, and, under minimal|∇ regularity| ∇ ∈ assumptions on the boundary,∈ also globally. Parallel results are derived in [27] for vectorial problems, namely 3 for N 2, but for the restricted range of powers p > 2 . The results of [27] and [28] rely upon the idea that, in the≥ nonlinear case, the role of the pointwise identity (1.1) can be performed by a pointwise inequality. The latter amounts to a bound from below for the square of the right-hand side of (1.3) by the square of the p 2 derivatives of u − u, plus an expression in divergence form. The restriction for the admissible values of p in the vectorial|∇ case| stems∇ from this pointwise inequality. In the present paper we offer an enhanced pointwise inequality in the same spirit, with best possible constant, for a class of nonlinear differential operators of the form div(a( u ) u). The relevant inequality holds under general assumptions on the function a, which also allow− growths|∇ that| ∇ are not necessarily of power type. Importantly, our inequality improves the available results even in the case when the operator is the p-Laplacian, p 2 namely when a(t)= t − . In particular, for this special choice, it entails the existence of a constant c> 0 such that p 2 2 2(p 2) T 1 2 2(p 2) 2 2 (1.4) div( u − u) div u − (∆u) u u + c u − u |∇ | ∇ ≥ |∇ | ∇ − 2 ∇|∇ | |∇ | |∇ | h i in u = 0 if and only either N = 1 and p> 1, or N 2 and p> 2(2 √2) 1.1715. The{∇ differential6 } inequality to be presented, in its general≥ version, is− the crucial≈ point of departure in our proof of the local and global W 1,2-regularity for the expression a( u ) u for systems of the form |∇ | ∇ (1.5) div(a( u ) u)= f in Ω. − |∇ | ∇ Regularity issues for equations and systems driven by non standard nonlinearities, encompassing (1.5), are nowadays the subject of a rich literature. A non exhaustive sample of contributions along this direction of research includes [2, 4, 6, 7, 14, 17, 21, 23, 24, 31, 34, 35, 38, 39, 43, 45, 54]. Let us incidentally note that system (1.5) is the Euler equation of the functional

(1.6) J(u)= B( u ) f u dx. |∇ | − · ZΩ Here, the dot “ ” stands for scalar product, and B : [0, ) [0, ) is the function deﬁned as · ∞ → ∞ t (1.7) B(t)= b(s) ds for t 0, ≥ Z0 where the function b : [0, ) [0, ) is given by ∞ → ∞ (1.8) b(t)= a(t)t for t> 0, and b(0) = 0. Under the assumptions to be imposed on a, the function B and the functional J turn out to be strictly convex. p 2 1 p In particular, if a(t) = t − , then B(t) = p t , and J agrees with the usual energy functional associated with the p-Laplace system (1.3). We shall focus on the case when N 2, the case of equations being already fully covered by the results of [28]. In particular, our regularity results≥ apply to the p-Laplacian system (1.3) for every (1.9) p> 2(2 √2) 1.1715. − ≈ 3 Hence, we extend the range of the admissible exponents p known until now, which was limited to p> 2 . A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 3

In the light of the pointwise inequality (1.4), the lower bound (1.9) for p is optimal for our approach to the second-order regularity of solutions to the p-Laplace system (1.3). The question of whether such a restriction is really indispensable for this regularity, or it can be dropped as in the case when N = 1, where every p> 1 is admitted, is an open challenging problem.

2. Main results The statement of the general diﬀerential inequality requires a few notations. Given a positive function a C1((0, )), we deﬁne the indices ∈ ∞ ta′(t) ta′(t) (2.1) ia = inf and sa = sup , t>0 a(t) t>0 a(t) p 2 where a′ stands for the derivative of a. Plainly, if a(t)= t − , then ia = sa = p 2. Moreover, we denote, for N 1 the continuously increasing function κ : [1,− ) R as ≥ N ∞ → (p 1)2 if p [1, 2) (2.2) κ1(p)= − ∈ 1 if p [2, ), ( ∈ ∞ if N = 1, and

1 2 4 1 8 (4 p) if p [1, 3 ) − 2− ∈ 4 (2.3) κN (p)= (p 1) if p [ , 2)  − ∈ 3 1 if p [2, ), ∈ ∞ if N 2.  ≥  Theorem 2.1. [General pointwise inequality] Let n 2 and N 1. Let Ω be an open set in Rn and let u C3(Ω, RN ). Assume that the function a C0([0, )) is≥ such that:≥ ∈ ∈ ∞ (2.4) a(t) > 0 for t> 0,

(2.5) i 1, a ≥− and (2.6) b C1([0, )), ∈ ∞ where b is the function deﬁned by (1.8). Then

(2.7) div a( u ) u 2 div a( u )2 (∆u)T u 1 u 2 + κ (i + 2)a( u )2 2u 2 |∇ | ∇ ≥ |∇ | ∇ − 2 ∇|∇ | N a |∇ | |∇ | h i in Ω, where κN is defined as in (2.2)-(2.3). Moreover, the constant κN (ia + 2) is sharp. If a is just defined in (0, ), a C1((0, )), and conditions (2.4) and (2.5) are fulfilled, then inequality (2.7) continues to hold in the set∞ u∈= 0 . ∞ {∇ 6 } Remark 2.2. Observe that the assumption (2.6) need not be fulfilled by the functions a appearing in the elliptic p 2 systems (1.5) to be considered. Such an assumpton fails, for instance, when a(t) = t − with 1

Corollary 2.3. [Pointwise inequality for the p-Laplacian] Let n 2 and N 1. Let Ω be an open set in Rn and let u C3(Ω, RN ). Assume that p 1. Then ≥ ≥ ∈ ≥ p 2 2 2(p 2) T 1 2 2(p 2) 2 2 (2.8) div( u − u) div u − (∆u) u u + κ (p) u − u |∇ | ∇ ≥ |∇ | ∇ − 2 ∇|∇ | N |∇ | |∇ | in u = 0 . Moreover, the constanthκ (p) is sharp. i {∇ 6 } N Notice that, if N = 1, then

(2.9) κ1(p) > 0 if p> 1, whereas, if N 2, ≥ (2.10) κ (p) > 0 if p> 2(2 √2). N − The gap between (2.9) and (2.10) is responsible for the different implications of inequality (2.7) in view of second-order L2-estimates for solutions to (2.11) div(a( u ) u)= f in Ω , − |∇ | ∇ according to whether N =1 or N 2. Indeed, inequality (2.7) is of use for this purpose only if κN (ia + 2) > 0. Since we are concerned with L2-estimates,≥ the datum f in (2.11) is assumed to be merely square integrable. Solutions in a suitably generalized sense have thus to be considered. For instance, the existence of standard weak solutions to the p-Laplace system (1.3) is only guaranteed if p 2n . In the scalar case, various definitions ≥ n+2 of solutions – entropy solutions, renormalized solutions, SOLA – that allow for right-hand sides that are just integrable functions, or even finite measures, are available in the literature, and turn out to be a posteriori equivalent. Note that these solutions need not be even weakly differentiable. The case of systems is more delicate and has been less investigated. A notion of solution, which is well tailored for our purposes and will be adopted, is patterned on the approach of [36]. Loosely speaking, the solutions in question are only approximately differentiable, and are pointwise limits of solutions to approximating problems with smooth right-hand sides. The outline of the derivation of the second-order L2-bounds for these solutions to system (2.11) via Theorem 2.1 is analogous to the one of [28]. However, new technical obstacles have to be faced, due to the non-polynomial growth of the coefficient a in the differential operator. In particular, an L1-estimate, of independent interest, for the expression a( u ) u for merely integrable data f is established. Such an estimate is already available in the literature for equations,|∇ | ∇ but seems to be new for systems, and its proof requires an ad hoc Sobolev type inequality in Orlicz spaces.

Our local estimate for system (2.11) reads as follows. In the statement, BR and B2R denote concentric balls, with radius R and 2R, respectively. Theorem 2.4. [Local estimates] Let Ω be an open set in Rn, with n 2, and let N 2. Assume that the function a : (0, ) (0, ) is continuously diﬀerentiable, and satisﬁes ≥ ≥ ∞ → ∞ (2.12) i > 2(1 √2) , a − and (2.13) s < . a ∞ Let f L2 (Ω, RN ) and let u be an approximable local solution to system (2.11). Then ∈ loc 1,2 N n (2.14) a( u ) u W (Ω, R × ), |∇ | ∇ ∈ loc and there exists a constant C = C(n,N,ia,sa) such that 1 (2.15) R− a( u ) u 2 RN×n + a( u ) u 2 RN×n |∇ | ∇ L (BR, ) ∇ |∇ | ∇ L (BR, ) n 1 C f 2 RN + R− 2 − a( u ) u 1 RN×n . ≤ k kL (B2R, ) k |∇ | ∇ kL (B2R, ) for any ball B Ω. 2R ⊂⊂ A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 5

n 1 n N n Remark 2.5. In particular, if Ω = R and, for instance, a( u ) u L (R , R × ), then passing to the limit in inequality (2.15) as R tells us that |∇ | ∇ ∈ →∞

(2.16) a( u ) u × C f 1 Rn RN . ∇ |∇ | ∇ L2(Rn,RN n) ≤ k kL ( , ) We next deal with global estimates for solutions to system (2.11), subject to Dirichlet homogeneous boundary conditions. Namely, we consider solutions to problems of the form

div(a( u ) u)= f in Ω (2.17) − |∇ | ∇ (u =0 on ∂Ω . As shown by classical counterexamples, yet in the linear case, global estimates involving second-order derivatives of solutions can only hold under suitable regularity assumptions on ∂Ω. Specifically, information on the (weak) curvatures of ∂Ω is relevant in this connection. Convexity of the domain Ω, which results in a positive 2,2 N n semidefinite second fundamental form of ∂Ω, is well known to ensure bounds in W (Ω, R × ) for the solution u to the homogeneous Dirichlet problem associated with the linear system (1.2) in terms of the L2(Ω, RN ) norm of f – see [37]. The following result provides us with an analogue for problem (2.17), for the same class of nonlinearities a as in Theorem 2.4. Theorem 2.6. [Global estimates in convex domains] Let Ω be any bounded convex open set in Rn, with n 2, and let N 2. Assume that the function a : (0, ) (0, ) is continuously differentiable and fulfills conditions≥ (2.12) and≥ (2.13). Let f L2(Ω, RN ) and let ∞u be→ an approximable∞ solution to the Dirichlet problem (2.17). Then ∈

1,2 N n (2.18) a( u ) u W (Ω, R × ), |∇ | ∇ ∈ and

(2.19) C f 2 RN a( u ) u 1,2 RN×n C f 2 RN 1k kL (Ω, ) ≤ k |∇ | ∇ kW (Ω, ) ≤ 2k kL (Ω, ) for some positive constants C1 = C1(n,N,ia,sa) and C2 = C2(N, ia,sa, Ω). The global assumption on the signature of the second fundamental form of ∂Ω entailed by the convexity of Ω can be replaced by local conditions on the relevant fundamental form. This is the subject of Theorem 2.7. The finest assumption on ∂Ω that we are able to allow for amounts to a decay estimate of the integral of its weak curvatures over subsets of ∂Ω whose diameter approaches zero, in terms of their capacity. Specifically, suppose that Ω is a bounded Lipschitz domain such that ∂Ω W 2,1. This means that the domain Ω is locally the subgraph of a Lipschitz continuous function of (n 1) variables,∈ which is also twice weakly differentiable. Denote by the weak second fundamental form on ∂Ω,− by its norm, and set B |B| n 1 d − (2.20) (r)= sup E |B| H for r (0, 1) . KΩ cap (E) ∈ E ∂Ω Br(x) R B1(x) ⊂ x ∩∂Ω ∈

Here, Br(x) stands for the ball centered at x, with radius r, the notation capB1(x)(E) is adopted for the capacity n 1 of the set E relative to the ball B1(x), and − is the (n 1)-dimensional Hausdorff measure. The decay we hinted to above consists in a smallness conditionH on the limi−t at as r 0+ of the function (r). The smallness → KΩ depends on Ω through its diameter dΩ and its Lipschitz characteristic LΩ. The latter quantity is defined as the maximum among the Lipschitz constants of the functions that locally describe the intersection of ∂Ω with balls centered on ∂Ω, and the reciprocals of their radii. Here, and in similar occurrences in what follows, the dependence of a constant on dΩ and LΩ is understood just via an upper bound for them. Theorem 2.7 also provides us with an ensuing alternate assumption on ∂Ω, which only depends on integra- bility properties of the weak curvatures of ∂Ω. Precisely, it requires the membership of in a suitable function space X(∂Ω) over ∂Ω defined in terms of weak type norms, and a smallness conditio|B|n on the decay of these 6 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA norms of over balls centered on ∂Ω. This membership will be denoted by ∂Ω W 2X. The relevant weak space is defined|B| as ∈ n 1, L − ∞(∂Ω) if n 3, (2.21) X(∂Ω) = 1, ≥ (L ∞ log L(∂Ω) if n = 2. n 1, n 1 1, Here, L − ∞(∂Ω) denotes the weak-L − (∂Ω) space, and L ∞ log L(∂Ω) denotes the weak-L log L(∂Ω) space (also called Marcinkiewicz spaces), with respect to the (n 1)-dimensional Hausdorff measure. − Theorem 2.7. [Global estimates under minimal boundary regularity] Let Ω be a bounded Lipschitz domain in Rn, n 2, such that ∂Ω W 2,1, and let N 2. Assume that the function a : (0, ) (0, ) is continuously differentiable≥ and fulfills∈ conditions (2.12≥) and (2.13). Let f L2(Ω, RN ) and∞ let→u be∞ an approximable solution to the Dirichlet problem (2.17). ∈ (i) There exists a constant c = c(n,N,ia,sa,LΩ, dΩ) such that, if

(2.22) lim Ω(r) < c, r 0+ K → 1,2 N n then a( u ) u W (Ω, R × ), and inequality (2.19) holds. (ii) Assume,|∇ | ∇ in addition,∈ that ∂Ω W 2X, where X(∂Ω) is the space defined by (2.21). There exists a constant ∈ c = c(n,N,ia,sa,LΩ, dΩ) such that, if (2.23) lim sup < c , + X(∂Ω Br (x)) r 0 x ∂Ω kBk ∩ → ∈ then a( u ) u W 1,2(Ω, RN n), and inequality (2.19) holds. |∇ | ∇ ∈ × Remark 2.8. We emphasize that the assumptions on ∂Ω in Theorem 2.7 are essentially sharp. For instance, the mere finiteness of the limit in (2.22) is not sufficient for the conclusion to hold. As shown in [47, 48], there exists a one-parameter family of domains Ω such that Ω(r) < for r (0, 1) and the solution to the homogeneous Dirichlet problem for (1.2), with a smooth right-handK side∞ f, belongs∈ to W 2,2(Ω) only for those values of the parameter which make the limit in (2.22) smaller than a critical (explicit) value. A similar phenomenon occurs in connection with assumption (2.23). An example from [40] applies to demon- 3 2 2, strate its optimality yet for the scalar p-Laplace equation. Actually, open sets Ω R , with ∂Ω W L ∞, ⊂ ∈ 3 are displayed where the solution u to the homogeneous Dirichlet problem for (1.3), with N = 1, p ( 2 , 2] p 2 1,2 ∈ and a smooth right-hand side f, is such that u − u / W (Ω). This lack of regularity is due to the fact that the limit in (2.23), though finite, is not|∇ small| ∇ enough.∈ Similarly, if n = 2 there exist open sets Ω, 2 1, with ∂Ω W L ∞ log L, for which the limit in (2.23) exceeds some threshold, and where the solution to the homogeneous∈ Dirichlet problem for (1.2), with a smooth right-hand side, does not belong to W 2,2(Ω) – see [47]. Remark 2.9. The one-parameter family of domains Ω mentioned in the first part of Remark 2.8 with regard 2 n 1, to condition (2.22) is such that ∂Ω / W L − ∞ if n 3. Hence, assumption (2.23) is not fulfilled even for those values of the parameter which render∈ (2.22) true.≥ This shows that the latter assumption is indeed weaker than (2.23) . 2,n 1 2 Remark 2.10. Condition (2.23) certainly holds if n 3 and ∂Ω W − , and if n = 2 and ∂Ω W L log L (and hence, if ∂Ω W 2,q for some q > 1). This is due≥ to the fact∈ that, under these assumptions,∈ the limit in (2.23) vanishes. In∈ particular, assumption (2.23) is satisfied if ∂Ω C2. ∈ 3. The pointwise inequality This section is devoted to the proof of Theorem 2.1, which is split in several lemmas. The point of departure is a pointwise identity, of possible independent use, stated in Lemma 3.1. Given a positive function a C1(0, ), we define the function Q : [0, ) R ∈ ∞ a ∞ → ta (t) (3.1) Q (t)= ′ for t> 0. a a(t) A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 7

Hence,

(3.2) ia = inf Qa(t) and sa = sup Qa(t), t>0 t>0 where ia and sa are the indices given by (2.1). Lemma 3.1. Let n, N, Ω and u be as in Theorem 2.1. Assume that the function a C0([0, )) and satisﬁes conditions (2.4)–(2.6). Then ∈ ∞

(3.3) div(a( u ) u) 2 = div a( u )2 (∆u)T u 1 u 2 |∇ | ∇ |∇ | ∇ − 2 ∇|∇ | h i 2 u + a( u )2 2u 2 + 2Q ( u ) u 2 + Q ( u )2 ∇ ( u )T in Ω, |∇ | |∇ | a |∇ | |∇|∇ || a |∇ | u ∇|∇ | " # |∇ | where the last two addends in square brackets on the right-hand side of equation (3.3) have to interpreted as 0 if u = 0. If ∇a is just deﬁned in (0, ), a C1((0, )), and conditions (2.4) and (2.5) are fulﬁlled, then inequality (2.7) continues to hold in the set∞ u∈= 0 . ∞ {∇ 6 } p 2 The next corollary follows from Lemma 3.1. applied with a(t)= t − . Corollary 3.2. Let n, N, Ω and u be as in Theorem 2.1. Assume that p 1. Then ≥ p 2 2 2(p 2) T 1 2 (3.4) div( u − u) = div u − (∆u) u u |∇ | ∇ |∇ | ∇ − 2 ∇|∇ | h i 2 2(p 2) 2 2 2 2 u T + u − u + 2(p 2) u + (p 2) ∇ ( u ) |∇ | |∇ | − |∇|∇ || − u ∇|∇ | " # |∇ | in u = 0 . {∇ 6 } Proof of Lemma 3.1. The following chain can be deduced via straightforward computations: 2 T 2 (3.5) div a( u ) u = a( u )∆u + a′( u ) u( u ) |∇ | ∇ |∇ | |∇ | ∇ ∇|∇ | 2 2 2 2 2 2 2 = a( u ) ∆u u + a( u ) u + |∇ | | | − |∇ | |∇ | |∇ | 2 T 2 T + a′( u ) u( u ) + 2a( u )a′( u )∆u u( u ) |∇ | |∇ ∇|∇ | |∇ | |∇ | ·∇ ∇|∇ | = a( u )2 div((∆u)T u) 1 div( u 2) + a( u )2 2u 2+ |∇ | ∇ − 2 ∇|∇ | |∇ | |∇ | 2 T 2 T + a′( u ) u( u ) + 2a( u )a′( u )∆u u( u ) . |∇ | ∇ ∇|∇ | |∇ | |∇ | ·∇ ∇|∇ | Notice that equation (3.5) also holds at the points where u = 0, provided the terms involving the factor |∇ | a′( u ) are intepreted as 0. This is due to the fact that all the terms in question also contain the factor u and,|∇ by| assumption (2.6), ∇ lim a′(t)t = 0. t 0+ → Moreover, 2 T 2 T T (3.6) a( u ) div((∆u) u) = div a( u ) (∆u) u 2a( u )a′( u )∆u u( u ) , |∇ | ∇ |∇ | ∇ − |∇ | |∇ | ·∇ ∇|∇ | and 1 2 2 1 2 2 2 (3.7) a( u ) div u = div a( u ) u 2a( u )a′( u ) u u . 2 |∇ | ∇|∇ | 2 |∇ | ∇|∇ | − |∇ | |∇ | |∇ ||∇|∇ || From equations (3.5)–(3.7) one deduces that (3.8) div(a( u ) u) 2 = div a( u )2(∆u)T u 1 div a( u )2 u 2 |∇ | ∇ |∇ | ∇ − 2 |∇ | ∇|∇ | 2 2 2 2 T 2 2 + a( u ) u + a′( u ) u( u ) + 2a( u )a′( u ) u u . |∇ | |∇ | |∇ | ∇ ∇|∇ | |∇ | |∇ | |∇ ||∇|∇ ||

8 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

If u = 0, then the last two addends on the right-hand side of equation (3.8) vanish. Hence, equation (3.3) follows.∇ Assume next that u = 0. Then, from equation (3.8) we obtain that ∇ 6 div(a( u ) u) 2 = div a( u )2(∆u)T u 1 div a( u )2 u 2 |∇ | ∇ |∇ | ∇ − 2 |∇ | ∇|∇ | 2 a ( u ) u 2 u a ( u ) u + a( u )2 2u 2 + ′ |∇ | |∇ | ∇ ( u )T + 2 ′ |∇ | |∇ | u 2 . |∇ | "|∇ | a( u ) u ∇|∇ | a( u ) |∇|∇ || # |∇ | |∇ | |∇ |

The proof of equation (3.3) is complete.

Having identity (3.3) at our disposal, the point is now to derive a sharp lower bound for the second addend on its right-hand side. This will be accomplished via Lemma 3.6 below. Its proof requires a delicate analysis of the quadratic form depending on the entries of the Hessian matrix 2u which appears in square brackets in the expression to be bounded. This analysis relies upon some critical∇ linear-algebraic steps that are presented in the next three lemmas. Rn n Rn n In what follows, sym× denotes the space of symmetric matrices in × . The dot “ ” is employed to denote scalar product of vectors or matrices, and the symbol “ ” for tensor product of vectors.· Also, I stands for the n n ⊗ identity matrix in R × . Lemma 3.3. Let ω Rn be such that ω = 1. Then, ∈ | | 2 1 2 1 2 1 2 (3.9) Hω ω Hω H = H ⊥ | | − 2 | · | − 2 | | − 2 | ω | n n for every H R , where H ⊥ = (I ω ω)H(I ω ω). ∈ sym× ω − ⊗ − ⊗ n n Proof. Let e1,...,en denote the canonical basis in R and let θ1,...,θn be an orthonormal basis of R such { } n n { } T that θ1 = ω. Let Q R × be the matrix whose columns are θ1,...,θn. Hence, ω = Qe1. Next, let R = Q HQ. Clearly, R Rn n.∈ Denote by r the entries of R. Computations show that ∈ sym× ij n n Hω 2 1 ω Hω 2 1 H 2 = Re 2 1 e Re 2 1 R 2 = r 2 1 r 2 1 r 2 | | − 2 | · | − 2 | | | 1| − 2 | 1 · 1| − 2 | | | i1| − 2 | 11| − 2 | ij| Xi=1 i,jX=1 n n n = 1 r 2 + 1 r 2 1 r 2 1 r 2 = 1 r 2 2 | 1j| 2 | i1| − 2 | 11| − 2 | ij| − 2 | ij| j=1 i=1 i,j=1 i,j 2 X X X X≥ = 1 (I e e )R(I e e ) 2 = 1 (I ω ω)H(I ω ω) 2. − 2 | − 1 ⊗ 1 − 1 ⊗ 1 | − 2 | − ⊗ − ⊗ | Hence, equation (3.9) follows. Given a vector ω Rn, define the set ∈ n n E(ω)= Hω : H R × , H 1 . ∈ sym | |≤ It is easily verified that E(ω) is a convex set in Rn for every ω Rn. Lemma 3.4 below tells us that, in fact, E(ω) is an ellipsoid, centered at 0 (which reduces to 0 if ω =∈ 0). This assertion will be verified by showing that, for each ω Rn, there exists a positive definite matrix{ } W Rn n such that E(ω) agrees with the ellipsoid ∈ ∈ sym× n 1 (3.10) F (W )= x R : x W − x 1 , ∈ · ≤ 1 where W − stands for the inverse of W . This is the content of Lemma 3.4 below. In its proof, we shall make use of the alternative representation (3.11) F (W )= x Rn : y x y Wy for every y Rn , ∈ · ≤ · ∈ which follows, for instance, via a maximization argumentp for the ratio of the two sides of the inequality in (3.11) for each given x Rn. Also, observe that, as a∈ consequence of equation (3.11), (3.12) x = x x √x W x for every x F (W ) 0 . | | · ≤ · ∈ \ { } b b b A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 9

Here, and in what follows, we adopt the notation x x = for x Rn 0 . x ∈ \ { } | | Lemma 3.4. Given ω Rn, let W (ω) Rn n be defined as ∈ ∈b sym× (3.13) W (ω)= 1 ω 2I + ω ω . 2 | | ⊗ Then W (ω) is positive definite, and (3.14) E(ω)= F (W (ω)). In particular, n n n (3.15) Hω H F W (ω) for every ω R and H R × . ∈ | | ∈ ∈ sym Proof. Equation (3.14) trivially holds if ω = 0. Thus, by a scaling argument, it suffices to consider the case when ω = 1. We begin showing that E(ω) F (W (ω)). One can verify that, since ω = 1, | | ⊂ | | 1 (3.16) W (ω)− = 2I ω ω. − ⊗ Let H Rn n be such that H 1. Owing to equation (3.16) and to Lemma 3.3, ∈ sym× | |≤ 1 2 2 2 (3.17) Hω W (ω)− Hω = 2 Hω ω Hω H 1. · | | − · ≤ | | ≤ This shows that Hω F (W (ω)). The inclusion E(ω) F (W (ω)) is thus established . Let us next prove that∈ F (W (ω)) E(ω). Let x F (W⊂(ω)). We have to detect a matrix H Rn n such that ⊂ ∈ ∈ sym× H 1 and x = Hω. To this purpose, consider the decomposition x = tω + sω , for suitable s,t R, where | | ≤ ⊥ ∈ ω ω and ω = 1. Since x F (W (ω)), one has that x W (ω) 1x 1. Furthermore, ⊥ ⊥ | ⊥| ∈ · − ≤ 1 2 2 2 2 2 x W (ω)− x = (tω + sω⊥) (2I ω ω)(tω + sω⊥) = 2(t + s ) t = t + 2s . · · − ⊗ − 2 2 Hence, t + 2s 1. We claim that the matrix H defined as H = tω ω + s (ω⊥ ω + ω ω⊥), has the desired properties. Indeed,≤ H Rn n, ⊗ ⊗ ⊗ ∈ sym× H 2 = tr(HT H)= t2 + 2s2 1 | | ≤ Hω = tω + sω⊥ = x. This proves that x E(ω). The inclusion F (W (ω)) E(ω) hence follows. ∈ ⊂ In view of the statement of the next lemma, we introduce the following notation. Given N vectors ωα Rn and N matrices Hα Rn n, with α = 1,...N, we set ∈ ∈ sym× N 2 N N 2 N (3.18) J = Hαωα , J = ωα Hβωβ , J = Hα 2. 0 · 1 | | α=1 α=1 β=1 α=1 X X X X

Lemma 3.5. Let N 2, 0 δ 1 and δ + σ 1. Assume that the vectors ωα Rn and the matrices ≥ ≤ ≤ 2 ≥ ∈ Hα Rn n, with α = 1,...N, satisfy the following constraints: ∈ sym× N (3.19) ωα 2 1, | | ≤ α=1 X N (3.20) Hα 2 1. | | ≤ α=1 X Then, 1 0 if δ [0, 3 ], (3.21) J δJ σJ 2 ∈ − 0 − 1 ≤ max 0, (δ+1) σ if δ ( 1 , 1 ]. ( 8δ − ∈ 3 2 n o 10 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

Proof. Given δ and σ as in the statement, set = J δJ σJ . Dδ,σ − 0 − 1 The quantities J , J and J are 1-homogeneous with respect to the quantity N H 2. Moreover, inequality 0 1 j=1 | j| (3.21) trivially holds if the latter quantity vanishes. Thereby, it suﬃces to prove this inequality under the P assumption that N H 2 = 1, namely that j=1 | j| (3.22) P J1 = 1.

N α α On setting ζ = α=1 H ω , one has that P N J = ζ 2 and J = ωα ζ 2. | | 0 | · | αX=1 Therefore, N (3.23) J ζ 2 ωα 2 ζ 2 = J. 0 ≤ | | | | ≤ | | αX=1 Owing to Lemma 3.4, Hαωα Hα F (W α) ∈ | | for α = 1,...,N, where W α = ωα 2 1 (Id + ωα ωα). Thus, by equations (3.15) and (3.12), | | 2 ⊗ 2 (3.24) Hαωα ζ Hαc ζ cW αζ = Hα ωα 1 + 1 ωα ζ · ≤ | | · | || | 2 2 | · | for α = 1,...,N. Since q q b b b c b N ζ = (ζ ζ)ζ = (Hαωα ζ)ζ, · · α=1 X equation (3.24) implies that b b b b

N N 2 ζ Hαωα ζ Hα ωα 1 + 1 ωα ζ . | |≤ · ≤ | || | 2 2 | · | α=1 α=1 q X X Hence, b c b

N 2 2 (3.25) ζ 2 1 Hα ωα 1 + 1 ωα ζ . | | ≤ 2 | || | 2 2 | · | α=1 X q N 2 b On setting J = ωα ζ , we obtain that c 0 α=1 | · | P N b b J = ωα 2 ωα ζ 2 and J = ζ 2J . 0 | | | · | 0 | | 0 α=1 X b 2 b b Note that J 1, inasmuch as J J = ζ . Moreover,c by equation (3.22), 0 ≤ 0 ≤ | | (3.26) = J δJ σ = ζ 2 1 δJ σ. b Dδ,σ − 0 − | | − 0 − From inequalities (3.25) and (3.26) we deduce that b N 2 N 2 (3.27) 1 Hα ωα 1 + 1 ωα ζ 1 δ ωα 2 ωα ζ 2 σ. Dδ,σ ≤ 2 | || | 2 2 | · | − | | | · | − α=1 α=1 X q X c b c b A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 11

Next, deﬁne the function with g : [0, 1]N [0, 1]N [0, 1]N R as × × → N N 2 g(h,s,t)= 1 h t 1+ s2 1 δ t2 s2 σ 2 α α α − α α − (3.28) α=1 α=1 X p X for (h,s,t) [0, 1]N [0, 1]N [0, 1]N , ∈ × × where h = (h1,...,hN ), s = (s1,...,sN ) and t = (t1,...,tN ). Inequality (3.27) then takes the form

g(( H1 ,..., HN ), ( ω1 ,..., ωN ), ( ω1 ζ ,..., ωN ζ )). Dδ,σ ≤ | | | | | | | | || · | | · | Our purpose is now to maximize the function g under the constraints c b c b N N (3.29) t2 1, h2 = 1. α ≤ α α=1 α=1 X X N 2 We claim that the maximum of g can only be attained if α=1 tα = 1. To verify this claim, it suﬃces to show that P (3.30) g(h,s,τt) g(h,s,t) for every (h,s,t) [0, 1]N [0, 1]N [0, 1]N and τ [0, 1]. ≤ ∈ × × ∈ Plainly,

N N 2 g(h,s,τt)= 1 τ 2 h t 1+ s2 1 τ 2δ t2 s2 σ 2 α α α − α α − α=1 α=1 X p X for (h,s,t) [0, 1]N [0, 1]N [0, 1]N and τ [0, 1]. Note that ∈ × × ∈ N N (3.31) 0 δ t2 s2 δ t2 = δ 1 . ≤ α α ≤ α ≤ 2 αX=1 αX=1 Thus, for each ﬁxed (h,s,t) [0, 1]n [0, 1]n [0, 1]n, we have that ∈ × × (3.32) g(h,s,τt)= c τ(1 c τ) β for τ [0, 1], 1 − 2 − ∈ 1 for suitable constants c1 0 and 0 c2 2 , depending on (h,s,t). Since the polynomial on the right-hand side of equation (3.32) is increasing≥ for≤ τ ≤[0, 1], inequality (3.30) follows. As a consequence, constraints (3.29) can be equivalently replaced by ∈

N N 2 2 (3.33) tα =1 and hα = 1. α=1 α=1 X X N 2 Let us maximize the function g(h,s,t) with respect to h, under the constraint α=1 hα = 1. Let (h1,...,hN ) be any point where the maximum is attained. Then, there exists a Langrange multiplier λ R such that P ∈ N N (3.34) t 1+ s2 h t 1+ s2 1 δ t2 s2 = 2λh for α = 1,...,N. α α γ γ γ − γ γ α γ=1 γ=1 p X q X Multiplying through equation (3.34) by hβ, and then subtracting equation (3.34), with α replaced by β, mul- tiplied by hα yield N n (3.35) h t 1+ s2 1 δ t2 s2 h t 1+ s2 h t 1+ s2 = 0 γ γ γ − γ γ β α α − α β β γ=1 γ=1 X q X p q 12 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA for α, β = 1,...,N. Owing to equation (3.31), we have that 1 δ n t2 s2 1 . Next, if N h t 1+ s2 = − γ=1 γ γ ≥ 2 γ=1 γ γ γ 0, then h1t1 = = hN tN = 0, whence δ,σ = σ 0, and inequality (3.21) holds trivially. Therefore,q we may · · · D − ≤ P P N 2 assume that γ=1 hγtγ 1+ sγ > 0 in what follows. Under this assumption, equation (3.35) tells us that q P 2 2 (3.36) hβtα 1+ sα = hαtβ 1+ sβ for α, β = 1,...,N. Combining equations (3.33p) and (3.36) yieldsq N N N 2 2 2 2 2 2 2 2 2 2 2 (3.37) tα(1 + sα)= tα(1 + sα) hβ = hα tβ(1 + sβ)= hα 1+ tβsβ βX=1 βX=1 βX=1 for α = 1,...,N. Hence,

N (3.38) h t 1+ s2 = h2 1+ t2 s2 α α α αv β β u β=1 p u X t for α = 1,...,N. From equations (3.28), (3.38) and (3.33) we deduce that 2 N N N g(h,s,t) 1 h2 1+ t2 s2 1 δ t2 s2 σ ≤ 2  αv β β − α α − α=1 u β=1 α=1 X u X X  N t N N = 1 1+ t2 s2 1 δ t2 s2 σ = ψ t2 s2 , 2 β β − α α − α α α=1 α=1 βX=1 X X where ψ : [0, 1] R is the function deﬁned as → ψ(r)= 1 (1 + r) 1 δr σ for r R. 2 − − ∈ Set ρ = N t2 s2 , and notice that ρ [0, 1], since 0 N t2 s2 N t2 = 1. Thereby, the maximum of j=α α α ∈ ≤ α=1 α α ≤ α=1 α the function g on [0, 1]N [0, 1]N [0, 1]N under constraints (3.33) agrees with the maximum of the function ψ P × × 1 P P on [0, 1]. It is easily veriﬁed that, if δ [0, 3 ], then maxr [0,1] ψ(r) = ψ(1). Hence, since we are assuming that δ + σ 1, ∈ ∈ ≥ ψ(1) = 1 δ σ 0. Dδ,σ ≤ − − ≤ 1 1 1 δ On the other hand, if δ ( 3 , 2 ], then maxr [0,1] ψ(r)= ψ( 2−δ ). Therefore, ∈ ∈ 1 δ (δ + 1)2 ψ − = σ. Dδ,σ ≤ 2δ 8δ − The proof of inequality (3.21) is complete.

Lemma 3.6. Let n, N, Ω and u be as in Theorem 2.1. Given p 1, let κN (p) be the constant deﬁned by (2.2)–(2.3). Then ≥ 2 2 2 u 2 (3.39) 2u + 2(p 2) u + (p 2)2 ∇ ( u )T κ (p) 2u |∇ | − ∇|∇ | − u ∇|∇ | ≥ N |∇ |

|∇ | in u = 0 . Moreover, the constant κN (p) is sharp in ( 3.39). {∇ 6 } Proof. Case N = 1. Inequality (3.39) trivially holds if p 2. Let us focus on the case when 1 p< 2. Notice that, on setting ≥ ≤ T ( u) n 2 n n ω = ∇ R and H = u R × u ∈ ∇ ∈ sym |∇ | A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 13 at any point in u = 0 , we have that {∇ 6 } 2 2 u 2 Hω 2 = u , ω Hω 2 = ∇ ( u )T , H 2 = 2u . | | ∇|∇ | | · | u ∇|∇ | | | |∇ |

|∇ | Therefore, by equation (3.9),

2 2 u 2 u 1 ∇ ( u )T + 1 2u . ∇|∇ | ≤ 2 u ∇|∇ | 2 |∇ |

|∇ | Consequently, the following chain holds:

2 2 2 u 2u + 2(p 2) u + (p 2)2 ∇ ( u )T |∇ | − ∇|∇ | − u ∇|∇ | |∇ | 2 2 u 1 + (p 2) 2u + (p 2) + (p 2)2 ∇ ( u )T ≥ − |∇ | − − u ∇|∇ | |∇ | 2 2 u (p 1) 2u + (p 1)(p 2) ∇ ( u )T ≥ − |∇ | − − u ∇|∇ | |∇ | 2 (p 1) + (p 1)(p 2) 2u ≥ − − − |∇ | 2 = (p 1)2 2u . − |∇ | Hence, inequality (3.39) follows. As far as the sharpness of the constant is concerned, if p 2, consider the function u : Rn 0 R given by ≥ \ { }→ u(x)= x for x Rn 0 . | | ∈ \ { } Since u = 0, equality holds in (3.39) for every x Rn 0 . On the other hand, if p [1, 2), consider the function∇|∇u :|Rn R defined as ∈ \ { } ∈ → u(x)= 1 x2 for x Rn. 2 1 ∈ One has that 2 2 2 u 2u = u = ∇ ( u )T =1 in Rn. |∇ | ∇|∇ | u ∇|∇ | |∇ | R n Hence, equality holds in (3.39) for every x 0 . ∈ \ { } Case N 2. It suffices to prove that inequality (3.39) holds at every point x u = 0 under the assumption that 2≥u(x) equals either 0 or 1. Indeed, if 2u(x) = 0 at some point∈ {∇x, then6 } the function given by |∇u | 2 |∇ | 6 u = 2 fulfills u(x) = 1. Hence, inequality (3.39) for u at the point x follows from the same inequality u(x) |∇ | applied|∇ to u| . If p 2, inequality (3.39) holds trivially. Thus, we may focus on the case when p [1, 2). In this case, we make use of≥ Lemma 3.5. Define ∈ α α u n α 2 α n n ω = ∇ R and H = u R × u ∈ ∇ ∈ sym |∇ | for α = 1,...,N, at any point in u = 0 . In particular, assumptions (3.19) and (3.20) are satisfied with this choice. Computations show that {∇ 6 } 2 2 u 2 (3.40) J = u , J = ∇ ( u )T , J = 2u , ∇|∇ | 0 u ∇|∇ | 1 |∇ |

|∇ | where J, J0 and J1 are deﬁned as in (3.18 ). 2 p 1 1 4 p Next, let δ = −2 . Notice that δ [0, 2 ], and that δ (0, 3 ] if and only if p [ 3 , 2). We next choose σ = 2 if 2 ∈ 2 ∈ ∈ 4 (δ+1) 1 (4 p) 4 p [ , 2), and σ = = − if p [1, ). Observe that δ + σ = 1 in the former case, and δ + σ > 1 in ∈ 3 8δ 16 2 p ∈ 3 the latter. Thus, the assumptions− on δ and σ of Theorem 3.5 are fulﬁlled. Furthermore, by our choice of σ, the 14 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

1 4 maximum on right-hand side of inequality (3.21) equals 0 when δ > 3 , namely when p [1, 3 ). From inequality (3.21) we infer that ∈ 2 p J − J + σJ . ≤ 2 0 1 This inequality is equivalent to J + 2(p 2)J + (p 2)2J (1 σ2(2 p))J . 1 − − 0 ≥ − − 1 Since 1 σ2(2 p)= (p), inequality (3.39) follows. − − K In order to prove the sharpness of the constant (p), let us distinguish the cases when p 2, p [ 4 , 2) and K ≥ ∈ 3 p [1, 4 ). ∈ 3 If p 2, consider the function u : Rn 0 RN given by ≥ \ { }→ u(x) = ( x , 0,..., 0) for x Rn 0 . | | ∈ \ { } Since u = 0, equality holds in (3.39) for every x Rn 0 . ∇|∇ | ∈ \ { } If p [ 4 , 2), consider the function u : Rn RN deﬁned as ∈ 3 → u(x) = ( 1 x2, 0,..., 0) for x Rn. 2 1 ∈ One has that 2 2 2 u 2u = u = ∇ ( u )T =1 in Rn. |∇ | ∇|∇ | u ∇|∇ |

Rn |∇ | Thus, equality holds in (3.39) for every x 0 . 4 p ∈ \ { } Rn If p [1, 3 ), set r0 = 2(2 p) . Let e1, e2 denote the ﬁrst two vectors of the canonical base of . Deﬁne ∈ − 1 t1 = √r0, ω = t1e1, t = √1 r , ω2 = t e , 2 − 0 2 2 2r h = 0 ,H1 = h e e , 1 1+ r 1 1 ⊗ 1 r 0 1 r h = − 0 ,H2 = h 1 e e + e e , 2 1+ r 2 √2 1 ⊗ 2 2 ⊗ 1 r 0 and ω3 = = ωN = 0, H3 = = HN = 0. Then · · · · · · N 2 2 (3.41) ωα 2 = ω1 + ω2 = 1. | | | | | | αX=1 Moreover, N 2 2 (3.42) J = Hα 2 = H1 + H2 = 1, 1 | | | | | | α=1 X 2 N 2 2 α α 1 1 2 2 2 1 1+ r0 1+ r0 (3.43) J = H ω = H ω + H ω = h1t1 + h2t2 e1 = e1 = , | | √2 2 2 α=1 r X N 2 2 r (1 + r ) (3.44) J = ωα (H 1ω1 + H2ω2) = ω1 (H1ω1 + H2ω2) = 0 0 . 0 | · | · 2 α=1 X Now, let u : Rn RN be a polynomial of degree two such that uα(0)T = ωα and 2uα = Hα for α = 1,...N. Formulas (3.40),→ combined with (3.42)–(3.44), tell us that ∇ ∇ 2 2 2 u 2 2u + 2(p 2) u + (p 2)2 ∇ ( u )T = 1 1 (4 p)2 = κ (p) 2u at 0. |∇ | − ∇|∇ | − u ∇|∇ | − 8 − N |∇ |

|∇ | Hence, equality holds in (3.39 ) for x = 0.

A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 15

We are now in a position to prove Theorem 2.1. Proof of Theorem 2.1. By Lemma 3.6, applied with p = Q ( u ) + 2, and the monotonicity of the function a |∇ | κN one has that u 2 (3.45) a( u )2 2u 2 + 2Q ( u ) u 2 + Q ( u )2 ∇ ( u )T |∇ | |∇ | a |∇ | |∇|∇ || a |∇ | u ∇|∇ | " # |∇ | κ Q ( u ) + 2 a( u )2 2 u 2 κ i + 2 a( u )2 2u 2 in u = 0 . ≥ N a |∇ | |∇ | |∇ | ≥ N a |∇ | |∇ | {∇ 6 } Inequality (2.7) holds at every point in the set u = 0 , owing to equation (3.3) and inequality (3.45). It also trivially holds at every point in the set u ={∇ 0 , since6 }κ i + 2 1. {∇ } N a ≤ In order to verify the optimality of the constant κN (ia + 2) in inequality (2.7), pick a function u and a point x from the proof of Lemma 3.6 such that u(x ) = 0 and equality holds in inequality (3.39) with u = u and 0 ∇ 0 6 p = ia + 2 at the point x0. Namely, 2 2 2 u(x ) 2 (3.46) 2u(x ) + 2i u (x ) + i2 ∇ 0 ( u (x ))T = κ (i + 2) 2u(x ) . |∇ 0 | a ∇|∇ | 0 a u(x ) ∇|∇ | 0 N a |∇ 0 | 0 |∇ | By the the deﬁnition of the index ia, given ε> 0 there exists t0 (0, ) such that ∈ ∞ (3.47) i Q (t ) i + ε. a ≤ a 0 ≤ a t0u Deﬁne , the function u = , so that u(0) = t0. From identity (3.3), equation (3.46) and inequality u(x0) |∇ | (3.47) we obtain that |∇ |

2 2 T 1 2 div(a( u ) u) div a( u ) (∆u) u 2 u (3.48) |∇ | ∇ − |∇ | ∇ − ∇|∇ | a( h u )2 2u 2 i |∇ | |∇ | x=x0 2 2 2 2 2 u(x0) T u(x0) + 2Qa(t0) u (x0) + Qa(t0) ∇ ( u (x0)) |∇ | |∇|∇ | | u(x0) ∇|∇ | = |∇ | 2 2 u(x0) |∇ | 2 2 2 2 2 u(x0) T u(x0) + 2Qa(t0) u (x0) + Qa(t0) ∇ ( u (x0)) |∇ | |∇|∇ | | u(x0) ∇|∇ | = |∇ | 2 2 u(x0) |∇ | 2 2 2 2 2 2 u(x0) T u(x0) + 2(ia + ε) u (x0) + (i + 2ε ia + ε )) ∇ ( u (x0)) |∇ | |∇|∇ | | a | | u(x0) ∇|∇ | |∇ | 2 2 ≤ u(x0) |∇ | 2 2 2 u(x0) T 2ε u (x0) + (2ε ia + ε )) ∇ ( u (x0)) |∇|∇ | | | | u(x0) ∇|∇ | = k (i +2)+ |∇ | N a 2 2 u(x0) |∇ | Hence, the optimality of the constant κN (ia + 2) in inequality (2.7) follows, owing to the arbitrariness of ε. 4. Function spaces An appropriate functional framework for the analysis of solutions to systems of the general form (1.5) is provided by the Orlicz-Sobolev spaces associated with the energy integral appearing in the functional (1.6). They consist in a generalization of the classical Sobolev spaces, where the role of powers in the definition of the norm is played by more general Young functions. Subsection 4.1 is devoted to some basic definitions and properties of Young functions and of Orlicz-Sobolev spaces. A Poincarétype inequality for functions in these spaces of use for our purposes is established as well. In Subsection 4.2 we collect specific properties of the Young function (and of perturbations of its) for the specific Orlicz-Sobolev ambient space associated with system (2.11). 16 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

4.1. Young functions and Orlicz-Sobolev spaces. A Young function A : [0, ) [0, ] is a convex ∞ → ∞ function such that A(0) = 0. The Young conjugate of a Young function A is the Young function A deﬁned as A(t) = sup st A(s) : s 0 for t 0. { − ≥ } ≥ e A Young function (and, more generally, an increasing function) A is said to belong to the class ∆2, or to satisfy e the ∆2-condition, if there exists a constant c> 1 such that (4.1) A(2t) cA(t) for t> 0. ≤ Let iA and sA be the indices associated with a continuously diﬀerentiable function A as in (2.1), with a replaced by A. Namely

tA′(t) tA′(t) (4.2) iA = inf and sA = sup . t>0 A(t) t>0 A(t) One has that A ∆ if and only if s < . The constant c in inequality (4.1) depends on s . Also, A ∆ if ∈ 2 A ∞ a ∈ 2 and only if iA > 1. The Orlicz space LA(Ω) is the Banach function space of those real-valued measurable functions u e: Ω : R whose Luxemburg norm → u u A = inf λ> 0 : A | | dx 1 k kL (Ω) λ ≤ ZΩ A N N A N n N n is finite. The Orlicz space L (Ω, R ) of R -valued functions and the Orlicz space L (Ω, R × ) of R × -valued functions are defined analogously. The Orlicz-Sobolev space W 1,A(Ω) is the Banach space (4.3) W 1,A(Ω) = u LA(Ω) : u is weakly differentiable in Ω and u LA(Ω, Rn) , { ∈ ∇ ∈ } and is equipped with the norm

u 1,A = u A + u A Rn . k kW (Ω) k kL (Ω) k∇ kL (Ω, ) 1,A 1,A 1,A The space Wloc (Ω) is defined accordingly. By W0 (Ω) we denote the subspace of W (Ω) of those functions 1,A Rn 1,A in W (Ω) whose extension by 0 outside Ω is weakly differentiable in the whole of . The notation (W0 (Ω))′ 1,A stands for the dual of W0 (Ω). If Ω has finite Lebesgue measure Ω , then the functional u LA(Ω,Rn) defines 1,A | | k∇ k a norm in W (Ω) equivalent to u 1,A . 0 k kW (Ω) The space C (Ω) is dense in W 1,A(Ω) if A ∆ . Moreover, W 1,A(Ω) is reflexive if both A ∆ and A ∆ , 0∞ 0 ∈ 2 0 ∈ 2 ∈ 2 and hence if iA > 1 and sA < . 1,A∞ RN RN 1,A RN 1,A RN The Orlicz-Sobolev space W (Ω, ) of -valued functions, its variants Wloc (Ω, ) and W0 (Ωe , ), 1,A RN and the space (W0 (Ω, ))′ are defined analogously. If Ω < and the Young function A ∆ , then the Poincarétype inequality | | ∞ ∈ 2 (4.4) A( u ) dx c A( u ) dx | | ≤ |∇ | ZΩ ZΩ 1,A holds for some constant c = c(n, Ω ,sa) and for every function u W0 (Ω). Inequality (4.4) follows, for instance, from [53, Lemma 3]. | | ∈ In order to bound lower-order terms appearing in our global estimate, we also need a stronger, yet non- 1,A optimal, Sobolev-Poincarétype inequality for functions in W0 (Ω) with an Orlicz target space smaller than LA(Ω). This is the subject of Theorem 4.1 below, which generalizes a version of the relevant inequality with optimal Orlicz target space from [23] (see also [22] for an equivalent form). Assume that the Young function A and the number σ> 1 satisfy the conditions 1 t σ−1 (4.5) dt < A(t) ∞ Z0 A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 17 and 1 t σ−1 (4.6) ∞ dt = . A(t) ∞ Z Then, we define the function Hσ : [0, ) [0, ) as ∞ → ∞ 1 1 s t σ−1 σ′ (4.7) H (s)= dt for s 0, σ A(t) ≥ Z0 and the Young function Aσ as 1 (4.8) A (t)= A(H− (t)) for t 0. σ σ ≥ Theorem 4.1. Let Ω be an open set in Rn with Ω < . Assume that the Young function A and the number σ n fulfill conditions (4.5) and (4.6). Then, there| | exists∞ a constant c = c(n,σ) such that ≥ u(x) (4.9) A dx A( u )dx σ 1 1 | | 1/σ Ω c Ω n − σ A( u )dy ! ≤ Ω |∇ | Z | | Ω |∇ | Z for every u W 1,A(Ω). R ∈ 0 Proof. By the Pólya-Szegöprinciple on the decrease of the functional on the right-hand side of inequality (4.9) 1,A under symmetric decreasing rearrangement of functions u W0 (Ω) (see [13]), it suffices to prove inequality (4.9) in the case when Ω is a ball and the trial functions∈ u are nonnegative and radially decreasing. As a consequence, this inequality will follow if we show that 1 Ω Ω ′ Ω | | ϕ(r)r− n dr (4.10) | | A s ds | | A(ϕ(s)) ds σ 1 1 Ω 1/σ ≤ 0 c Ω n −Rσ | | A(ϕ(r))dr ! 0 Z | | 0 Z for a suitable constant c as in the statement R and for every measurable function ϕ : (0, Ω ) [0, ). Let S be the linear operator defined as | | → ∞

Ω 1 | | ′ (4.11) Sϕ(s)= ϕ(r)r− n dr for s (0, Ω ), ∈ | | Zs for every measurable function ϕ : (0, Ω ) R that makes the integral on the right-hand side converge. One has that | | → 1 ′ 1 Ω ′ Ω ′ Ω σ ′ | | ′ σ | | σ | | σ ′ σ n′ (4.12) Sϕ Lσ (0, Ω ) = Sϕ(s) ds s− ϕ(r) dr ds k k | | 0 | | ≤ 0 s | | Z Z1 Z ′ ′ Ω σ σ 1 1 | | ′ ϕ L1(0, Ω ) s− n ds = c Ω n − σ ϕ L1(0, Ω ) ≤ k k | | | | k k | | Z0 for a suitable constant c = c(n,σ) and for every ϕ L1(0, Ω ). Also, by the Hardy-Littlewood inequality for rearrangements, ∈ | |

Ω 1 Ω 1 | | ′ | | ′ (4.13) Sϕ L∞(0, Ω ) ϕ(r) r− n dr ϕ∗(r)r− n dr k k | | ≤ | | ≤ Z0 Z0 1 1 Ω 1 1 1 | | ′ Ω n − σ ϕ∗(r)r− σ dr = Ω n − σ ϕ Lσ,1(0, Ω ) ≤ | | | | k k | | Z0 σ,1 σ,1 for every ϕ L (0, Ω ). Here, ϕ∗ denotes the decreasing rearrangement of ϕ, and L (0, Ω ) is the Lorentz space whose∈ norm is| deﬁned| by the last integral in equation (4.13). Owing to equations (4.12| )| and (4.13), the interpolation theorem established in [23, Theorem 4] can be applied to deduce inequality (4.10).

The next lemma tells us that the assumptions of Theorem 4.1 are certainly fulfilled if A satisfies the ∆2- condition, provided that σ is sufficiently large. 18 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

Lemma 4.2. Let A be a continuously differentiable Young function satisfying the ∆2-condition and let σ>sA. Then conditions (4.5) and (4.6) are fulfilled. A(t) Proof. Owing to the definition of sa, one verifies via differentiation that the function tsA is non-increasing. Thus, (4.14) A(t) A(1)tsA if t (0, 1], ≥ ∈ and (4.15) A(t) A(1)tsB if t [1, ). ≤ ∈ ∞ Equations (4.5) and (4.6) follow from (4.14) and (4.15), respectively. 4.2. Young functions built upon the function a. Given a continuously differentiable function a : (0, ) ∞ → (0, ) such that ia 1, let b and B the functions defined by (1.8) and (1.7). Our assumption on ia ensures that∞b is a non-decreasing≥ − function, and hence B is a Young function. One has that

(4.16) ib = ia +1 and sb = sa + 1. Also (4.17) i i +1 and s s + 1. B ≥ b B ≤ b Thus, if s < , then the functions b and B satisfy the ∆ -conditon, and if i > 1, then the function B a ∞ 2 a − satisﬁes the ∆2-conditon. Hence, if s < , then for every λ> 1, there exists a constant c = c(λ, s ) > 1 such that e a ∞ a (4.18) b(λt) cb(t) for t 0, ≤ ≥ and (4.19) B(λt) cB(t) for t 0. ≤ ≥ Moreover,

(4.20) tb′(t) (s + 1)b(t) for t> 0, ≤ a and (4.21) B(t) tb(t) (s + 2)B(t) for t> 0. ≤ ≤ a Since B(b(t)) B(2t) for t 0, there exists a constant c = c(s ) such that ≤ ≥ a (4.22) B(b(t)) cB(t) for t 0. e ≤ ≥ Finally, if ia > 1 and sa < , then − ∞ e (4.23) a(1) min tia ,tsa a(t) a(1) max tia ,tsa for t> 0. { }≤ ≤ { } If the function a is as above and ε> 0, we define the function a : [0, ) (0, ) as ε ∞ → ∞ (4.24) a (t)= a( t2 + ε2) for t 0. ε ≥ The functions bε and Bε are defined as in (1.8) andp (1.7), with a replaced by aε. Lemma 4.3. Assume that the function a : (0, ) (0, ) is continuously differentiable in (0, ) and that i > 1 and s < . Let ε> 0 and let a be the∞ function→ ∞ defined by (4.24). Then ∞ a − a ∞ ε (4.25) i min i , 0 and s max s , 0 , aε ≥ { a } aε ≤ { a } where iaε and saε are defined as in (2.1), with a replaced by aε. Let b, B, bε and Bε be the functions defined above. Then there exist constants c1, c2, c3, depending only on sa, such that (4.26) c B(t) c B(ε) a (t)t2 c (B(t)+ B(ε)) for t 0. 1 − 2 ≤ ε ≤ 3 ≥ A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 19

Moreover, there exists a constant c = c(sa) such that (4.27) B (t) c(B(t)+ B(ε)) for t 0, ε ≤ ≥ and (4.28) B(b (t)) c(B(t)+ B(ε)) for t 0. ε ≤ ≥ Proof. Property (4.25) can be verified by straightforward computations. Consider equation (4.26). One has e that (4.29) a (t)t2 a(t + ε)t2 (s + 2)B(t + ε) (s + 2)(B(2t)+ B(2ε)) c(B(t)+ B(ε)) for t 0, ε ≤ ≤ a ≤ a ≤ ≥ for some constant c = c(sa), where the second inequality holds by (4.21) and the last one by (4.1). This proves the second inequality in (4.26). As for the first one, observe that (4.30) B(t) B(t + ε) B(2t)+ B(2ε) cB(t)+ cB(ε) for t 0, ≤ ≤ ≤ ≥ for some constant c = c(sa), where we have made use of inequality (4.1) again. Now, t t t (4.31) B(t)= a(τ)τ dτ a(τ + ε)(τ + ε) dτ a(2 τ 2 + ε2)2 τ 2 + ε2 dτ 0 ≤ 0 ≤ 0 Z t Z Z p p 2 2 2 2 2 2 2 2 2 2 c a( τ + ε ) τ + ε dτ c t a( t + ε ) t + ε = c aε(t)t t + ε for t 0, ≤ 0 ≤ ≥ Z p p p p p for some constant c = c(sa), where the third inequality is due to (4.18). On the other hand, (4.32) a (t)t t2 + ε2 √2a (t)t2 if t ε, ε ≤ ε ≥ and p (4.33) a (t)t t2 + ε2 √2a (ε)ε2 = √2a(√2ε)ε2 cB(ε) if 0 t ε, ε ≤ ε ≤ ≤ ≤ for some constant c = c(sap), where the last inequality holds thanks to (4.21). Combining inequalities (4.31)– (4.33) yields B(t) ca (t)t2 + cB(ε) for t 0, ≤ ε ≥ for some constant c = c(sa). Hence, the first inequality in (4.26) follows. Inequality (4.27) holds because of the first inequality in (4.21), applied with B replaced by Bε, and of the second inequality in (4.26). Inequality (4.28) is a consequence of the following chain: (4.34) B(b (t)) = B(a( t2 + ε2)t) B(b( t2 + ε2)) ε ≤ B(b(tp+ ε)) cB(t + ε) pc′(B(t)+ B(ε)) for t 0, e ≤ e ≤ e ≤ ≥ for some constants c and c′ depending on sa. Notice, that we have made use of property (4.22) in last but one inequality, and of property (4.1) ine the last inequality. Lemma 4.4. Assume that the function a : (0, ) (0, ) is continuously differentiable in (0, ) and that i > 1 and s < . Let ε > 0 and let a be the∞ function→ ∞ defined by (4.24). Let M > 0. Then there∞ exists a a − a ∞ ε constant c = c(ia,sa, ε, M) such that (4.35) P Q c a (P )P a (Q)Q | − |≤ | ε − ε | for every P,Q RN n such that P M and Q M. ∈ × | |≤ | |≤

Proof. By [33, Lemma 21], there exists a positive constant c = c(iaε ,saε ) such that 2 (4.36) c a ( P + Q )+ a′ ( P + Q )( P + Q ) P Q (a ( P )P a ( Q )Q) (P Q) ε | | | | ε | | | | | | | | | − | ≤ ε | | − ε | | · − N n for every P,Q R × . Hence, via inequalities (4.25), ∈ (4.37) c(1 + min i , 0 )a ( P + Q ) P Q a ( P )P a ( Q )Q { a } ε | | | | | − | ≤ | ε | | − ε | | | 20 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA for every P,Q RN n. Inequality (4.4) hence follows, since ∈ × a ( P + Q ) min a(t) : ε t 2M 2 + ε2 > 0 ε | | | | ≥ ≤ ≤ if P M and Q M, and (1+min i , 0 ) > 0. p | |≤ | |≤ { a } One more function associated with a function a as above and to a number ε> 0 will be needed in our proofs. The function in question is denoted by V : RN n RN n and is defined as ε × → × N n (4.38) V (P )= a ( P )P for P R × . ε ε | | ∈ Lemma 4.5. Assume that the function a : (0p, ) (0, ) is continuously differentiable and such that ia > 1 and s < . Let ε> 0 and let a be the function∞ → defined∞ by (4.24). Then − a ∞ ε (4.39) a ( P )P a( P )P as ε 0+, ε | | → | | → N n uniformly for P in any compact subset of R × . Moreover,

2 N n (4.40) (a ( P )P a ( Q )Q) (P Q) V (P ) V (Q) for P,Q R × , ε | | − ε | | · − ≈ ε − ε ∈ where the relation means that the two sides are bounded by each other, up to positive multiplicative constants ≈ depending only on ia and sa. Proof. Fix any 0 <ℓ

1+min ia,0 1+min ia,0 (4.42) aε( P )P aε(1) P { } max a(t) P { }. | | | |≤ | | ≤ t [1,√2] | || | ∈ Now, let L> 0. Fix any σ > 0. By inequality (4.42), there exists ℓ> 0 such that

(4.43) a ( P )P a( P )P a ( P )P + a( P )P σ | ε | | − | | | ≤ | ε | | | | | | |≤ for every ε [0, 1], provided that P < ℓ. On the other hand, inequality (4.41) ensures that there exists ε (0, 1) such∈ that | | 0 ∈ (4.44) a ( P )P a( P )P <σ | ε | | − | | | if ℓ P L. From inequalities (4.43) and (4.44) we deduce that, if 0 ε < ε , then ≤ | |≤ ≤ 0 (4.45) a ( P )P a( P )P <σ if P L. | ε | | − | | | | |≤ This shows that the limit (4.39) holds unifromly for P L. | |≤ As far as equation (4.40) is concerned, it follows from [32, Lemma 41] that, if iaε > 1 and saε < , then the ratio of the two sides of this equation is bounded from below and from above by positive− constants∞ depending only on a lower bound for iaε and an upper bound for saε . Owing to inequalities (4.25) and to our assumption that i > 1 and s < , we have that i min i , 0 > 0 and s max s , 0 < for every ε> 0. This a − a ∞ aε ≥ { a } aε ≤ { a } ∞ implies that equation (4.40) actually holds up to equivalence constants depending only on ia and sa. A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 21

5. Second-order regularity: local solutions The definiton of generalized local solution to the system (5.1) div(a( u ) u)= f in Ω − |∇ | ∇ that will be adopted is inspired by the results of [36], and involves the notion of approximate differentiability. Recall that a measurable function u :Ω RN is said to be approximately differentiable at x Ω if there exists a matrix ap u(x) RN n such that,→ for every ε> 0, ∈ ∇ ∈ × 1 y Br(x) : u(y) u(x) ap u(x)(y x) > ε lim { ∈ r | − − ∇ − | } = 0. r 0+ rn → N n If u is approximately differentiable at every point in Ω, then the function ap u :Ω R × is measurable. q RN ∇ → Assume that a is as in Theorem 2.4 and let f Lloc(Ω, ) for some q 1. An approximately differentiable RN ∈ ≥ 1 function u :Ω is called a local approximable solution to system (5.1) if a( ap u ) ap u Lloc(Ω), and → RN q RN | ∇ | | ∇ |∈ there exist a sequence fk C∞(Ω, ), with fk f in Lloc(Ω, ), and a corresponding sequence of local weak solutions u to{ the} systems⊂ → { k} (5.2) div(a( u ) u )= f in Ω , − |∇ k| ∇ k k such that (5.3) u u and u ap u a.e. in Ω, k → ∇ k → ∇ and

(5.4) lim a( uk ) uk dx = a( ap u ) ap u dx k ′ |∇ | |∇ | ′ | ∇ | | ∇ | →∞ ZΩ ZΩ for every open set Ω′ Ω. In what follows, we shall denote ap u simply by u. ⊂⊂ ∇ 1 R∇N 1,B RN Weak solutions to system (5.1) are deﬁned in a standard way if f Lloc(Ω, ) (W0 (Ω, ))′, where B 1,B∈ RN ∩ is the Young function deﬁned via (1.7). Namely, a function u Wloc (Ω, ) is called a local weak solution to this system if ∈

(5.5) a( u ) u ϕ dx = f ϕ dx ′ |∇ | ∇ ·∇ ′ · ZΩ ZΩ for every open set Ω Ω, and every function ϕ W 1,B(Ω , RN ). ′ ⊂⊂ ∈ 0 ′ Inequality (2.7) enters the proof of Theorem 2.4 through Lemma 5.1 below. The latter will be applied to solutions to systems which approximate system (2.11), and involve regularized differential operators and smooth right-hand sides. Lemma 5.1 can be deduced from Theorem 2.1 and inequality (2.10), along the same lines as in the proof of [27, Theorem 3.1, Inequality (3.4)]. The details are omitted, for brevity. We seize this opportunity to point out an incorrect dependence on the radius R of the constants in that inequality, due to a flaw in the scaling argument in the derivation of [27, Inequality (3.43)]. Lemma 5.1. Let n 2, N 2, and let Ω be an open set in Rn. Assume that the function a C1([0, )) ≥ ≥ ∈ ∞ satisfies conditions (2.4)–(2.6). Then there exists a constant C = C(n,N,ia,sa), such that 1 (5.6) R− a( u ) u 2 RN×n + a( u ) u 2 RN×n |∇ | ∇ L (BR, ) ∇ |∇ | ∇ L (BR, ) n 1 C div(a( u) u) 2 RN + R− 2 − a( u ) u 1 RN×n ≤ k |∇ | ∇ kL (B2R, ) k |∇ | ∇ kL (B2R, ) for every function u C3(Ω, RN ) and any ball B Ω. ∈ 2R ⊂⊂ Proof of Theorem 2.4. Let us temporarily assume that N (5.7) f C∞(Ω, R ) , ∈ and that u is a local weak solution to system (5.1). Observe that, thanks to equations (2.12) and (4.25), (5.8) i > 2(1 √2) . aε − 22 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

Let B Ω and, given ε (0, 1), let u u + W 1,B(B , RN ) be the weak solution to the Dirichlet problem 2R ⊂⊂ ∈ ε ∈ 0 2R div(a ( u ) u )= f in B (5.9) − ε |∇ ε| ∇ ε 2R (uε = u on ∂B2R . We claim that N (5.10) u C∞(B , R ). ε ∈ 2R RN n Actually, as a consequence of [34, Corollary 5.5], uε Lloc∞ (B2R, × ) and there exists a constant C, independent of ε, such that ∇ ∈

(5.11) u ∞ RN×n C. k∇ εkL (BR, ) ≤ α RN n The same result also tells us that aε( uε ) uε Cloc(B2R, × ) for some α (0, 1). Therefore, by inequality α |∇RN |n∇ ∈ 1,α ∈ (4.35), we have that uε Cloc(B2R, × ) as well. Hence, aε( uε ) Cloc (B2R), and by the Schauder theory ∇ ∈ 2,α RN |∇ | ∈ for linear elliptic systems, uε Cloc (B2R, ). An iteration argument relying upon the the Schauder theory again yields property (5.10). ∈ We claim that

(5.12) B( u ) dx C B( f ) dx + B( u ) dx + B(ε) |∇ ε| ≤ | | |∇ | ZB2R ZB2R ZB2R e 1,B RN for some constant C = C(n,N,sa,R) and for ε (0, 1). Indeed, choosing uε u W0 (B2R, ) as a test function in the weak formulation of problem (5.9∈) results in − ∈

(5.13) a ( u ) u ( u u) dx = f (u u) dx . ε |∇ ε| ∇ ε · ∇ ε −∇ · ε − ZB2R ZB2R The Poincar´einequality (4.4) implies that

(5.14) B( u u ) dx C B( u u ) dx | ε − | ≤ |∇ ε −∇ | ZB2R ZB2R for some constant C = C(n,sa,R). Fix δ (0, 1). From equation (5.13), the ﬁrst inequality in (4.26), and inequalities (5.14) , (4.22) and (4.27) one obtains∈ that

(5.15) c B( u ) dx f u u dx + C a ( u ) u u dx + CRnB(ε) 1 |∇ ε| ≤ | || ε − | ε |∇ ε| |∇ ε||∇ | ZB2R ZB2R ZB2R C B( f ) dx + δ B( u u ) dx ≤ 1 | | | ε − | ZB2R ZB2R + δ eB (a ( u ) u ) dx + C B ( u ) dx + CRnB(ε) ε ε |∇ ε| |∇ ε| 1 ε |∇ | ZB2R ZB2R C Be( f ) dx + δC B( u ) dx + C B( u ) dx ≤ 1 | | 2 |∇ ε| 3 |∇ | ZB2R ZB2R ZB2R + δC e B ( u ) dx + C B ( u ) dx + CRnB(ε) 4 ε |∇ ε| 1 ε |∇ | ZB2R ZB2R C B( f ) dx + δC B( u ) dx + C B( u ) dx + CRnB(ε) ≤ 1 | | 5 |∇ ε| 6 |∇ | ZB2R ZB2R ZB2R for suitable constants C2, C4 and C5 dependinge on n,N,sa,R, and constants C1, C3 and C6 depending also on δ. Inequality (5.12) follows from (5.15), on choosing δ small enough. Coupling inequality (5.12) with the Poincar´einequality (4.4) tells us that the family u is bounded in { ε} A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 23

1,B N W (B2R, R ). Since under assumptions (2.12) and (2.13) this space is reﬂexive, there exist a sequence εk and a function v W 1,B(B , RN ) such that ε 0+ and { } ∈ 2R k → 1,B RN (5.16) uεk ⇀ v in W (B2R, ).

Choosing the test function uεk u for system (2.11), and subtracting the resultant equation from (5.13) enables us to deduce that, given any δ− > 0,

(5.17) a ( u ) u a ( u ) u ( u u) dx εk |∇ εk | ∇ εk − εk |∇ | ∇ · ∇ εk −∇ ZB2R = a( u ) u a ( u ) u ( u u) dx |∇ | ∇ − εk |∇ | ∇ · ∇ εk −∇ ZB2R δ B( u )+ B( u ) dx + C B a( u ) u a ( u ) u dx ≤ |∇ εk | |∇ | | |∇ | ∇ − εk |∇ | ∇ | ZB2R ZB2R for some constant C = C(δ, sa). Owing to equation (4.40), theree exists a constant c = c(ia,sa) such that (5.18) V ( u ) V ( u) 2 dx 2 V ( u ) V ( u) 2 dx + 2 V ( u) V ( u) 2 dx | εk ∇ εk − ∇ | ≤ | εk ∇ εk − εk ∇ | | εk ∇ − ∇ | ZB2R ZB2R ZB2R c a ( u ) u a ( u ) u ( u u) dx + 2 V ( u) V ( u) 2 dx. ≤ εk |∇ εk | ∇ εk − εk |∇ | ∇ · ∇ εk −∇ | εk ∇ − ∇ | ZB2R ZB2R Combining equations (5.18), (5.17) and (5.12) yields

(5.19) V ( u ) V ( u) 2 dx δc B( f ) dx + B( u ) dx + B(ε) | εk ∇ εk − ∇ | ≤ | | |∇ | ZB2R ZB2R ZB2R + c Be a( u ) u a ( u ) u dx | |∇ | ∇ − εk |∇ | ∇ | ZB2R + 2 eV ( u) V ( u) 2 dx | εk ∇ − ∇ | ZB2R for some constant c = c(n,N,R,ia,sa). Inequalities (4.22) and (4.28) entail that (5.20) B a( u ) u a ( u ) u c(B( u )+ B(ε )) a.e. in B , | |∇ | ∇ − εk |∇ | ∇ | ≤ |∇ | k 2R for some constant c = c(sa). Furthermore, from inequality (4.26) one infers that e (5.21) V ( u) 2 c(B( u )+ B(ε )) a.e. in B , | εk ∇ | ≤ |∇ | k 2R for some constant c = c(sa). Thanks to inequalities (5.20) and (5.21), and to property (4.39), the last two integrals on the right-hand side of inequality (5.19) tend to 0 as k , via the dominated convergence theorem. Owing to the same theorem, equation (5.19) implies that → ∞

2 (5.22) lim Vεk ( uεk ) V ( u) dx δc k | ∇ − ∇ | ≤ →∞ ZB2R for every δ (0, 1). Thereby, ∈ 2 N n (5.23) V ( u ) V ( u) in L (B , R × ), εk ∇ εk → ∇ 2R and, on passing to a subsequence, still indexed by k, (5.24) V ( u ) V ( u) a.e. in B . εk ∇ εk → ∇ 2R 1 An analogous argument as in [35, Lemma 4.8] shows that the function (ε, P ) Vε− (P ) is continuous. Thus, one can deduce from equation (5.24) that 7→ (5.25) u u a.e. in B . ∇ εk →∇ 2R Hence, equation (5.16) implies that v = u and 1,B RN (5.26) uεk ⇀ u in W (B2R, ). 24 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

Inequalities (4.26) and (5.12), and the monotonicity of the function bεk , yield

2 (5.27) aεk ( uεk ) uεk dx aεk ( uεk ) uεk dx + aεk ( uεk ) uεk dx B2R |∇ | |∇ | ≤ uε 1 B2R |∇ | |∇ | B2R |∇ | |∇ | Z Z{|∇ k |≤ }∩ Z cRnb (1) + c B( u ) dx + cRnB(ε ) C ≤ εk ∇ εk k ≤ ZB2R for some constants c and C independent of k. Thanks to assumption (5.8) and to property (4.25), Lemma 5.1 can be applied with a replaced by aεk . The use of inequality (5.6) of this lemma for the function uεk , and the equation in (5.9), ensure that

(5.28) a ( u ) u 1,2 RN×n k ε |∇ εk | ∇ εk kW (BR, ) n n 1 C f 2 RN + (R− 2 + R− 2 − ) a ( u ) u 1 RN×n , ≤ k kL (B2R, ) k εk |∇ εk | ∇ εk kL (B2R, ) for some constant C = C(n,N,ia,sa). Owing to inequalities (5.27) and (5.28), the sequence aεk ( uεk ) uεk 1,2 N n 1,2 N n { |∇ | ∇ } is bounded in W (BR, R × ). Thus, there exists a function U W (BR, R × ), and a subsequence of εk , still indexed by k, such that ∈ { } (5.29) a ( u ) u U in L2(B , RN n) εk |∇ εk | ∇ εk → R × and a ( u ) u ⇀ U in W 1,2(B , RN n). εk |∇ εk | ∇ εk R × Combining property (4.39) with equations (5.11), (5.25) and (5.29) yields 1,2 N n (5.30) a( u ) u = U W (B , R × ). |∇ | ∇ ∈ R On passing to the limit as k , from equations (5.28), (5.29) and (5.30) we infer that →∞ n n 1 (5.31) a( u ) u 1,2 RN×n C f 2 RN + (R− 2 + R− 2 − ) a( u ) u 1 RN×n . k |∇ | ∇ kW (BR, ) ≤ k kL (B2R, ) k |∇ | ∇ kL (B2R, ) It remains to remove assumption (5.7). Suppose that f L2 (Ω, RN ). Let u be an approximable local solution ∈ loc to equation (2.11), and let fk and uk be as in the definition of this kind of solution. Applying inequality (5.31) to the function u tells us that a( u ) u W 1,2(B , RN n), and k |∇ k| ∇ k ∈ R × (5.32) a( u ) u 1,2 RN×n k |∇ k| ∇ kkW (BR, ) n n 1 C f 2 RN + (R− 2 + R− 2 − ) a( u ) u 1 RN×n , ≤ k kkL (B2R, ) k |∇ k| ∇ kkL (B2R, ) for some constant C independent of k. Hence, by equation (5.4), the sequence a( uk ) uk is bounded in 1,2 N n { |∇ | ∇ }1,2 N n W (BR, R × ). Thereby, there exist a subsequence, still indexed by k, and a function U W (BR, R × ), such that ∈ (5.33) a( u ) u U in L2(B , RN n) and a( u ) u ⇀ U in W 1,2(B , RN n). |∇ k| ∇ k → R × |∇ k| ∇ k R × By assumption (5.3), we have that u u a.e. in Ω. Hence, thanks to properties (5.33), ∇ k →∇ 1,2 N n (5.34) a( u ) u = U W (B , R × ) . |∇ | ∇ ∈ R Inequality (2.15) follows on passing to the limit as k in (5.32), via (5.4), (5.33) and (5.34). →∞ 6. Second-order regularity: Dirichlet problems Generalized solutions, in the approximable sense, to the Dirichlet problem div(a( u ) u)= f in Ω (6.1) − |∇ | ∇ (u =0 on ∂Ω , are defined in analogy with the local solutions introduced in Section 5. Assume that a is as in Theorems 2.6 and 2.7 and let f Lq(Ω, RN ) for some q 1. An approximately differentiable function u :Ω RN is called an approximable∈ solution to the Dirichlet≥ problem (6.1) if there → A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 25

RN q RN exists a sequence fk C0∞(Ω, ) such that fk f in L (Ω, ), and the sequence uk of weak solutions to the Dirichlet problems{ }⊂ → { } div(a( u ) u )= f in Ω (6.2) − |∇ k| ∇ k k (uk =0 on ∂Ω satisﬁes (6.3) u u and u ap u a.e. in Ω. k → ∇ k → ∇ As above, in what follows ap u will simply be denoted by u. ∇ 1 RN ∇ 1,B RN 1,B RN Recall that, under the assumption that f L (Ω, ) (W0 (Ω, ))′, a function u W0 (Ω, ) is called a weak solution to the Dirichlet problem∈ (6.1) if ∩ ∈

(6.4) a( u ) u ϕ dx = f ϕ dx |∇ | ∇ ·∇ · ZΩ ZΩ for every ϕ W 1,B(Ω, RN ). A unique weak solution to problem (6.1) exists whenever Ω < . ∈ 0 | | ∞ Before accomplishing the proof of our global estimates, we recall the notions of capacity and of Marcinkiewicz spaces that enter conditions (2.22) and (2.23), respectively, in the statement of Theorem 2.7. The capacity cap (E) of a set E Ω relative to Ω is deﬁned as Ω ⊂ (6.5) cap (E) = inf v 2 dx : v C0,1(Ω), v 1 on E . Ω |∇ | ∈ 0 ≥ ZΩ 0,1 Here, C0 (Ω) denotes the space of Lipschitz continuous, compactly supported functions in Ω. q, The Marcinkiewicz space L ∞(∂Ω) is the Banach function space endowed with the norm deﬁned as 1 q (6.6) ψ Lq,∞(∂Ω) = sup s ψ∗∗(s) k k s (0, n−1(∂Ω)) ∈ H 1 s for a measurable function ψ on ∂Ω. Here, ψ∗∗(s) = s 0 ψ∗(r) dr for s > 0, where ψ∗ denotes the decreasing 1, ∫ rearrangement of ψ. The Marcinkiewicz space L ∞ log L(∂Ω) is equipped with the norm given by

∞ C (6.7) ψ L1, log L(∂Ω) = sup s log 1+ s ψ∗∗(s), k k s (0, n−1(∂Ω)) ∈ H for any constant C > n 1(∂Ω). Diﬀerent constants C result in equivalent norms in (6.7). H − The next lemma stands with respect to Theorems 2.6 and 2.7 that Lemma 5.1 stands to Theorem 2.4. It follows from Theorem 2.1 and inequality (2.10), via the same proof of [28, Theorem 3.1, Part (ii)]. Lemma 6.1. Let n 2, N 2, and let Ω be a bounded open set in Rn with ∂Ω C2. Assume that a is a function as in Theorem≥ 2.1≥ , which also fulﬁlls conditions (2.12) and (2.13). There∈ exists a constant c = c(n,N,ia,sa,LΩ, dΩ) such that, if (6.8) (r) (r) for r (0, 1), KΩ ≤ K ∈ for some function : (0, 1) [0, ) satisfying K → ∞ (6.9) lim (r) < c , r 0+ K → then

(6.10) a( u ) u 1,2 RN×n C div(a( u ) u) 2 RN + a( u ) u 1 RN×n k |∇ | ∇ kW (Ω, ) ≤ k |∇ | ∇ kL (Ω, ) k |∇ | ∇ kL (Ω, ) for some constant C = C(n,N,i ,s ,L , d , ), and for every function u C3(Ω, RN ) C2(Ω, RN ) such that a a Ω Ω K ∈ ∩ (6.11) u = 0 on ∂Ω. In particular, if Ω is convex, then inequality (6.10) holds whatever is, and the constant C in (6.10) only KΩ depends on n,N,ia,sa,LΩ, dΩ. 26 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

The following gradient bound for solutions to the Dirichlet problem (6.1) is needed to deal with lower-order terms appearing in our global estimates. Proposition 6.2. Assume that n 2, N 2. Let Ω be an open set in Rn such that Ω < . Assume that the function a : [0, ) [0, ) is continuously≥ ≥ differentiable in (0, ) and fulfills conditions| | ∞(2.12) and (2.13). 1 R∞N → 1,B∞ RN ∞ Let f L (Ω, ) (W0 (Ω, ))′ and let u be the weak solution to the Dirichlet problem (6.1). Then, there exists∈ a constant C∩= C(n,N,i ,s , Ω ) such that a a | | (6.12) a( u ) u 1 RN×n C f 1 RN . k |∇ | ∇ kL (Ω, ) ≤ k kL (Ω, ) The same conclusion holds if f L1(Ω, RN ) and u is an approximable solution to the Dirichlet problem (6.1). ∈ Proof. Assume that f L1(Ω, RN ) (W 1,B(Ω, RN )) and that u is the weak solution to the Dirichlet problem ∈ ∩ 0 ′ (6.1). Given t> 0, let T (u):Ω RN be the function defined by t → u in u t (6.13) T (u)= u {| |≤ } t t in u >t . u {| | }  | | Then T (u) W 1,B(Ω, RN ), and t ∈ 0  u a.e. in u t (6.14) T (u)= ∇t u u {| |≤ } ∇ t  I u a.e. in u >t  u − u ⊗ u ∇ {| | } | | | | | | Observe that  a( P )P (I ω ω)P 0 N n | | ·N − ⊗ ≥ for every matrix P R × and any vector ω R such that ω 1. Thus, on making use of Tt(u) as a test function ϕ in equation∈ (6.4), one deduces that∈ | |≤

2 (6.15) a( u ) u dx a( u ) u Tt(u) dx = f Tt(u) dx u t |∇ | |∇ | ≤ Ω |∇ | ∇ ·∇ Ω · Z{| |≤ } Z Z u = f u dx + f t dx t f L1(Ω,RN ). u t · u>t · u ≤ k k Z{| |≤ } Z{| } | | Hence, by the ﬁrst inequality in (4.21),

(6.16) B( u ) dx t f L1(Ω,RN ). u t |∇ | ≤ k k Z{| |≤ } 1,B On the other hand, the chain rule for vector-valued functions ensures that the function u W0 (Ω), and u u a.e. in Ω. Inequality (6.16) thus implies that | | ∈ |∇ | ≥ |∇| ||

(6.17) B( u ) dx t f L1(Ω) for t> 0. u max s + 2,n . Hence, t | | ∈ { a } σ> max sB,n , inasmuch as iB ib +1= ia + 2. Owing to Lemma 4.2, the assumptions of Theorem 4.1 are fulﬁlled,{ with A} replaced by B and≤ with this choice of σ. An application of the the Orlicz-Sobolev inequality (4.9) to the function T ( u ) tells us that t | | T ( u ) (6.18) B t dx B( (T ( u )) )dx, σ | | | | 1/σ t Ω C B( T ( u ) )dy ! ≤ Ω |∇ | | | Z Ω |∇ t | | | Z 1 1 where C = c Ω n − σ . Here, Bσ denotesR the function deﬁned as in (4.7)–(4.8), with A replaced by B. One has that | |

(6.19) B( Tt( u ) )dx = B( u )dx for t> 0, Ω |∇ | | | u

(6.20) T ( u ) = t in u t , | t | | | {| |≥ } and (6.21) T ( u ) t = u t for t> 0. {| t | | |≥ } {| |≥ } Thus,

t Tt( u ) (6.22) u t Bσ 1 Bσ | | | | 1/σ dx |{| |≥ }| C( B( u )dy) σ ≤ u t u 0. Hence, by (6.17), t (6.23) u t Bσ 1 t f L1(Ω,RN ) for t> 0. |{| |≥ }| C(t f 1 RN ) σ ≤ k k k kL (Ω, ) From inequality (6.16) we deduce that

1 t f L1(Ω,RN ) (6.24) B( u ) > s, u t B( u ) dx k k for t> 0 and s> 0. |{ |∇ | | |≤ }| ≤ s B( u )>s, u t |∇ | ≤ s Z{ |∇ | | |≤ } Coupling inequalities (6.24) and (6.23) yields (6.25) B( u ) >s u >t + B( u ) > s, u t |{ |∇ | }| ≤ |{| | }| |{ |∇ | | |≤ }| t f L1(Ω,RN ) t f L1(Ω,RN ) 1k k 1 + k k for t> 0 and s> 0. ′ ≤ B (Ct σ /(t f 1 RN ) σ ) s σ k kL (Ω, ) ′ The choice t = 1 f 1/σ B 1(s) σ in inequality (6.25) results in C k kL1(Ω,RN ) σ− ′ σ ′ 2 f 1 1 σ L (Ω,RN ) Bσ− (s) (6.26) B( u ) >s k k ′ for s> 0. |{ |∇ | }| ≤ Cσ s 1 Next, set s = B(b− (τ)) in (6.26) and make use of (4.8) to obtain that ′ σ ′ 2 f 1 1 σ k kL (Ω,RN ) Hσ(b− (τ)) (6.27) b( u ) >τ σ′ 1 for τ > 0, |{ |∇ | }| ≤ C B(b− (τ)) where Hσ is deﬁned as in (4.7), with A replaced by B. Thanks to inequality (6.27), 1 σ′ ∞ σ′ σ′ ∞ Hσ(b− (τ)) (6.28) b( u ) dx = b( u ) >τ dτ λb( Ω ) + 2C− f 1 N dτ |∇ | |{ |∇ | }| ≤ | | k kL (Ω,R ) B(b 1(τ)) ZΩ Z0 Zλ − for λ> 0. Owing to inequalities (4.20) and (4.21), and to Fubinis’s theorem, the following chain holds: 1 σ′ σ′ ∞ Hσ(b− (τ)) ∞ Hσ(s) (6.29) 1 dτ (sa + 1) b(s)ds B(b (τ)) ≤ −1 sB(s) Zλ − Zb (λ) 1 s − ∞ b(s) t σ 1 = (sa + 1) dt ds −1 sB(s) B(t) Zb (λ) Z0 1 s − ∞ 1 t σ 1 (sa + 1)(sa + 2) 2 dt ds ≤ −1 s B(t) Zb (λ) Z0 −1 1 1 b (λ) − − t σ 1 ∞ ds ∞ t σ 1 ∞ ds = (sa + 1)(sa + 2) 2 dt + 2 dt B(t) −1 s −1 B(t) s Z0 Zb (λ) Zb (λ) Zt 28 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

−1 1 1 b (λ) − − 1 t σ 1 ∞ t σ 1 dt = (sa + 1)(sa + 2) dt + b 1(λ) B(t) −1 B(t) t − Z0 Zb (λ) −1 1 1 b (λ) σ−1 σ−1 σ′ 1 1 ∞ 1 dt (sa + 1)(sa + 2) 1 dt + ≤ b (λ) b(t) −1 b(t) t − Z0 Zb (λ) s +1+ε for λ> 0. The function t a is increasing for every ε> 0. Hence, if 0 <ε<σ s 2, then b(t) − a − − 1 − 1 b 1(λ) b 1(λ) s +1+ε 1 1 σ−1 1 t a σ−1 sa+1+ε (6.30) dt = t− σ−1 dt b 1(λ) b(t) b 1(λ) b(t) − Z0 − Z0 1 −1 1 sa+1+ε − b (λ) 1 b− (λ) σ 1 sa+1+ε σ 1 1 t− σ−1 dt = − λ− σ−1 for λ> 0. ≤ b 1(λ) λ σ s 2 ε − Z0 − a − − tε On the other hand, if 0 <ε

1 1 1 σ−1 ε σ−1 1 ε σ−1 ∞ 1 dt ∞ t ε 1 b− (λ) ∞ ε 1 (6.31) = t− σ−1 − dt t− σ−1 − dt −1 b(t) t −1 b(t) ≤ λ −1 Zb (λ) Zb (λ) Zb (λ) σ 1 1 = − λ− σ−1 for λ> 0. ε

Inequalities (6.29)–(6.31) entail that there exists a constant c = c(σ, ia,sa) such that 1 σ′ ∞ Hσ(b− (τ)) 1 (6.32) dτ cλ− σ−1 for λ> 0. B(b 1(τ)) ≤ Zλ − Inequality (6.12) follows from (6.28) and (6.32), with the choice λ = f 1 RN . k kL (Ω, ) The assertion about the case when f L1(Ω, RN ) and u is an approximable solution to the Dirichlet problem ∈ (6.1) follows on applying inequality (6.12) with f and u replaced by the functions fk and uk appearing in the definition of approximable solutions, and passing to the limit as k in the resultant inequality. Fatou’s lemma plays a role here. → ∞ A last preliminary result, proved in [27, Lemma 5.2], is needed in an approximation argument for the domain Ω in our proof of Theorem 2.7. Lemma 6.3. Let Ω be a bounded Lipschitz domain in Rn, n 2 such that ∂Ω W 2,1. Assume that the function (r), defined as in (2.20), is finite-valued for r (0, 1). Then≥ there exist positive∈ constants r and C and a KΩ ∈ 0 sequence of bounded open sets Ωm , such that ∂Ωm C∞, Ω Ωm, limm Ωm Ω = 0, the Hausdorff distance between Ω and Ω tends{ to}0 as m , ∈ ⊂ →∞ | \ | m →∞ (6.33) L CL , d Cd Ωm ≤ Ω Ωm ≤ Ω and (6.34) (r) C (r) KΩm ≤ KΩ for r (0, r ) and m N. ∈ 0 ∈

Proof of Theorem 2.7. It suﬃces to prove Part (i). Part (ii) will then follow, since, by [27, Lemmas 3.5 and 3.7],

(6.35) Ω(r) C sup X(∂Ω Br (x)) for r (0, r0), K ≤ x ∂Ω kBk ∩ ∈ ∈ for suitable constants r0 and C depending on n, LΩ and dΩ. We split the proof in three separate steps, where approximation arguments for the diﬀerential operator, the A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 29 domain and the datum on the right-hand side of the system, respectively, are provided. Step 1. Assume that the additional conditions N (6.36) f C∞(Ω, R ) , ∈ 0 and

(6.37) ∂Ω C∞ ∈ are in force. Given ε (0, 1), we denote by u the weak solution to the system ∈ ε div(a ( u ) u )= f in Ω (6.38) − ε |∇ ε| ∇ ε (uε =0 on ∂Ω , where aε is the function deﬁned by (4.24). An application of [26, Theorem 2.1] tells us that

(6.39) u ∞ RN×n C k∇ εkL (Ω, ) ≤ for some constant C independent of ε. Let us notice that the statement of [26, Theorem 2.1] yields inequality (6.39) under the assumption that the function aε be either increasing or decreasing; such an additional assumption can however be dropped via a slight variant in the proof. Inequality (6.39) implies that, for each ε (0, 1), ∈

(6.40) c a ( u ) c in Ω 1 ≤ ε |∇ ε| ≤ 2 for suitable positive constants c1 and c2. A classical result by Elcrat and Meyers [9, Theorem 8.2] enables us to deduce, via properties (6.36), (6.37) and (6.40), that u W 2,2(Ω, RN ). Consequently, u W 1,2(Ω, Rn) W 1, (Ω, RN ) W 2,2(Ω, RN ). One can then ε ∈ ε ∈ 0 ∩ ∞ ∩ ﬁnd a sequence u C (Ω, RN ) C2(Ω, RN ) such that u =0 on ∂Ω for k N, and { k}⊂ ∞ ∩ k ∈ (6.41) u u in W 1,2(Ω, RN ), u u in W 2,2(Ω, RN ), u u a.e. in Ω, k → ε 0 k → ε ∇ k →∇ ε as k – see e.g. [15, Chapter 2, Corollary 3]. One also has that →∞ (6.42) u ∞ RN×n C u ∞ RN×n k∇ kkL (Ω, ) ≤ k∇ εkL (Ω, ) for some constant C independent of k, and, by the chain rule for vector-valued Sobolev functions [46, Theorem 2.1], u 2u a.e. in Ω. Moreover, [26, Equation (6.12)] tells us that |∇|∇ k|| ≤ |∇ k| (6.43) div(a ( u ) u ) f in L2(Ω, RN ), − ε |∇ k| ∇ k → as k . Assumption (2.22) enables us to apply inequality (6.10), with a replaced by a and u replaced by → ∞ ε uk, to deduce that

(6.44) a ( u ) u 1,2 RN×n C div(a ( u ) u ) 2 RN + a ( u ) u 1 RN×n k ε |∇ k| ∇ kkW (Ω, ) ≤ k ε |∇ k| ∇ k kL (Ω, ) k ε |∇ k| ∇ kkL (Ω, ) for k N, and for some constant C = C(n,N,i ,s ,L , d , ). Notice that this constant actually depends on ∈ a a Ω Ω KΩ the function aε only through ia and sa, and it is hence independent of ε, owing to (4.25). Equations (6.42)–(6.44) 1,2 N n ensure that the sequence aε( uk ) uk is bounded in W (Ω, R × ). Therefore, there exist a subsequence of u , still denoted by {u |∇, and| a∇ function} U W 1,2(Ω, RN n) such that { k} { k} ε ∈ × (6.45) a ( u ) u U in L2(Ω, RN n), a ( u ) u ⇀ U in W 1,2(Ω, RN n). ε |∇ k| ∇ k → ε × ε |∇ k| ∇ k ε × Equation (6.41) entails that u u a.e. in Ω. As a consequence, ∇ k →∇ ε (6.46) a ( u ) u a ( u ) u a.e. in Ω. ε |∇ k| ∇ k → ε |∇ ε| ∇ ε From equations (6.46) and (6.45) one infers that 1,2 N n (6.47) a ( u ) u = U W (Ω, R × ) . ε |∇ ε| ∇ ε ε ∈ Furthermore, passing to the limit as k in (6.44) yields →∞ (6.48) a ( u ) u 1,2 RN×n C f 2 RN + a ( u ) u 1 RN×n . k ε |∇ ε| ∇ εkW (Ω, ) ≤ k kL (Ω, ) k ε |∇ ε| ∇ εkL (Ω, ) 30 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

Here, equations (6.45) and (6.47) have been exploited to pass to the limit on the left-hand side, and equations (6.42) and (6.43) on the right-hand side. Combining equations (6.48) and (6.39) tells us that

(6.49) a ( u ) u 1,2 RN×n C k ε |∇ ε| ∇ εkW (Ω, ) ≤ for some constant C, independent of ε. By the last inequality, the family of functions aε( uε ) uε is 1,2 N n { |∇ | ∇ } uniformly bounded in W (Ω, R × ) for ε (0, 1). Therefore, there exist a sequence εm converging to 0 and a function U W 1,2(Ω, RN n) such that ∈ { } ∈ × (6.50) a ( u ) u U in L2(Ω, RN n), a ( u ) u ⇀ U in W 1,2(Ω, RN n). εm |∇ εm | ∇ εm → × εm |∇ εm | ∇ εm × An argument parallel to that of the proof of (5.25) yields (6.51) u u a.e. in Ω. ∇ εm →∇ We omit the details, for brevity. Let us just point out that, in this argument, one has to make use of the inequality

(6.52) B( u ) dx C B( f ) dx + B(ε ) , |∇ εm | ≤ | | m ZΩ ZΩ instead of (5.12). Inequality (6.52) easily follows on choosinge uεm as a test function in the deﬁnition of weak solution to problem (6.38), with ε = εm. Coupling equations (6.50) and (6.51) implies that 1,2 N n (6.53) a( u ) u = U W (Ω, R × ) . |∇ | ∇ ∈ On the other hand, on exploiting equations (6.51) and (6.39), the dominated convergence theorem for Lebesgue integrals and inequality (6.12) (applied with a and u replaced by aεm and uεm ) one can deduce that

(6.54) lim aε ( uε ) uε 1 RN×n = a( u ) u 1 RN×n C f 2 RN m m m m L (Ω, ) L (Ω, ) L (Ω, ) →∞ k |∇ | ∇ k k |∇ | ∇ k ≤ k k for some constant C = C(n,N,i ,s , Ω ). Combining equations (6.48), (6.50), (6.53) and (6.54) yields a a | | (6.55) a( u ) u 1,2 RN×n C f 2 RN k |∇ | ∇ kW (Ω, ) ≤ k kL (Ω, ) for some constant C = C(n,N,i ,s ,L , d , ). a a Ω Ω KΩ Step 2. Assume now that the temporary condition (6.36) is still in force, but Ω is just as in the statement. Let Ωm be a sequence of open sets approximating Ω in the sense of Lemma 6.3. For each m N, denote by um the{ weak} solution to the Dirichlet problem ∈ div(a( u ) u )= f in Ω (6.56) − |∇ m| ∇ m m (um =0 on ∂Ωm , where f is continued by 0 outside Ω. Owing to our assumptions on Ωm, inequality (6.55) holds for um. Thereby, there exists a constant C(n,N,i ,s ,L , d , ) such that a a Ω Ω KΩ (6.57) a( u ) u 1,2 RN×n a( u ) u 1,2 RN×n C f 2 RN = C f 2 RN . k |∇ m| ∇ mkW (Ω, ) ≤ k |∇ m| ∇ mkW (Ωm, ) ≤ k kL (Ωm, ) k kL (Ω, ) Observe that the dependence of the constant C on the speciﬁed quantities, and, in particular, its independence of m, is due to properties (6.33) and (6.34). 1,2 N n Thanks to (6.57), the sequence a( um ) um is bounded in W (Ω, R × ), and hence there exists a subsequence, still denoted by u and{ |∇ a function| ∇ U} W 1,2(Ω, RN n), such that { m} ∈ × (6.58) a( u ) u U in L2(Ω, RN n), |∇ m| ∇ m → × a( u ) u ⇀ U in W 1,2(Ω, RN n). |∇ m| ∇ m × 1,α RN We now notice that there exists α (0, 1), independent of m, such that um Cloc (Ω, ), and for every open set Ω Ω with a smooth boundary∈ ∈ ′ ⊂⊂ (6.59) u 1,α ′ RN C, k mkC (Ω , ) ≤ A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 31 for some C, independent of m. To verify this assertion, one can make use of [34, Corollary 5.5] and of inequality (4.35) to deduce that, for each open set Ω′ as above, there exists a constant C, independent of m, such that

(6.60) u α ′ RN×n C. k∇ mkC (Ω , ) ≤ Since the function f satisﬁes assumption (6.36), a basic energy estimate for weak solutions tells us that

(6.61) B( u ) dx C |∇ m| ≤ ZΩm for some constant C independent of m. Thus, as a consequence of the Poincar´einequality (4.4),

(6.62) B( u ) dx C , | m| ≤ ZΩm where the constant C is independent of m, for Ωm is uniformly bounded. Inequalities (6.60) and (6.62), via a Sobolev type inequality, tell us that | |

(6.63) u ∞ ′ RN C k mkL (Ω , ) ≤ for some constant C independent of m. Inequality (6.59) follows from (6.60) and (6.63). On passing, if necessary, to another subsequence, we deduce from inequality (6.59) that there exists a function v C1(Ω, RN ) such that ∈ (6.64) u v and u v in Ω. m → ∇ m →∇ Hence, (6.65) a( u ) u a( v ) v in Ω. |∇ m| ∇ m → |∇ | ∇ Owing to equations (6.65) and (6.58), 1,2 N n (6.66) a( v ) v = U W (Ω, R × ) . |∇ | ∇ ∈ RN Next, we pick a test function ϕ C0∞(Ω, ) (continued by 0 outside Ω) in the definition of weak solution to problem (6.56). Passing to the limit∈ as m in the resulting equation yields, via (6.58) and (6.66), →∞ (6.67) a( v ) v ϕ dx = f ϕ dx . |∇ | ∇ ·∇ · ZΩ ZΩ Inequality (6.61) tells us that B( u ) dx C for some constant C independent of m. Therefore, this in- Ω |∇ m| ≤ equality is still true if u is replaced with v. Consequently, thanks to inequality (4.22), B(a( v ) v ) dx < m R Ω |∇ | |∇ | RN 1,B RN . Thus, since under our assumptions on a the space C0∞(Ω, ) is dense in W0 (ΩR , ), equation (6.67) ∞ 1,B RN e holds for every function ϕ W0 (Ω, ) as well. Hence, v is a weak solution to the Dirichlet problem (2.17), and, inasmuch as such a solution∈ is unique, v = u. Moreover, by equations (6.57) and (6.58), there exists a constant C = C(n,N,i ,s ,L , d , ) such that a a Ω Ω KΩ (6.68) a( u ) u 1,2 RN×n C f 2 RN . k |∇ | ∇ kW (Ω, ) ≤ k kL (Ω, ) Step 3. Finally, assume that both Ω and f are as in the statement. The definition of approximable solution entails that there exists a sequence f C (Ω, RN ), such that f f in L2(Ω, RN ) and the sequence of weak { k}⊂ 0∞ k → solutions u W 1,B(Ω, RN ) to problems (6.2), fulfills u u and u u a.e. in Ω. An application of { k}⊂ 0 k → ∇ k →∇ inequality (6.68) with u and f replaced by u and f , tells us that a( u ) u W 1,2(Ω, RN n), and k k |∇ k| ∇ k ∈ × (6.69) a( u ) u 1,2 RN×n C f 2 RN C f 2 RN k |∇ k| ∇ kkW (Ω, ) ≤ 1k kkL (Ω, ) ≤ 2k kL (Ω, ) for some constants C1 and C2, depending on N, ia, sa and Ω. Therefore, the sequence a( uk ) uk is bounded 1,2 N n { |∇ | ∇ 1},2 N n in W (Ω, R × ), whence there exists a subsequence, still indexed by k, and a function U W (Ω, R × ) such that ∈ (6.70) a( u ) u U in L2(Ω, RN n), a( u ) u ⇀ U in W 1,2(Ω, RN n). |∇ k| ∇ k → × |∇ k| ∇ k × 32 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

1,2 N n Inasmuch as uk u a.e. in Ω, one hence deduces that a( u ) u = U W (Ω, R × ). Thereby, the second inequality∇ in→ (2.19 ∇ ) follows from equations (6.69) and (6.70|∇).| The∇ ﬁrst inequality∈ in (2.19) holds trivially. The proof is complete.

Proof of Theorem 2.6. The proof parallels that of Theorem 2.7. However, Step 2 requires a variant. The sequence Ωm of bounded sets, with smooth boundaries, coming into play in this step has to be chosen in such a way that{ they} are convex and approximate Ω from outside with respect to the Hausdorff distance. Inequalities (6.33) automatically hold in this case. Moreover, inequality (6.34) is not needed, inasmuch as the constant C in (6.10) does not depend on the function if Ω is convex. KΩ Compliance with Ethical Standards Funding. This research was partly funded by: (i) German Research Foundation (DFG) through the CRC 1283 in Bielefeld University (A. Kh.Balci and L. Diening); (ii) Research Project of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: theoretical aspects and applications”, grant number 201758MTR2 (A. Cianchi); (iii) GNAMPA of the Italian INdAM - National Institute of High Mathematics (grant number not available) (A. Cianchi); (iv) RUDN University Strategic Academic Leadership Program (V. Maz’ya). Conflict of Interest. The authors declare that they have no conflict of interest.

References [1] E. Acerbi and N. Fusco. “Regularity for minimizers of nonquadratic functionals: the case 1 Berlin, 2002, pp. xii+441. [10] S. Bernstein. “Sur la nature analytique des solutions des équations aux dérivées partielles du second ordre”. In: Math. Ann. 59.1-2 (1904), pp. 20–76. [11] D. Breit, A. Cianchi, L. Diening, T. Kuusi, and S. Schwarzacher. “Pointwise Calderón-Zygmund gradient estimates for the p-Laplace system”. In: J. Math. Pures Appl. (9) 114 (2018), pp. 146–190. [12] D. Breit, A. Cianchi, L. Diening, and S. Schwarzacher. “Global Schauder estimates for the p-Laplace system”. In: arXiv e-prints (Mar. 2019), arXiv:1903.12496. [13] J. E. Brothers and W. P. Ziemer. “Minimal rearrangements of Sobolev functions”. In: J. Reine Angew. Math. 384 (1988), pp. 153–179. A POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 33

[14] M. Bul´ıˇcek, E. Maringová, B. Stroffolini, and A. Verde. “A boundary regularity result for minimizers of variational integrals with nonstandard growth”. In: Nonlinear Anal. 177.part A (2018), pp. 153–168. [15] V. I. Burenkov. Sobolev spaces on domains. Vol. 137. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998, p. 312. [16] S.-S. Byun, P. Shin, and Y. Youn. “Fractional differentiability results for nonlinear measure data problems α with coefficients in Cγ ”. In: J. Differential Equations 270 (2021), pp. 390–434. [17] M. Carozza, J. Kristensen, and A. Passarelli di Napoli. “Regularity of minimizers of autonomous convex variational integrals”. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13.4 (2014), pp. 1065–1089. [18] A. Cellina. “The regularity of solutions to some variational problems, including the p-Laplace equation for 2 p< 3”. In: ESAIM Control Optim. Calc. Var. 23.4 (2017), pp. 1543–1553. [19] A. Cellina.≤ “The regularity of solutions to some variational problems, including the p-Laplace equation for 3 p< 4”. In: Discrete Contin. Dyn. Syst. 38.8 (2018), pp. 4071–4085. [20] Y. Z.≤ Chen and E. DiBenedetto. “Boundary estimates for solutions of nonlinear degenerate parabolic systems”. In: J. Reine Angew. Math. 395 (1989), pp. 102–131. [21] I. Chlebicka. “A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces”. In: Nonlinear Anal. 175 (2018), pp. 1–27. [22] A. Cianchi. “A sharp embedding theorem for Orlicz-Sobolev spaces”. In: Indiana Univ. Math. J. 45.1 (1996), pp. 39–65. [23] A. Cianchi. “Boundedness of solutions to variational problems under general growth conditions”. In: Comm. Partial Differential Equations 22.9-10 (1997), pp. 1629–1646. [24] A. Cianchi. “Local boundedness of minimizers of anisotropic functionals”. In: Ann. Inst. H. Poincaré Anal. Non Linéaire 17.2 (2000), pp. 147–168. [25] A. Cianchi and V. Maz’ya. “Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems”. In: J. Eur. Math. Soc. (JEMS) 16.3 (2014), pp. 571–595. [26] A. Cianchi and V. G. Maz’ya. “Global boundedness of the gradient for a class of nonlinear elliptic systems”. In: Arch. Ration. Mech. Anal. 212.1 (2014), pp. 129–177. [27] A. Cianchi and V. G. Maz’ya. “Optimal second-order regularity for the p-Laplace system”. In: J. Math. Pures Appl. (9) 132 (2019), pp. 41–78. [28] A. Cianchi and V. G. Maz’ya. “Second-order two-sided estimates in nonlinear elliptic problems”. In: Arch. Ration. Mech. Anal. 229.2 (2018), pp. 569–599. [29] F. Crispo, C. R. Grisanti, and P. Maremonti. “On the high regularity of solutions to the p-Laplacian boundary value problem in exterior domains”. In: Ann. Mat. Pura Appl. (4) 195.3 (2016), pp. 821–834. [30] L. Damascelli and B. Sciunzi. “Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations”. In: J. Differential Equations 206.2 (2004), pp. 483–515. [31] T. Di Marco and P. Marcellini. “A-priori gradient bound for elliptic systems under either slow or fast growth conditions”. In: Calc. Var. Partial Differential Equations 59.4 (2020), Paper No. 120, 26. [32] L. Diening, M. Fornasier, R. Tomasi, and M. Wank. “A relaxed Kaˇcanov iteration for the p-Poisson problem”. In: Numer. Math. 145.1 (2020), pp. 1–34. [33] L. Diening and F. Ettwein. “Fractional Estimates for Non-Differentiable Elliptic Systems with general Growth”. In: Forum Mathematicum 20.3 (2008), pp. 523–556. [34] L. Diening, P. Kaplický, and S. Schwarzacher. “BMO estimates for the p-Laplacian”. In: Nonlinear Anal. 75.2 (2012), pp. 637–650. [35] L. Diening, B. Stroffolini, and A. Verde. “Everywhere regularity of functionals with ϕ-growth”. In: Manuscripta Math. 129.4 (2009), pp. 449–481. [36] G. Dolzmann, N. Hungerbühler, and S. Müller. “Non-linear elliptic systems with measure-valued right hand side”. In: Math. Z. 226.4 (1997), pp. 545–574. [37] P. Grisvard. Elliptic problems in nonsmooth domains. Vol. 24. Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA, 1985, pp. xiv+410. [38] P. Gwiazda, I. Skrzypczak, and A. Zatorska-Goldstein. “Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space”. In: J. Differential Equations 264.1 (2018), pp. 341–377. 34 KH.BALCI, CIANCHI, DIENING, AND MAZ’YA

[39] P. Harjulehto, P. Hästö, and O. Toivanen. “Hölder regularity of quasiminimizers under generalized growth conditions”. In: Calc. Var. Partial Differential Equations 56.2 (2017), Paper No. 22, 26. [40] I. Krol′ and V. G. Maz′ ja. “The absence of the continuity and Hölder continuity of the solutions of quasilinear elliptic equations near a nonregular boundary”. In: Trudy Moskov. Mat. Obˇsˇc. 26 (1972), pp. 75–94. [41] T. Kuusi and G. Mingione. “Linear potentials in nonlinear potential theory”. In: Arch. Ration. Mech. Anal. 207.1 (2013), pp. 215–246. [42] T. Kuusi and G. Mingione. “Vectorial nonlinear potential theory”. In: J. Eur. Math. Soc. (JEMS) 20.4 (2018), pp. 929–1004. [43] G. M. Lieberman. “The natural generalization of the natural conditions of Ladyzhenskaya and Ural′tseva for elliptic equations”. In: Comm. Partial Differential Equations 16.2-3 (1991), pp. 311–361. [44] H. Lou. “On singular sets of local solutions to p-Laplace equations”. In: Chin. Ann. Math. Ser. B 29.5 (2008), pp. 521–530. [45] P. Marcellini. “Regularity for elliptic equations with general growth conditions”. In: J. Differential Equa- tions 105.2 (1993), pp. 296–333. [46] M. B. Marcus and V. J. Mizel. “Absolute continuity on tracks and mappings of Sobolev spaces”. In: Arch. Rational Mech. Anal. 45 (1972), pp. 294–320. 2 [47] V. G. Maz′ ja. “Solvability in W˙ 2 of the Dirichlet problem in a region with a smooth irregular boundary”. In: Vestnik Leningrad. Univ. 22.7 (1967), pp. 87–95. [48] V. G. Maz′ ja. “The coercivity of the Dirichlet problem in a domain with irregular boundary”. In: Izv. Vysˇs. Uˇcebn. Zaved. Matematika 4(131) (1973), pp. 64–76. [49] G. Mingione. “Gradient estimates below the duality exponent”. In: Math. Ann. 346.3 (2010), pp. 571–627. [50] M. Miskiewicz. “Fractional differentiability for solutions of the inhomogeneous p-Laplace system”. In: Proc. Amer. Math. Soc. 146.7 (2018), pp. 3009–3017. [51] J. Simon. “Régularitéde solutions de problèmes nonlinéaires”. In: C. R. Acad. Sci. Paris Sér. A-B 282.23 (1976), Aii, A1351–A1354. [52] S. L. Sobolev. “On almost periodicity of solutions to the wave equation”. In: Dokl. AN SSSR (Russian) 48 (1945), pp. 570–573. [53] G. Talenti. “Boundedness of minimizers”. In: Hokkaido Math. J. 19.2 (1990), pp. 259–279. [54] G. Talenti. “Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces”. In: Ann. Mat. Pura Appl. (4) 120 (1979), pp. 160–184. [55] K. Uhlenbeck. “Regularity for a class of non-linear elliptic systems”. In: Acta Math. 138.3-4 (1977), pp. 219–240. [56] N. N. Ural′ ceva. “Degenerate quasilinear elliptic systems”. In: Zap. Nauˇcn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), pp. 184–222.

Fakultat¨ fur¨ Mathematik, University Bielefeld, Universitatsstrasse¨ 25, 33615 Bielefeld, Germany Email address: [email protected]

Dipartimento di Matematica e Informatica “U. Dini”, Universita` di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy Email address: [email protected]

Fakultat¨ fur¨ Mathematik, University Bielefeld, Universitatsstrasse¨ 25, 33615 Bielefeld, Germany Email address: [email protected]

Department of Mathematics, Linkoping¨ University,E-581 83 Linkoping,¨ Sweden and Peoples Friendship Univer- sity of Russia (RUDN University), 6 Miklukho-Maklay St, Moscow, 117198, Russian Federation Email address: [email protected]