A pointwise differential inequality and second-order regularity for nonlinear elliptic systems
Anna Kh. Balci, Andrea Cianchi, Lars Diening, Vladimir Maz'ya
aa r X i v : . [ m a t h . A P ] F e b A POINTWISE DIFFERENTIAL INEQUALITY AND SECOND-ORDER REGULARITYFOR NONLINEAR ELLIPTIC SYSTEMS
ANNA KH.BALCI, ANDREA CIANCHI, LARS DIENING, AND VLADIMIR MAZ’YA
Abstract.
A sharp pointwise differential inequality for vectorial second-order partial differential operators, withUhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions tononlinear elliptic systems in domains in R n are derived. Both local and global estimates are established. Minimalassumptions on the boundary of the domain are required for the latter. In the special case of the p -Laplace system,our conclusions broaden the range of the admissible values of the exponent p previously known. Introduction
A classical identity, which links the Laplacian ∆ u of a vector-valued function u ∈ C (Ω , R N ) to its Hessian ∇ u , tells us that(1.1) | ∆ u | = div (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17) + |∇ u | in Ω,where Ω is an open set in R n . Here, and in what follows, n ≥ N ≥
1, and the gradient ∇ u of a function u : Ω → R N is regarded as the matrix in R N × n whose rows are the gradients in R × n of the components u , . . . , u N of u . Moreover, the suffix “ T ” stands for transpose.Identity (1.1) can be found as early as more than one century ago in [10] for n = 2 – see also [52, 37]. It hasapplications, for instance, in the second-order L -regularity theory for solutions to the Poisson system for theLaplace operator(1.2) − ∆ u = f in Ω.Indeed, identity (1.1) enables one to bound the integral of |∇ u | over some set in Ω by the integral of | ∆ u | over the same set, plus a boundary integral involving the expression under the divergence operator. Of course,since the equations in the linear system (1.2) are uncoupled, its theory is reduced to that of its single equations.The second-order regularity theory of nonlinear equations and systems is much less developed, yet for thebasic p -Laplace equation or system(1.3) − div ( |∇ u | p − ∇ u ) = f in Ω,where p > div ” denotes the R N -valued divergence operator. Standard results concern weak differentia-bility properties of the expression |∇ u | p − ∇ u . They trace back to [55] for p >
2, and to [1, 20] for every p > p -Laplacian type equa-tions and systems is often most neatly described in terms of the expression |∇ u | p − ∇ u appearing under thedivergence operator in (1.3). This surfaces, for instance, from BMO and H¨older bounds of [34], potential esti-mates of [41], rearrangement inequalities of [25], pointwise oscillation estimates of [11], regularity results up tothe boundary of [12]. Further results in this connection can be found e.g. [3, 26, 42].Differentiability properties of |∇ u | p − ∇ u have customarily been detected under strong regularity assump-tions on the right-hand side f . This is the case of [44], where local solutions are considered. High regularityof the right-hand side is also assumed [30], where results for boundary value problems can be found under Mathematics Subject Classifications:
Keywords:
Quasilinear elliptic systems, second-order derivatives, p -Laplacian, Dirichlet problems, local solutions, capacity, convexdomains, Lorentz spaces. 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 hasbeen studied in [51], and in the more recent contributions [3, 16, 19, 49, 50]. The question of fractional-orderregularity of the quantity |∇ u | p − ∇ u , when N = 1 and the right-hand side of equation (1.3) is in divergenceform, is addressed in [5], where, in particular, sharp results are obtained for n = 2.Optimal second-order L -estimates for solutions to a class of problems, including (1.3) for every p >
1, in thescalar case, have recently been established in [28]. Loosely speaking, these estimates tell us that |∇ u | p − ∇ u ∈ W , if and only if f ∈ L . Such a property is shown to hold both locally, and, under minimal regularityassumptions on the boundary, also globally. Parallel results are derived in [27] for vectorial problems, namelyfor N ≥
2, but for the restricted range of powers p > . 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. Thelatter amounts to a bound from below for the square of the right-hand side of (1.3) by the square of thederivatives of |∇ u | p − ∇ 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 holdsunder 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,namely when a ( t ) = t p − . In particular, for this special choice, it entails the existence of a constant c > (cid:12)(cid:12) div ( |∇ u | p − ∇ u ) (cid:12)(cid:12) ≥ div h |∇ u | p − (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i + c |∇ u | p − |∇ u | (1.4)in {∇ u = 0 } if and only either N = 1 and p >
1, or N ≥ p > − √ ≈ . W , -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), arenowadays the subject of a rich literature. A non exhaustive sample of contributions along this direction ofresearch 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 ) = Z Ω B ( |∇ u | ) − f · u dx. Here, the dot “ · ” stands for scalar product, and B : [0 , ∞ ) → [0 , ∞ ) is the function defined as(1.7) B ( t ) = Z t b ( s ) ds for t ≥ b : [0 , ∞ ) → [0 , ∞ ) is given by(1.8) b ( t ) = a ( t ) t for t > b (0) = 0.Under the assumptions to be imposed on a , the function B and the functional J turn out to be strictly convex.In particular, if a ( t ) = t p − , then B ( t ) = p t p , and J agrees with the usual energy functional associated withthe 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 > − √ ≈ . . Hence, we extend the range of the admissible exponents p known until now, which was limited to p > . 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 thesecond-order regularity of solutions to the p -Laplace system (1.3). The question of whether such a restrictionis really indispensable for this regularity, or it can be dropped as in the case when N = 1, where every p > Main results
The statement of the general differential inequality requires a few notations. Given a positive function a ∈ C ((0 , ∞ )), we define the indices(2.1) i a = inf t> ta ′ ( t ) a ( t ) and s a = sup t> ta ′ ( t ) a ( t ) , where a ′ stands for the derivative of a . Plainly, if a ( t ) = t p − , then i a = s a = p − N ≥ κ N : [1 , ∞ ) → R as(2.2) κ ( p ) = ( ( p − if p ∈ [1 , p ∈ [2 , ∞ ) , if N = 1, and(2.3) κ N ( p ) = − (4 − p ) if p ∈ [1 , )( p − if p ∈ [ , p ∈ [2 , ∞ ) , if N ≥ Theorem 2.1. [General pointwise inequality]
Let n ≥ and N ≥ . Let Ω be an open set in R n and let u ∈ C (Ω , R N ) . Assume that the function a ∈ C ([0 , ∞ )) is such that: (2.4) a ( t ) > for t > , (2.5) i a ≥ − , and (2.6) b ∈ C ([0 , ∞ )) , where b is the function defined by (1.8) . Then (cid:12)(cid:12) div (cid:0) a ( |∇ u | ) ∇ u (cid:1)(cid:12)(cid:12) ≥ div h a ( |∇ u | ) (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i + κ N ( i a + 2) a ( |∇ u | ) |∇ u | (2.7) in Ω , where κ N is defined as in (2.2) - (2.3) . Moreover, the constant κ N ( i a + 2) is sharp.If a is just defined in (0 , ∞ ) , a ∈ C ((0 , ∞ )) , and conditions (2.4) and (2.5) are fulfilled, then inequality (2.7) continues to hold in the set {∇ u = 0 } . Remark 2.2.
Observe that the assumption (2.6) need not be fulfilled by the functions a appearing in the ellipticsystems (1.5) to be considered. Such an assumpton fails, for instance, when a ( t ) = t p − with 1 < p <
2. Thiscalls for a regularization argument for a in our applications of inequality (2.7) to the solutions to the systemsin question. The solutions to the regularized systems will also enjoy the smoothness properties required on thefunction u in Theorem 2.1. On the other hand, the functions a in the original systems satisfy the conditionsrequired in the last part of the statement of Theorem 2.1 for the validity of inequality (2.7) outside the set {∇ u = 0 } of critical points of the function u .Specializing Theorem 2.1 to the case in which a ( t ) = t p − yields the following inequality for the p -Laplaceoperator we alluded to in Section 1. KH.BALCI, CIANCHI, DIENING, AND MAZ’YA
Corollary 2.3. [Pointwise inequality for the p -Laplacian] Let n ≥ and N ≥ . Let Ω be an open set in R n and let u ∈ C (Ω , R N ) . Assume that p ≥ . Then (cid:12)(cid:12) div ( |∇ u | p − ∇ u ) (cid:12)(cid:12) ≥ div h |∇ u | p − (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i + κ N ( p ) |∇ u | p − |∇ u | (2.8) in {∇ u = 0 } . Moreover, the constant κ N ( p ) is sharp. Notice that, if N = 1, then(2.9) κ ( p ) > p > , whereas, if N ≥ κ N ( p ) > p > − √ . The gap between (2.9) and (2.10) is responsible for the different implications of inequality (2.7) in view ofsecond-order L -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 κ N ( i a + 2) > L -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 standardweak solutions to the p -Laplace system (1.3) is only guaranteed if p ≥ nn +2 . In the scalar case, various definitionsof solutions – entropy solutions, renormalized solutions, SOLA – that allow for right-hand sides that are justintegrable functions, or even finite measures, are available in the literature, and turn out to be a posterioriequivalent. Note that these solutions need not be even weakly differentiable. The case of systems is moredelicate and has been less investigated. A notion of solution, which is well tailored for our purposes and will beadopted, is patterned on the approach of [36]. Loosely speaking, the solutions in question are only approximatelydifferentiable, and are pointwise limits of solutions to approximating problems with smooth right-hand sides.The outline of the derivation of the second-order L -bounds for these solutions to system (2.11) via Theorem2.1 is analogous to the one of [28]. However, new technical obstacles have to be faced, due to the non-polynomialgrowth of the coefficient a in the differential operator. In particular, an L -estimate, of independent interest,for the expression a ( |∇ u | ) ∇ u for merely integrable data f is established. Such an estimate is already availablein the literature for equations, but seems to be new for systems, and its proof requires an ad hoc Sobolev typeinequality in Orlicz spaces.Our local estimate for system (2.11) reads as follows. In the statement, B R and B R denote concentric balls,with radius R and 2 R , respectively. Theorem 2.4. [Local estimates]
Let Ω be an open set in R n , with n ≥ , and let N ≥ . Assume that thefunction a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable, and satisfies (2.12) i a > − √ , and (2.13) s a < ∞ . Let f ∈ L (Ω , R N ) and let u be an approximable local solution to system (2.11) . Then (2.14) a ( |∇ u | ) ∇ u ∈ W , (Ω , R N × n ) , and there exists a constant C = C ( n, N, i a , s a ) such that R − (cid:13)(cid:13) a ( |∇ u | ) ∇ u (cid:13)(cid:13) L ( B R , R N × n ) + (cid:13)(cid:13) ∇ (cid:0) a ( |∇ u | ) ∇ u (cid:1)(cid:13)(cid:13) L ( B R , R N × n ) (2.15) ≤ C (cid:16) k f k L ( B R , R N ) + R − n − k a ( |∇ u | ) ∇ u k L ( B R , R N × n ) (cid:17) . for any ball B R ⊂⊂ Ω . POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 5
Remark 2.5.
In particular, if Ω = R n and, for instance, a ( |∇ u | ) ∇ u ∈ L ( R n , R N × n ), then passing to the limitin inequality (2.15) as R → ∞ tells us that(2.16) (cid:13)(cid:13) ∇ (cid:0) a ( |∇ u | ) ∇ u (cid:1)(cid:13)(cid:13) L ( R n , R N × n ) ≤ C k f k L ( R n , R N ) . We next deal with global estimates for solutions to system (2.11), subject to Dirichlet homogeneous boundaryconditions. Namely, we consider solutions to problems of the form(2.17) ( − div ( a ( |∇ u | ) ∇ u ) = f in Ω u = 0 on ∂ Ω . As shown by classical counterexamples, yet in the linear case, global estimates involving second-order deriva-tives 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 positivesemidefinite second fundamental form of ∂ Ω, is well known to ensure bounds in W , (Ω , R N × n ) for the solution u to the homogeneous Dirichlet problem associated with the linear system (1.2) in terms of the L (Ω , R N )norm of f – see [37]. The following result provides us with an analogue for problem (2.17), for the same classof nonlinearities a as in Theorem 2.4. Theorem 2.6. [Global estimates in convex domains]
Let Ω be any bounded convex open set in R n , with n ≥ , and let N ≥ . Assume that the function a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable and fulfillsconditions (2.12) and (2.13) . Let f ∈ L (Ω , R N ) and let u be an approximable solution to the Dirichlet problem (2.17) . Then (2.18) a ( |∇ u | ) ∇ u ∈ W , (Ω , R N × n ) , and (2.19) C k f k L (Ω , R N ) ≤ k a ( |∇ u | ) ∇ u k W , (Ω , R N × n ) ≤ C k f k L (Ω , R N ) for some positive constants C = C ( n, N, i a , s a ) and C = C ( N, i a , s a , Ω) . The global assumption on the signature of the second fundamental form of ∂ Ω entailed by the convexity ofΩ 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 ofits 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 , . This means that the domain Ω is locallythe subgraph of a Lipschitz continuous function of ( n −
1) variables, which is also twice weakly differentiable.Denote by B the weak second fundamental form on ∂ Ω, by |B| its norm, and set(2.20) K Ω ( r ) = sup E ⊂ ∂ Ω ∩ B r ( x ) x ∈ ∂ Ω R E |B| d H n − cap B ( x ) ( E ) for r ∈ (0 , . Here, B r ( x ) stands for the ball centered at x , with radius r , the notation cap B ( x ) ( E ) is adopted for the capacityof the set E relative to the ball B ( x ), and H n − is the ( n − r → + of the function K Ω ( r ). The smallnessdepends on Ω through its diameter d Ω and its Lipschitz characteristic L Ω . The latter quantity is defined asthe maximum among the Lipschitz constants of the functions that locally describe the intersection of ∂ Ω withballs centered on ∂ Ω, and the reciprocals of their radii. Here, and in similar occurrences in what follows, thedependence of a constant on d Ω and L Ω 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 |B| in a suitable functionspace X ( ∂ Ω) over ∂ Ω defined in terms of weak type norms, and a smallness condition on the decay of these
KH.BALCI, CIANCHI, DIENING, AND MAZ’YA norms of |B| over balls centered on ∂ Ω. This membership will be denoted by ∂ Ω ∈ W X . The relevant weakspace is defined as(2.21) X ( ∂ Ω) = ( L n − , ∞ ( ∂ Ω) if n ≥ L , ∞ log L ( ∂ Ω) if n = 2.Here, L n − , ∞ ( ∂ Ω) denotes the weak- L n − ( ∂ Ω) space, and L , ∞ log L ( ∂ Ω) denotes the weak- L log L ( ∂ Ω) space(also called Marcinkiewicz spaces), with respect to the ( n − Theorem 2.7. [Global estimates under minimal boundary regularity]
Let Ω be a bounded Lipschitzdomain in R n , n ≥ , such that ∂ Ω ∈ W , , and let N ≥ . Assume that the function a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable and fulfills conditions (2.12) and (2.13) . Let f ∈ L (Ω , R N ) and let u be anapproximable solution to the Dirichlet problem (2.17) .(i) There exists a constant c = c ( n, N, i a , s a , L Ω , d Ω ) such that, if (2.22) lim r → + K Ω ( r ) < c, then a ( |∇ u | ) ∇ u ∈ W , (Ω , R N × n ) , and inequality (2.19) holds.(ii) Assume, in addition, that ∂ Ω ∈ W X , where X ( ∂ Ω) is the space defined by (2.21) . There exists a constant c = c ( n, N, i a , s a , L Ω , d Ω ) such that, if (2.23) lim r → + (cid:16) sup x ∈ ∂ Ω kBk X ( ∂ Ω ∩ B r ( x )) (cid:17) < c , then a ( |∇ u | ) ∇ u ∈ W , (Ω , R N × 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 K Ω ( r ) < ∞ for r ∈ (0 ,
1) and the solution to thehomogeneous Dirichlet problem for (1.2), with a smooth right-hand side f , belongs to W , (Ω) only for thosevalues 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-strate its optimality yet for the scalar p -Laplace equation. Actually, open sets Ω ⊂ R , with ∂ Ω ∈ W L , ∞ ,are displayed where the solution u to the homogeneous Dirichlet problem for (1.3), with N = 1, p ∈ ( , f , is such that |∇ u | p − ∇ u / ∈ W , (Ω). This lack of regularity is due to thefact that the limit in (2.23), though finite, is not small enough. Similarly, if n = 2 there exist open sets Ω,with ∂ Ω ∈ W L , ∞ log L , for which the limit in (2.23) exceeds some threshold, and where the solution to thehomogeneous Dirichlet problem for (1.2), with a smooth right-hand side, does not belong to W , (Ω) – see [47]. Remark 2.9.
The one-parameter family of domains Ω mentioned in the first part of Remark 2.8 with regardto condition (2.22) is such that ∂ Ω / ∈ W L n − , ∞ if n ≥
3. Hence, assumption (2.23) is not fulfilled even forthose values of the parameter which render (2.22) true. This shows that the latter assumption is indeed weakerthan (2.23) .
Remark 2.10.
Condition (2.23) certainly holds if n ≥ ∂ Ω ∈ W ,n − , and if n = 2 and ∂ Ω ∈ W L log L (and hence, if ∂ Ω ∈ W ,q for some q > ∂ Ω ∈ C .3. The pointwise inequality
This section is devoted to the proof of Theorem 2.1, which is split in several lemmas. The point of departureis a pointwise identity, of possible independent use, stated in Lemma 3.1.Given a positive function a ∈ C (0 , ∞ ), we define the function Q a : [0 , ∞ ) → R (3.1) Q a ( t ) = ta ′ ( t ) a ( t ) for t > POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 7
Hence,(3.2) i a = inf t> Q a ( t ) and s a = sup t> Q a ( t ) , where i a and s a 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 ∈ C ([0 , ∞ )) and satisfiesconditions (2.4) – (2.6) . Then (cid:12)(cid:12) div ( a ( |∇ u | ) ∇ u ) (cid:12)(cid:12) = div h a ( |∇ u | ) (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i (3.3) + a ( |∇ u | ) " |∇ u | + 2 Q a ( |∇ u | ) |∇|∇ u || + Q a ( |∇ u | ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) in Ω ,where the last two addends in square brackets on the right-hand side of equation (3.3) have to interpreted as if ∇ u = 0 .If a is just defined in (0 , ∞ ) , a ∈ C ((0 , ∞ )) , and conditions (2.4) and (2.5) are fulfilled, then inequality (2.7) continues to hold in the set {∇ u = 0 } . The next corollary follows from Lemma 3.1. applied with a ( t ) = t p − . Corollary 3.2.
Let n , N , Ω and u be as in Theorem 2.1. Assume that p ≥ . Then (cid:12)(cid:12) div ( |∇ u | p − ∇ u ) (cid:12)(cid:12) = div h |∇ u | p − (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i (3.4) + |∇ u | p − " |∇ u | + 2( p − |∇|∇ u || + ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) in {∇ u = 0 } .Proof of Lemma 3.1. The following chain can be deduced via straightforward computations: (cid:12)(cid:12) div (cid:0) a ( |∇ u | ) ∇ u (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) a ( |∇ u | )∆ u + a ′ ( |∇ u | ) ∇ u ( ∇|∇ u | ) T (cid:12)(cid:12) (3.5) = a ( |∇ u | ) (cid:0) | ∆ u | − |∇ u | (cid:1) + a ( |∇ u | ) |∇ u | ++ a ′ ( |∇ u | ) |∇ u ( ∇|∇ u | ) T (cid:12)(cid:12) + 2 a ( |∇ u | ) a ′ ( |∇ u | )∆ u · ∇ u ( ∇|∇ u | ) T = a ( |∇ u | ) (cid:16) div((∆ u ) T ∇ u ) − div( ∇|∇ u | ) (cid:17) + a ( |∇ u | ) |∇ u | ++ a ′ ( |∇ u | ) (cid:12)(cid:12) ∇ u ( ∇|∇ u | ) T (cid:12)(cid:12) + 2 a ( |∇ u | ) a ′ ( |∇ u | )∆ u · ∇ u ( ∇|∇ u | ) T . 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 t → + a ′ ( t ) t = 0 . Moreover, a ( |∇ u | ) div((∆ u ) T ∇ u ) = div (cid:0) a ( |∇ u | ) (∆ u ) T ∇ u (cid:1) − a ( |∇ u | ) a ′ ( |∇ u | )∆ u · ∇ u ( ∇|∇ u | ) T , (3.6)and a ( |∇ u | ) div (cid:0) ∇|∇ u | (cid:1) = div (cid:0) a ( |∇ u | ) ∇|∇ u | (cid:1) − a ( |∇ u | ) a ′ ( |∇ u | ) |∇ u ||∇|∇ u || . (3.7)From equations (3.5)–(3.7) one deduces that (cid:12)(cid:12) div ( a ( |∇ u | ) ∇ u ) (cid:12)(cid:12) = div (cid:0) a ( |∇ u | ) (∆ u ) T ∇ u (cid:1) − div (cid:0) a ( |∇ u | ) ∇|∇ u | (cid:1) (3.8) + a ( |∇ u | ) |∇ u | + a ′ ( |∇ u | ) (cid:12)(cid:12) ∇ u ( ∇|∇ u | ) T (cid:12)(cid:12) + 2 a ( |∇ u | ) a ′ ( |∇ u | ) |∇ u ||∇|∇ u || . 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 (cid:12)(cid:12) div ( a ( |∇ u | ) ∇ u ) (cid:12)(cid:12) = div (cid:0) a ( |∇ u | ) (∆ u ) T ∇ u (cid:1) − div (cid:0) a ( |∇ u | ) ∇|∇ u | (cid:1) + a ( |∇ u | ) " |∇ u | + (cid:18) a ′ ( |∇ u | ) |∇ u | a ( |∇ u | ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 a ′ ( |∇ u | ) |∇ u | a ( |∇ u | ) |∇|∇ u || . The proof of equation (3.3) is complete. (cid:3)
Having identity (3.3) at our disposal, the point is now to derive a sharp lower bound for the second addendon its right-hand side. This will be accomplished via Lemma 3.6 below. Its proof requires a delicate analysis ofthe quadratic form depending on the entries of the Hessian matrix ∇ u which appears in square brackets inthe expression to be bounded. This analysis relies upon some critical linear-algebraic steps that are presentedin the next three lemmas.In what follows, R n × n sym denotes the space of symmetric matrices in R n × n . The dot “ · ” is employed to denotescalar product of vectors or matrices, and the symbol “ ⊗ ” for tensor product of vectors. Also, I stands for theidentity matrix in R n × n . Lemma 3.3.
Let ω ∈ R n be such that | ω | = 1 . Then, | Hω | − | ω · Hω | − | H | = − | H ω ⊥ | (3.9) for every H ∈ R n × n sym , where H ω ⊥ = ( I − ω ⊗ ω ) H ( I − ω ⊗ ω ) .Proof. Let { e , . . . , e n } denote the canonical basis in R n and let { θ , . . . , θ n } be an orthonormal basis of R n suchthat θ = ω . Let Q ∈ R n × n be the matrix whose columns are θ , . . . , θ n . Hence, ω = Qe . Next, let R = Q T HQ .Clearly, R ∈ R n × n sym . Denote by r ij the entries of R . Computations show that | Hω | − | ω · Hω | − | H | = | Re | − | e · Re | − | R | = n X i =1 | r i | − | r | − n X i,j =1 | r ij | = n X j =1 | r j | + n X i =1 | r i | − | r | − n X i,j =1 | r ij | = − X i,j ≥ | r ij | = − | ( I − e ⊗ e ) R ( I − e ⊗ e ) | = − | ( I − ω ⊗ ω ) H ( I − ω ⊗ ω ) | . Hence, equation (3.9) follows. (cid:3)
Given a vector ω ∈ R n , define the set E ( ω ) = (cid:8) Hω : H ∈ R n × n sym , | H | ≤ (cid:9) . It is easily verified that E ( ω ) is a convex set in R n for every ω ∈ R n . Lemma 3.4 below tells us that, in fact, E ( ω ) is an ellipsoid, centered at 0 (which reduces to { } if ω = 0). This assertion will be verified by showingthat, for each ω ∈ R n , there exists a positive definite matrix W ∈ R n × n sym such that E ( ω ) agrees with the ellipsoid F ( W ) = (cid:8) x ∈ R n : x · W − x ≤ (cid:9) , (3.10)where W − stands for the inverse of W . This is the content of Lemma 3.4 below. In its proof, we shall makeuse of the alternative representation F ( W ) = (cid:8) x ∈ R n : y · x ≤ p y · W y for every y ∈ R n (cid:9) , (3.11)which follows, for instance, via a maximization argument for the ratio of the two sides of the inequality in(3.11) for each given x ∈ R n .Also, observe that, as a consequence of equation (3.11), | x | = x · b x ≤ √ b x · W b x for every x ∈ F ( W ) \ { } .(3.12) POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 9
Here, and in what follows, we adopt the notation b x = x | x | for x ∈ R n \ { } . Lemma 3.4.
Given ω ∈ R n , let W ( ω ) ∈ R n × n sym be defined as W ( ω ) = (cid:0) | ω | I + ω ⊗ ω (cid:1) . (3.13) Then W ( ω ) is positive definite, and E ( ω ) = F ( W ( ω )) . (3.14) In particular, Hω ∈ | H | F (cid:0) W ( ω ) (cid:1) for every ω ∈ R n and H ∈ R n × n sym . (3.15) Proof.
Equation (3.14) trivially holds if ω = 0. Thus, by a scaling argument, it suffices to consider the casewhen | ω | = 1. We begin showing that E ( ω ) ⊂ F ( W ( ω )). One can verify that, since | ω | = 1, W ( ω ) − = 2 I − ω ⊗ ω. (3.16)Let H ∈ R n × n sym be such that | H | ≤
1. Owing to equation (3.16) and to Lemma 3.3, Hω · W ( ω ) − Hω = 2 | Hω | − (cid:12)(cid:12) ω · Hω (cid:12)(cid:12) ≤ | H | ≤ . (3.17)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 ∈ R n × n sym such that | H | ≤ 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 ) − x ≤
1. Furthermore, x · W ( ω ) − x = ( tω + sω ⊥ ) · (2 I − ω ⊗ ω )( tω + sω ⊥ ) = 2( t + s ) − t = t + 2 s . Hence, t + 2 s ≤
1. We claim that the matrix H defined as H = t ω ⊗ ω + s ( ω ⊥ ⊗ ω + ω ⊗ ω ⊥ ), has the desiredproperties. Indeed, H ∈ R n × n sym , | H | = tr( H T H ) = t + 2 s ≤ Hω = tω + sω ⊥ = x. This proves that x ∈ E ( ω ). The inclusion F ( W ( ω )) ⊂ E ( ω ) hence follows. (cid:3) In view of the statement of the next lemma, we introduce the following notation. Given N vectors ω α ∈ R n and N matrices H α ∈ R n × n sym , with α = 1 , . . . N , we set J = (cid:12)(cid:12)(cid:12)(cid:12) N X α =1 H α ω α (cid:12)(cid:12)(cid:12)(cid:12) , J = N X α =1 (cid:12)(cid:12)(cid:12)(cid:12) ω α · N X β =1 H β ω β (cid:12)(cid:12)(cid:12)(cid:12) , J = N X α =1 | H α | . (3.18) Lemma 3.5.
Let N ≥ , ≤ δ ≤ and δ + σ ≥ . Assume that the vectors ω α ∈ R n and the matrices H α ∈ R n × n sym , with α = 1 , . . . N , satisfy the following constraints: N X α =1 | ω α | ≤ , (3.19) N X α =1 | H α | ≤ . (3.20) Then, J − δJ − σJ ≤ ( if δ ∈ [0 , ] , max n , ( δ +1) δ − σ o if δ ∈ ( , ] . (3.21) Proof.
Given δ and σ as in the statement, set D δ,σ = J − δJ − σJ . The quantities J , J and J are 1-homogeneous with respect to the quantity P Nj =1 | H j | . Moreover, inequality(3.21) trivially holds if the latter quantity vanishes. Thereby, it suffices to prove this inequality under theassumption that P Nj =1 | H j | = 1, namely that J = 1 . (3.22)On setting ζ = P Nα =1 H α ω α , one has that J = | ζ | and J = N X α =1 | ω α · ζ | . Therefore, J ≤ | ζ | N X α =1 | ω α | ≤ | ζ | = J. (3.23)Owing to Lemma 3.4, H α ω α ∈ | H α | F ( W α )for α = 1 , . . . , N , where W α = | ω α | (Id + c ω α ⊗ c ω α ). Thus, by equations (3.15) and (3.12), H α ω α · b ζ ≤ | H α | qb ζ · W α b ζ = | H α || ω α | q + | c ω α · b ζ | (3.24)for α = 1 , . . . , N . Since ζ = ( ζ · b ζ ) b ζ = N X α =1 ( H α ω α · b ζ ) b ζ, equation (3.24) implies that | ζ | ≤ N X α =1 (cid:12)(cid:12) H α ω α · b ζ (cid:12)(cid:12) ≤ N X α =1 | H α || ω α | q + | c ω α · b ζ | . Hence, | ζ | ≤ (cid:18) N X α =1 | H α || ω α | q + | c ω α · b ζ | (cid:19) . (3.25)On setting b J = P Nα =1 | ω α · b ζ | , we obtain that b J = N X α =1 | ω α | | c ω α · b ζ | and J = | ζ | b J . Note that b J ≤
1, inasmuch as J ≤ J = | ζ | . Moreover, by equation (3.22), D δ,σ = J − δJ − σ = | ζ | (cid:0) − δ b J (cid:1) − σ. (3.26)From inequalities (3.25) and (3.26) we deduce that D δ,σ ≤ (cid:18) N X α =1 | H α || ω α | q + | c ω α · b ζ | (cid:19) (cid:16) − δ N X α =1 | ω α | | c ω α · b ζ | (cid:17) − σ. (3.27) POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 11
Next, define the function with g : [0 , N × [0 , N × [0 , N → R as g ( h, s, t ) = (cid:16) N X α =1 h α t α p s α (cid:17) (cid:18) − δ (cid:16) N X α =1 t α s α (cid:17)(cid:19) − σ for ( h, s, t ) ∈ [0 , N × [0 , N × [0 , N ,(3.28)where h = ( h , . . . , h N ), s = ( s , . . . , s N ) and t = ( t , . . . , t N ). Inequality (3.27) then takes the form D δ,σ ≤ g (( | H | , . . . , | H N | ) , ( | ω | , . . . , | ω N | ) , ( || c ω · b ζ | , . . . , | c ω N · b ζ | )) . Our purpose is now to maximize the function g under the constraints N X α =1 t α ≤ , N X α =1 h α = 1 . (3.29)We claim that the maximum of g can only be attained if P Nα =1 t α = 1. To verify this claim, it suffices to showthat g ( h, s, τ t ) ≤ g ( h, s, t ) for every ( h, s, t ) ∈ [0 , N × [0 , N × [0 , N and τ ∈ [0 , . (3.30)Plainly, g ( h, s, τ t ) = τ (cid:16) N X α =1 h α t α p s α (cid:17) (cid:18) − τ δ (cid:16) N X α =1 t α s α (cid:17)(cid:19) − σ for ( h, s, t ) ∈ [0 , N × [0 , N × [0 , N and τ ∈ [0 , ≤ δ (cid:16) N X α =1 t α s α (cid:17) ≤ δ (cid:16) N X α =1 t α (cid:17) = δ ≤ . (3.31)Thus, for each fixed ( h, s, t ) ∈ [0 , n × [0 , n × [0 , n , we have that g ( h, s, τ t ) = c τ (1 − c τ ) − β for τ ∈ [0 , c ≥ ≤ c ≤ , depending on ( h, s, t ). Since the polynomial on the right-handside of equation (3.32) is increasing for τ ∈ [0 , N X α =1 t α = 1 and N X α =1 h α = 1 . (3.33)Let us maximize the function g ( h, s, t ) with respect to h , under the constraint P Nα =1 h α = 1. Let ( h , . . . , h N )be any point where the maximum is attained. Then, there exists a Langrange multiplier λ ∈ R such that t α p s α (cid:18) N X γ =1 h γ t γ q s γ (cid:19)(cid:18) − δ (cid:16) N X γ =1 t γ s γ (cid:17)(cid:19) = 2 λh α for α = 1 , . . . , N .(3.34)Multiplying through equation (3.34) by h β , and then subtracting equation (3.34), with α replaced by β , mul-tiplied by h α yield (cid:18) N X γ =1 h γ t γ q s γ (cid:19)(cid:18) − δ (cid:16) n X γ =1 t γ s γ (cid:17)(cid:19)(cid:16) h β t α p s α − h α t β q s β (cid:17) = 0(3.35) for α, β = 1 , . . . , N . Owing to equation (3.31), we have that (cid:0) − δ (cid:0) P nγ =1 t γ s γ (cid:1)(cid:1) ≥ . Next, if P Nγ =1 h γ t γ q s γ =0, then h t = · · · = h N t N = 0, whence D δ,σ = − σ ≤
0, and inequality (3.21) holds trivially. Therefore, we mayassume that P Nγ =1 h γ t γ q s γ > h β t α p s α = h α t β q s β (3.36)for α, β = 1 , . . . , N . Combining equations (3.33) and (3.36) yields t α (1 + s α ) = t α (1 + s α ) N X β =1 h β = h α N X β =1 t β (1 + s β ) = h α (cid:18) N X β =1 t β s β (cid:19) (3.37)for α = 1 , . . . , N . Hence, h α t α p s α = h α vuut N X β =1 t β s β (3.38)for α = 1 , . . . , N . From equations (3.28), (3.38) and (3.33) we deduce that g ( h, s, t ) ≤ N X α =1 h α vuut N X β =1 t β s β (cid:18) − δ (cid:16) N X α =1 t α s α (cid:17)(cid:19) − σ = (cid:18) N X β =1 t β s β (cid:19)(cid:18) − δ (cid:16) N X α =1 t α s α (cid:17)(cid:19) − σ = ψ (cid:18) N X α =1 t α s α (cid:19) , where ψ : [0 , → R is the function defined as ψ ( r ) = (1 + r ) (cid:0) − δr (cid:1) − σ for r ∈ R .Set ρ = P Nj = α t α s α , and notice that ρ ∈ [0 , ≤ P Nα =1 t α s α ≤ P Nα =1 t α = 1. Thereby, the maximum ofthe function g on [0 , N × [0 , N × [0 , N under constraints (3.33) agrees with the maximum of the function ψ on [0 , δ ∈ [0 , ], then max r ∈ [0 , ψ ( r ) = ψ (1). Hence, since we are assuming that δ + σ ≥ D δ,σ ≤ ψ (1) = 1 − δ − σ ≤ . On the other hand, if δ ∈ ( , ], then max r ∈ [0 , ψ ( r ) = ψ ( − δ δ ). Therefore, D δ,σ ≤ ψ (cid:16) − δ δ (cid:17) = ( δ + 1) δ − σ. The proof of inequality (3.21) is complete. (cid:3)
Lemma 3.6.
Let n , N , Ω and u be as in Theorem 2.1. Given p ≥ , let κ N ( p ) be the constant defined by (2.2) – (2.3) . Then |∇ u | + 2( p − (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) + ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) ≥ κ N ( p ) |∇ u | (3.39) in {∇ u = 0 } . Moreover, the constant κ N ( p ) is sharp in (3.39) .Proof. Case N = 1 . Inequality (3.39) trivially holds if p ≥
2. Let us focus on the case when 1 ≤ p <
2. Noticethat, on setting ω = ( ∇ u ) T |∇ u | ∈ R n and H = ∇ u ∈ R n × n sym POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 13 at any point in {∇ u = 0 } , we have that | Hω | = (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) , | ω · Hω | = (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) , | H | = |∇ u | . Therefore, by equation (3.9), (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) + |∇ u | . Consequently, the following chain holds: |∇ u | + 2( p − (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) + ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:0) p − (cid:1) |∇ u | + (cid:0) ( p −
2) + ( p − (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) ≥ ( p − |∇ u | + ( p − p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:0) ( p −
1) + ( p − p − (cid:1) |∇ u | = ( p − |∇ u | . Hence, inequality (3.39) follows.As far as the sharpness of the constant is concerned, if p ≥
2, consider the function u : R n \ { } → R given by u ( x ) = | x | for x ∈ R n \ { } .Since ∇|∇ u | = 0, equality holds in (3.39) for every x ∈ R n \ { } . On the other hand, if p ∈ [1 , u : R n → R defined as u ( x ) = x for x ∈ R n .One has that |∇ u | = (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) = 1 in R n .Hence, equality holds in (3.39) for every x ∈ R n \ { } . Case N ≥ . It suffices to prove that inequality (3.39) holds at every point x ∈ {∇ u = 0 } under the assumptionthat |∇ u ( x ) | equals either 0 or 1. Indeed, if |∇ u ( x ) | 6 = 0 at some point x , then the function given by u = u |∇ u ( x ) | fulfills |∇ u ( x ) | = 1. Hence, inequality (3.39) for u at the point x follows from the same inequalityapplied to u .If p ≥
2, inequality (3.39) holds trivially. Thus, we may focus on the case when p ∈ [1 , ω α = ∇ u α |∇ u | ∈ R n and H α = ∇ u α ∈ R n × n sym for α = 1 , . . . , N , at any point in {∇ u = 0 } . In particular, assumptions (3.19) and (3.20) are satisfied with thischoice. Computations show that(3.40) J = (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) , J = (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) , J = |∇ u | , where J , J and J are defined as in (3.18).Next, let δ = − p . Notice that δ ∈ [0 , ], and that δ ∈ (0 , ] if and only if p ∈ [ , σ = p if p ∈ [ , σ = ( δ +1) δ =
116 (4 − p ) − p if p ∈ [1 , ). Observe that δ + σ = 1 in the former case, and δ + σ > δ and σ of Theorem 3.5 are fulfilled. Furthermore, by our choice of σ , the maximum on right-hand side of inequality (3.21) equals 0 when δ > , namely when p ∈ [1 , ). From inequality(3.21) we infer that J ≤ − p J + σJ . This inequality is equivalent to J + 2( p − J + ( p − J ≥ (1 − σ − p )) J . Since 1 − σ − p ) = K ( p ), inequality (3.39) follows.In order to prove the sharpness of the constant K ( p ), let us distinguish the cases when p ≥ p ∈ [ ,
2) and p ∈ [1 , ).If p ≥
2, consider the function u : R n \ { } → R N given by u ( x ) = ( | x | , , . . . ,
0) for x ∈ R n \ { } .Since ∇|∇ u | = 0, equality holds in (3.39) for every x ∈ R n \ { } .If p ∈ [ , u : R n → R N defined as u ( x ) = ( x , , . . . ,
0) for x ∈ R n .One has that |∇ u | = (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) = 1 in R n .Thus, equality holds in (3.39) for every x ∈ R n \ { } .If p ∈ [1 , ), set r = p − p ) . Let e , e denote the first two vectors of the canonical base of R n . Define t = √ r , ω = t e ,t = √ − r , ω = t e ,h = r r r , H = h e ⊗ e ,h = r − r r , H = h √ (cid:0) e ⊗ e + e ⊗ e (cid:1) , and ω = · · · = ω N = 0, H = · · · = H N = 0. Then N X α =1 | ω α | = | ω | + | ω | = 1 . (3.41)Moreover, J = N X α =1 | H α | = | H | + | H | = 1 , (3.42) J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X α =1 H α ω α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | H ω + H ω | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) h t + 1 √ h t (cid:17) e (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)r r e (cid:12)(cid:12)(cid:12)(cid:12) = 1 + r , (3.43) J = N X α =1 | ω α · ( H ω + H ω ) | = (cid:12)(cid:12) ω · ( H ω + H ω ) (cid:12)(cid:12) = r (1 + r )2 . (3.44)Now, let u : R n → R N be a polynomial of degree two such that ∇ u α (0) T = ω α and ∇ u α = H α for α = 1 , . . . N .Formulas (3.40), combined with (3.42)–(3.44), tell us that |∇ u | + 2( p − (cid:12)(cid:12) ∇|∇ u | (cid:12)(cid:12) + ( p − (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) = 1 − (4 − p ) = κ N ( p ) |∇ u | at 0.Hence, equality holds in (3.39) for x = 0. (cid:3) 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 a ( |∇ u | ) + 2, and the monotonicity of the function κ N one has that(3.45) a ( |∇ u | ) " |∇ u | + 2 Q a ( |∇ u | ) |∇|∇ u || + Q a ( |∇ u | ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u |∇ u | ( ∇|∇ u | ) T (cid:12)(cid:12)(cid:12)(cid:12) ≥ κ N (cid:0) Q a ( |∇ u | ) + 2 (cid:1) a ( |∇ u | ) |∇ u | ≥ κ N (cid:0) i a + 2 (cid:1) a ( |∇ u | ) |∇ u | in {∇ u = 0 } .Inequality (2.7) holds at every point in the set {∇ u = 0 } , owing to equation (3.3) and inequality (3.45). It alsotrivially holds at every point in the set {∇ u = 0 } , since κ N (cid:0) i a + 2 (cid:1) ≤ κ N ( i a + 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 p = i a + 2 at the point x . Namely, |∇ u ( x ) | + 2 i a (cid:12)(cid:12) ∇|∇ u | ( x ) (cid:12)(cid:12) + i a (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) |∇ u ( x ) | ( ∇|∇ u | ( x )) T (cid:12)(cid:12)(cid:12)(cid:12) = κ N ( i a + 2) |∇ u ( x ) | . (3.46)By the the definition of the index i a , given ε > t ∈ (0 , ∞ ) such that i a ≤ Q a ( t ) ≤ i a + ε. (3.47)Define , the function u = t u |∇ u ( x ) | , so that |∇ u (0) | = t . From identity (3.3), equation (3.46) and inequality(3.47) we obtain that (cid:12)(cid:12) div ( a ( |∇ u | ) ∇ u ) (cid:12)(cid:12) − div h a ( |∇ u | ) (cid:16) (∆ u ) T ∇ u − ∇|∇ u | (cid:17)i a ( |∇ u | ) |∇ u | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x (3.48) = |∇ u ( x ) | + 2 Q a ( t ) |∇|∇ u | ( x ) | + Q a ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) |∇ u ( x ) | ( ∇|∇ u | ( x )) T (cid:12)(cid:12)(cid:12)(cid:12) |∇ u ( x ) | = |∇ u ( x ) | + 2 Q a ( t ) |∇|∇ u | ( x ) | + Q a ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) |∇ u ( x ) | ( ∇|∇ u | ( x )) T (cid:12)(cid:12)(cid:12)(cid:12) |∇ u ( x ) | ≤ |∇ u ( x ) | + 2( i a + ε ) |∇|∇ u | ( x ) | + ( i a + 2 ε | i a | + ε )) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) |∇ u ( x ) | ( ∇|∇ u | ( x )) T (cid:12)(cid:12)(cid:12)(cid:12) |∇ u ( x ) | = k N ( i a + 2) + 2 ε |∇|∇ u | ( x ) | + (2 ε | i a | + ε )) (cid:12)(cid:12)(cid:12)(cid:12) ∇ u ( x ) |∇ u ( x ) | ( ∇|∇ u | ( x )) T (cid:12)(cid:12)(cid:12)(cid:12) |∇ u ( x ) | Hence, the optimality of the constant κ N ( i a + 2) in inequality (2.7) follows, owing to the arbitrariness of ε . (cid:3) Function spaces
An appropriate functional framework for the analysis of solutions to systems of the general form (1.5) isprovided 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 definitionof the norm is played by more general Young functions. Subsection 4.1 is devoted to some basic definitionsand properties of Young functions and of Orlicz-Sobolev spaces. A Poincar´e type inequality for functions inthese spaces of use for our purposes is established as well. In Subsection 4.2 we collect specific properties ofthe Young function (and of perturbations of its) for the specific Orlicz-Sobolev ambient space associated withsystem (2.11).
Young functions and Orlicz-Sobolev spaces.
A Young function A : [0 , ∞ ) → [0 , ∞ ] is a convexfunction such that A (0) = 0. The Young conjugate of a Young function A is the Young function e A defined as e A ( t ) = sup { st − A ( s ) : s ≥ } for t ≥ A is said to belong to the class ∆ , or to satisfythe ∆ -condition, if there exists a constant c > A (2 t ) ≤ cA ( t ) for t > i A and s A be the indices associated with a continuously differentiable function A as in (2.1), with a replacedby A . Namely(4.2) i A = inf t> tA ′ ( t ) A ( t ) and s A = sup t> tA ′ ( t ) A ( t ) . One has that A ∈ ∆ if and only if s A < ∞ . The constant c in inequality (4.1) depends on s a . Also, e A ∈ ∆ ifand only if i A > L A (Ω) is the Banach function space of those real-valued measurable functions u : Ω : → R whose Luxemburg norm k u k L A (Ω) = inf (cid:26) λ > Z Ω A (cid:18) | u | λ (cid:19) dx ≤ (cid:27) is finite. The Orlicz space L A (Ω , R N ) of R N -valued functions and the Orlicz space L A (Ω , R N × n ) of R N × n -valuedfunctions are defined analogously.The Orlicz-Sobolev space W ,A (Ω) is the Banach space(4.3) W ,A (Ω) = { u ∈ L A (Ω) : u is weakly differentiable in Ω and ∇ u ∈ L A (Ω , R n ) } , and is equipped with the norm k u k W ,A (Ω) = k u k L A (Ω) + k∇ u k L A (Ω , R n ) . The space W ,A loc (Ω) is defined accordingly. By W ,A (Ω) we denote the subspace of W ,A (Ω) of those functionsin W ,A (Ω) whose extension by 0 outside Ω is weakly differentiable in the whole of R n . The notation ( W ,A (Ω)) ′ stands for the dual of W ,A (Ω). If Ω has finite Lebesgue measure | Ω | , then the functional k∇ u k L A (Ω , R n ) definesa norm in W ,A (Ω) equivalent to k u k W ,A (Ω) .The space C ∞ (Ω) is dense in W ,A (Ω) if A ∈ ∆ . Moreover, W ,A (Ω) is reflexive if both A ∈ ∆ and e A ∈ ∆ ,and hence if i A > s A < ∞ .The Orlicz-Sobolev space W ,A (Ω , R N ) of R N -valued functions, its variants W ,A loc (Ω , R N ) and W ,A (Ω , R N ),and the space ( W ,A (Ω , R N )) ′ are defined analogously.If | Ω | < ∞ and the Young function A ∈ ∆ , then the Poincar´e type inequality(4.4) Z Ω A ( | u | ) dx ≤ c Z Ω A ( |∇ u | ) dx holds for some constant c = c ( n, | Ω | , s a ) and for every function u ∈ W ,A (Ω). Inequality (4.4) follows, forinstance, from [53, Lemma 3].In order to bound lower-order terms appearing in our global estimate, we also need a stronger, yet non-optimal, Sobolev-Poincar´e type inequality for functions in W ,A (Ω) with an Orlicz target space smaller than L A (Ω). This is the subject of Theorem 4.1 below, which generalizes a version of the relevant inequality withoptimal Orlicz target space from [23] (see also [22] for an equivalent form).Assume that the Young function A and the number σ > Z (cid:18) tA ( t ) (cid:19) σ − dt < ∞ POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 17 and(4.6) Z ∞ (cid:18) tA ( t ) (cid:19) σ − dt = ∞ . Then, we define the function H σ : [0 , ∞ ) → [0 , ∞ ) as(4.7) H σ ( s ) = (cid:18) Z s (cid:18) tA ( t ) (cid:19) σ − dt (cid:19) σ ′ for s ≥ A σ as(4.8) A σ ( t ) = A ( H − σ ( t )) for t ≥ Theorem 4.1.
Let Ω be an open set in R n 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 (4.9) Z Ω A σ | u ( x ) | c | Ω | n − σ (cid:0) R Ω A ( |∇ u | ) dy (cid:1) /σ ! dx ≤ Z Ω A ( |∇ u | ) dx for every u ∈ W ,A (Ω) .Proof. By the P´olya-Szeg¨o principle on the decrease of the functional on the right-hand side of inequality (4.9)under symmetric decreasing rearrangement of functions u ∈ W ,A (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 aconsequence, this inequality will follow if we show that(4.10) Z | Ω | A σ R | Ω | s ϕ ( r ) r − n ′ drc | Ω | n − σ (cid:0) R | Ω | A ( ϕ ( r )) dr (cid:1) /σ ! ds ≤ Z | Ω | A ( ϕ ( s )) ds for a suitable constant c as in the statement and for every measurable function ϕ : (0 , | Ω | ) → [0 , ∞ ). Let S bethe linear operator defined as(4.11) Sϕ ( s ) = Z | Ω | s ϕ ( r ) r − n ′ dr for s ∈ (0 , | Ω | ),for every measurable function ϕ : (0 , | Ω | ) → R that makes the integral on the right-hand side converge. Onehas that k Sϕ k L σ ′ (0 , | Ω | ) = (cid:18) Z | Ω | | Sϕ ( s ) | σ ′ ds (cid:19) σ ′ ≤ (cid:18) Z | Ω | s − σ ′ n ′ (cid:18) Z | Ω | s | ϕ ( r ) | dr (cid:19) σ ′ ds (cid:19) σ ′ (4.12) ≤ k ϕ k L (0 , | Ω | ) (cid:18) Z | Ω | s − σ ′ n ′ ds (cid:19) σ ′ = c | Ω | n − σ k ϕ k L (0 , | Ω | ) for a suitable constant c = c ( n, σ ) and for every ϕ ∈ L (0 , | Ω | ). Also, by the Hardy-Littlewood inequality forrearrangements, k Sϕ k L ∞ (0 , | Ω | ) ≤ Z | Ω | | ϕ ( r ) | r − n ′ dr ≤ Z | Ω | ϕ ∗ ( r ) r − n ′ dr (4.13) ≤ | Ω | n − σ Z | Ω | ϕ ∗ ( r ) r − σ ′ dr = | Ω | n − σ k ϕ k L σ, (0 , | Ω | ) for every ϕ ∈ L σ, (0 , | Ω | ). Here, ϕ ∗ denotes the decreasing rearrangement of ϕ , and L σ, (0 , | Ω | ) is the Lorentzspace whose norm is defined by the last integral in equation (4.13). Owing to equations (4.12) and (4.13), theinterpolation theorem established in [23, Theorem 4] can be applied to deduce inequality (4.10). (cid:3) The next lemma tells us that the assumptions of Theorem 4.1 are certainly fulfilled if A satisfies the ∆ -condition, provided that σ is sufficiently large. Lemma 4.2.
Let A be a continuously differentiable Young function satisfying the ∆ -condition and let σ > s A .Then conditions (4.5) and (4.6) are fulfilled.Proof. Owing to the definition of s a , one verifies via differentiation that the function A ( t ) t sA is non-increasing.Thus,(4.14) A ( t ) ≥ A (1) t s A if t ∈ (0 , A ( t ) ≤ A (1) t s B if t ∈ [1 , ∞ ).Equations (4.5) and (4.6) follow from (4.14) and (4.15), respectively. (cid:3) Young functions built upon the function a . Given a continuously differentiable function a : (0 , ∞ ) → (0 , ∞ ) such that i a ≥ −
1, let b and B the functions defined by (1.8) and (1.7). Our assumption on i a ensuresthat b is a non-decreasing function, and hence B is a Young function.One has that(4.16) i b = i a + 1 and s b = s a + 1.Also(4.17) i B ≥ i b + 1 and s B ≤ s b + 1.Thus, if s a < ∞ , then the functions b and B satisfy the ∆ -conditon, and if i a > −
1, then the function e B satisfies the ∆ -conditon.Hence, if s a < ∞ , then for every λ >
1, there exists a constant c = c ( λ, s a ) > b ( λt ) ≤ cb ( t ) for t ≥ B ( λt ) ≤ cB ( t ) for t ≥ tb ′ ( t ) ≤ ( s a + 1) b ( t ) for t > B ( t ) ≤ tb ( t ) ≤ ( s a + 2) B ( t ) for t > e B ( b ( t )) ≤ B (2 t ) for t ≥
0, there exists a constant c = c ( s a ) such that(4.22) e B ( b ( t )) ≤ cB ( t ) for t ≥ i a > − s a < ∞ , then(4.23) a (1) min { t i a , t s a } ≤ a ( t ) ≤ a (1) max { t i a , t s a } for t > a is as above and ε >
0, we define the function a ε : [0 , ∞ ) → (0 , ∞ ) as(4.24) a ε ( t ) = a ( p t + ε ) for t ≥ b ε and B ε are defined as in (1.8) and (1.7), with a replaced by a ε . Lemma 4.3.
Assume that the function a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable in (0 , ∞ ) and that i a > − and s a < ∞ . Let ε > and let a ε be the function defined by (4.24) . Then (4.25) i a ε ≥ min { i a , } and s a ε ≤ max { s a , } , where i a ε and s a ε are defined as in (2.1) , with a replaced by a ε .Let b , B , b ε and B ε be the functions defined above. Then there exist constants c , c , c , depending only on s a ,such that (4.26) c B ( t ) − c B ( ε ) ≤ a ε ( t ) t ≤ c ( B ( t ) + B ( ε )) for t ≥ . POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 19
Moreover, there exists a constant c = c ( s a ) such that (4.27) B ε ( t ) ≤ c ( B ( t ) + B ( ε )) for t ≥ ,and (4.28) e B ( b ε ( t )) ≤ c ( B ( t ) + B ( ε )) for t ≥ .Proof. Property (4.25) can be verified by straightforward computations. Consider equation (4.26). One hasthat a ε ( t ) t ≤ a ( t + ε ) t ≤ ( s a + 2) B ( t + ε ) ≤ ( s a + 2)( B (2 t ) + B (2 ε )) ≤ c ( B ( t ) + B ( ε )) for t ≥ c = c ( s a ), where the second inequality holds by (4.21) and the last one by (4.1). This provesthe second inequality in (4.26). As for the first one, observe that B ( t ) ≤ B ( t + ε ) ≤ B (2 t ) + B (2 ε ) ≤ cB ( t ) + cB ( ε ) for t ≥ c = c ( s a ), where we have made use of inequality (4.1) again. Now, B ( t ) = Z t a ( τ ) τ dτ ≤ Z t a ( τ + ε )( τ + ε ) dτ ≤ Z t a (2 p τ + ε )2 p τ + ε dτ (4.31) ≤ c Z t a ( p τ + ε ) p τ + ε dτ ≤ c t a ( p t + ε ) p t + ε = c a ε ( t ) t p t + ε for t ≥ c = c ( s a ), where the third inequality is due to (4.18). On the other hand, a ε ( t ) t p t + ε ≤ √ a ε ( t ) t if t ≥ ε ,(4.32)and a ε ( t ) t p t + ε ≤ √ a ε ( ε ) ε = √ a ( √ ε ) ε ≤ cB ( ε ) if 0 ≤ t ≤ ε ,(4.33)for some constant c = c ( s a ), where the last inequality holds thanks to (4.21). Combining inequalities (4.31)–(4.33) yields B ( t ) ≤ ca ε ( t ) t + cB ( ε ) for t ≥ c = c ( s a ). 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 B ε , and of thesecond inequality in (4.26).Inequality (4.28) is a consequence of the following chain: e B ( b ε ( t )) = e B ( a ( p t + ε ) t ) ≤ e B ( b ( p t + ε ))(4.34) ≤ e B ( b ( t + ε )) ≤ cB ( t + ε ) ≤ c ′ ( B ( t ) + B ( ε )) for t ≥ c and c ′ depending on s a . Notice, that we have made use of property (4.22) in last but oneinequality, and of property (4.1) in the last inequality. (cid:3) Lemma 4.4.
Assume that the function a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable in (0 , ∞ ) and that i a > − and s a < ∞ . Let ε > and let a ε be the function defined by (4.24) . Let M > . Then there exists aconstant c = c ( i a , s a , ε, M ) such that (4.35) | P − Q | ≤ c | a ε ( P ) P − a ε ( Q ) Q | for every P, Q ∈ R N × n such that | P | ≤ M and | Q | ≤ M .Proof. By [33, Lemma 21], there exists a positive constant c = c ( i a ε , s a ε ) such that(4.36) c (cid:2) a ε ( | P | + | Q | ) + a ′ ε ( | P | + | Q | )( | P | + | Q | ) (cid:3) | P − Q | ≤ ( a ε ( | P | ) P − a ε ( | Q | ) Q ) · ( P − Q )for every P, Q ∈ R N × n . Hence, via inequalities (4.25),(4.37) c (1 + min { i a , } ) a ε ( | P | + | Q | ) | P − Q | ≤ | a ε ( | P | ) P − a ε ( | Q | ) Q | for every P, Q ∈ R N × n . Inequality (4.4) hence follows, since a ε ( | P | + | Q | ) ≥ min (cid:8) a ( t ) : ε ≤ t ≤ p M + ε (cid:9) > | P | ≤ M and | Q | ≤ M , and (1 + min { i a , } ) > (cid:3) One more function associated with a function a as above and to a number ε > V ε : R N × n → R N × n and is defined as(4.38) V ε ( P ) = p a ε ( | P | ) P for P ∈ R N × n . Lemma 4.5.
Assume that the function a : (0 , ∞ ) → (0 , ∞ ) is continuously differentiable and such that i a > − and s a < ∞ . Let ε > and let a ε be the function defined by (4.24) . Then (4.39) a ε ( | P | ) P → a ( | P | ) P as ε → + ,uniformly for P in any compact subset of R N × n .Moreover, (4.40) ( a ε ( | P | ) P − a ε ( | Q | ) Q ) · ( P − Q ) ≈ (cid:12)(cid:12) V ε ( P ) − V ε ( Q ) (cid:12)(cid:12) for P, Q ∈ R N × n ,where the relation ≈ means that the two sides are bounded by each other, up to positive multiplicative constantsdepending only on i a and s a .Proof. Fix any 0 < ℓ < L and assume that ε ∈ [0 , a ∈ C (0 , ∞ ), if ℓ ≤ | P | ≤ L then | a ε ( | P | ) P − a ( | P | ) P | ≤ | P || a ε ( | P | ) − a ( | P | ) | (4.41) ≤ max t ∈ [ ℓ, √ L +1] | a ′ ( t ) | ( p | P | + ε − | P | ) ≤ max t ∈ [ ℓ, √ L +1] | a ′ ( t ) | ε. Moreover, if | P | ≤
1, then, by the second inequality in (4.23) applied with a replaced by a ε and by the firstinequality in (4.25),(4.42) | a ε ( | P | ) P | ≤ a ε (1) | P | { i a , } ≤ max t ∈ [1 , √ | a ( t ) || P | { i a , } . Now, let
L >
0. Fix any σ >
0. By inequality (4.42), there exists ℓ > | a ε ( | P | ) P − a ( | P | ) P | ≤ | a ε ( | P | ) P | + | a ( | P | ) P | ≤ σ for every ε ∈ [0 , | P | < ℓ . On the other hand, inequality (4.41) ensures that there exists ε ∈ (0 ,
1) such that(4.44) | a ε ( | P | ) P − a ( | P | ) P | < σ if ℓ ≤ | P | ≤ L . From inequalities (4.43) and (4.44) we deduce that, if 0 ≤ ε < ε , then(4.45) | a ε ( | P | ) P − a ( | P | ) P | < σ if | P | ≤ L .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 i a ε > − s a ε < ∞ , then theratio of the two sides of this equation is bounded from below and from above by positive constants dependingonly on a lower bound for i a ε and an upper bound for s a ε . Owing to inequalities (4.25) and to our assumptionthat i a > − s a < ∞ , we have that i a ε ≥ min { i a , } > s a ε ≤ max { s a , } < ∞ for every ε >
0. Thisimplies that equation (4.40) actually holds up to equivalence constants depending only on i a and s a . (cid:3) POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 21 Second-order regularity: local solutions
The definiton of generalized local solution to the system(5.1) − div ( a ( |∇ u | ) ∇ u ) = f in Ωthat will be adopted is inspired by the results of [36], and involves the notion of approximate differentiability.Recall that a measurable function u : Ω → R N is said to be approximately differentiable at x ∈ Ω if thereexists a matrix ap ∇ u ( x ) ∈ R N × n such that, for every ε > r → + (cid:12)(cid:12) { y ∈ B r ( x ) : r | u ( y ) − u ( x ) − ap ∇ u ( x )( y − x ) | > ε } (cid:12)(cid:12) r n = 0 . If u is approximately differentiable at every point in Ω, then the function ap ∇ u : Ω → R N × n is measurable.Assume that a is as in Theorem 2.4 and let f ∈ L q loc (Ω , R N ) for some q ≥
1. An approximately differentiablefunction u : Ω → R N is called a local approximable solution to system (5.1) if a ( | ap ∇ u | ) | ap ∇ u | ∈ L (Ω), andthere exist a sequence { f k } ⊂ C ∞ (Ω , R N ), with f k → f in L q loc (Ω , R N ), and a corresponding sequence of localweak solutions { u k } to the systems(5.2) − div ( a ( |∇ u k | ) ∇ u k ) = f k in Ω , such that(5.3) u k → u and ∇ u k → ap ∇ u a.e. in Ω,and(5.4) lim k →∞ Z Ω ′ a ( |∇ u k | ) |∇ u k | dx = Z Ω ′ a ( | ap ∇ u | ) | ap ∇ u | dx for every open set Ω ′ ⊂⊂ Ω. In what follows, we shall denote ap ∇ u simply by ∇ u .Weak solutions to system (5.1) are defined in a standard way if f ∈ L (Ω , R N ) ∩ ( W ,B (Ω , R N )) ′ , where B is the Young function defined via (1.7). Namely, a function u ∈ W ,B loc (Ω , R N ) is called a local weak solution tothis system if(5.5) Z Ω ′ a ( |∇ u | ) ∇ u · ∇ ϕ dx = Z Ω ′ f · ϕ dx for every open set Ω ′ ⊂⊂ Ω, and every function ϕ ∈ W ,B (Ω ′ , R N ).Inequality (2.7) enters the proof of Theorem 2.4 through Lemma 5.1 below. The latter will be applied tosolutions to systems which approximate system (2.11), and involve regularized differential operators and smoothright-hand sides. Lemma 5.1 can be deduced from Theorem 2.1 and inequality (2.10), along the same lines as inthe proof of [27, Theorem 3.1, Inequality (3.4)]. The details are omitted, for brevity. We seize this opportunityto point out an incorrect dependence on the radius R of the constants in that inequality, due to a flaw in thescaling argument in the derivation of [27, Inequality (3.43)]. Lemma 5.1.
Let n ≥ , N ≥ , and let Ω be an open set in R n . Assume that the function a ∈ C ([0 , ∞ )) satisfies conditions (2.4) – (2.6) . Then there exists a constant C = C ( n, N, i a , s a ) , such that (5.6) R − (cid:13)(cid:13) a ( |∇ u | ) ∇ u (cid:13)(cid:13) L ( B R , R N × n ) + (cid:13)(cid:13) ∇ (cid:0) a ( |∇ u | ) ∇ u (cid:1)(cid:13)(cid:13) L ( B R , R N × n ) ≤ C (cid:16) k div ( a ( |∇ u | ) ∇ u ) k L ( B R , R N ) + R − n − k a ( |∇ u | ) ∇ u k L ( B R , R N × n ) (cid:17) for every function u ∈ C (Ω , R N ) and any ball B R ⊂⊂ Ω .Proof of Theorem 2.4. Let us temporarily assume that(5.7) f ∈ C ∞ (Ω , R N ) , 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 a ε > − √ . Let B R ⊂⊂ Ω and, given ε ∈ (0 , u ε ∈ u + W ,B ( B R , R N ) be the weak solution to the Dirichlet problem(5.9) ( − div ( a ε ( |∇ u ε | ) ∇ u ε ) = f in B R u ε = u on ∂B R . We claim that u ε ∈ C ∞ ( B R , R N ) . (5.10)Actually, as a consequence of [34, Corollary 5.5], ∇ u ε ∈ L ∞ loc ( B R , R N × n ) and there exists a constant C ,independent of ε , such that(5.11) k∇ u ε k L ∞ ( B R , R N × n ) ≤ C. The same result also tells us that a ε ( |∇ u ε | ) ∇ u ε ∈ C α loc ( B R , R N × n ) for some α ∈ (0 , ∇ u ε ∈ C α loc ( B R , R N × n ) as well. Hence, a ε ( |∇ u ε | ) ∈ C ,α loc ( B R ), and by the Schauder theoryfor linear elliptic systems, u ε ∈ C ,α loc ( B R , R N ). An iteration argument relying upon the the Schauder theoryagain yields property (5.10).We claim that Z B R B ( |∇ u ε | ) dx ≤ C (cid:18) Z B R e B ( | f | ) dx + Z B R B ( |∇ u | ) dx + B ( ε ) (cid:19) (5.12)for some constant C = C ( n, N, s a , R ) and for ε ∈ (0 , u ε − u ∈ W ,B ( B R , R N ) as a testfunction in the weak formulation of problem (5.9) results in Z B R a ε ( |∇ u ε | ) ∇ u ε · ( ∇ u ε − ∇ u ) dx = Z B R f · ( u ε − u ) dx . (5.13)The Poincar´e inequality (4.4) implies that(5.14) Z B R B ( | u ε − u | ) dx ≤ C Z B R B ( |∇ u ε − ∇ u | ) dx for some constant C = C ( n, s a , R ).Fix δ ∈ (0 , c Z B R B ( |∇ u ε | ) dx ≤ Z B R | f || u ε − u | dx + C Z B R a ε ( |∇ u ε | ) |∇ u ε ||∇ u | dx + CR n B ( ε )(5.15) ≤ C Z B R e B ( | f | ) dx + δ Z B R B ( | u ε − u | ) dx + δ Z B R e B ε ( a ε ( |∇ u ε | ) |∇ u ε | ) dx + C Z B R B ε ( |∇ u | ) dx + CR n B ( ε ) ≤ C Z B R e B ( | f | ) dx + δC Z B R B ( |∇ u ε | ) dx + C Z B R B ( |∇ u | ) dx + δC Z B R B ε ( |∇ u ε | ) dx + C Z B R B ε ( |∇ u | ) dx + CR n B ( ε ) ≤ C Z B R e B ( | f | ) dx + δC Z B R B ( |∇ u ε | ) dx + C Z B R B ( |∇ u | ) dx + CR n B ( ε )for suitable constants C , C and C depending on n, N, s a , R , and constants C , C and C depending also on δ . Inequality (5.12) follows from (5.15), on choosing δ small enough.Coupling inequality (5.12) with the Poincar´e inequality (4.4) tells us that the family { u ε } is bounded in POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 23 W ,B ( B R , R N ). Since under assumptions (2.12) and (2.13) this space is reflexive, there exist a sequence { ε k } and a function v ∈ W ,B ( B R , R N ) such that ε k → + and(5.16) u ε k ⇀ v in W ,B ( B R , R N ).Choosing the test function u ε k − u for system (2.11), and subtracting the resultant equation from (5.13) enablesus to deduce that, given any δ > Z B R (cid:0) a ε k ( |∇ u ε k | ) ∇ u ε k − a ε k ( |∇ u | ) ∇ u (cid:1) · ( ∇ u ε k − ∇ u ) dx (5.17) = Z B R (cid:0) a ( |∇ u | ) ∇ u − a ε k ( |∇ u | ) ∇ u (cid:1) · ( ∇ u ε k − ∇ u ) dx ≤ δ Z B R B ( |∇ u ε k | ) + B ( |∇ u | ) dx + C Z B R e B (cid:0) | a ( |∇ u | ) ∇ u − a ε k ( |∇ u | ) ∇ u | (cid:1) dx for some constant C = C ( δ, s a ). Owing to equation (4.40), there exists a constant c = c ( i a , s a ) such that Z B R | V ε k ( ∇ u ε k ) − V ( ∇ u ) | dx ≤ Z B R | V ε k ( ∇ u ε k ) − V ε k ( ∇ u ) | dx + 2 Z B R | V ε k ( ∇ u ) − V ( ∇ u ) | dx (5.18) ≤ c Z B R (cid:0) a ε k ( |∇ u ε k | ) ∇ u ε k − a ε k ( |∇ u | ) ∇ u (cid:1) · ( ∇ u ε k − ∇ u ) dx + 2 Z B R | V ε k ( ∇ u ) − V ( ∇ u ) | dx. Combining equations (5.18), (5.17) and (5.12) yields Z B R | V ε k ( ∇ u ε k ) − V ( ∇ u ) | dx ≤ δc (cid:18) Z B R e B ( | f | ) dx + Z B R B ( |∇ u | ) dx + B ( ε ) (cid:19) (5.19) + c Z B R e B (cid:0) | a ( |∇ u | ) ∇ u − a ε k ( |∇ u | ) ∇ u | (cid:1) dx + 2 Z B R | V ε k ( ∇ u ) − V ( ∇ u ) | dx for some constant c = c ( n, N, R, i a , s a ). Inequalities (4.22) and (4.28) entail that e B (cid:0) | a ( |∇ u | ) ∇ u − a ε k ( |∇ u | ) ∇ u | (cid:1) ≤ c ( B ( |∇ u | ) + B ( ε k )) a.e. in B R ,(5.20)for some constant c = c ( s a ). Furthermore, from inequality (4.26) one infers that | V ε k ( ∇ u ) | ≤ c ( B ( |∇ u | ) + B ( ε k )) a.e. in B R ,(5.21)for some constant c = c ( s a ). Thanks to inequalities (5.20) and (5.21), and to property (4.39), the last twointegrals on the right-hand side of inequality (5.19) tend to 0 as k → ∞ , via the dominated convergencetheorem. Owing to the same theorem, equation (5.19) implies that(5.22) lim k →∞ Z B R | V ε k ( ∇ u ε k ) − V ( ∇ u ) | dx ≤ δc for every δ ∈ (0 , V ε k ( ∇ u ε k ) → V ( ∇ u ) in L ( B R , R N × n ),and, on passing to a subsequence, still indexed by k ,(5.24) V ε k ( ∇ u ε k ) → V ( ∇ u ) a.e. in B R .An analogous argument as in [35, Lemma 4.8] shows that the function ( ε, P ) V − ε ( P ) is continuous. Thus,one can deduce from equation (5.24) that(5.25) ∇ u ε k → ∇ u a.e. in B R .Hence, equation (5.16) implies that v = u and(5.26) u ε k ⇀ u in W ,B ( B R , R N ). Inequalities (4.26) and (5.12), and the monotonicity of the function b ε k , yield Z B R a ε k ( |∇ u ε k | ) |∇ u ε k | dx ≤ Z {|∇ u εk |≤ }∩ B R a ε k ( |∇ u ε k | ) |∇ u ε k | dx + Z B R a ε k ( |∇ u ε k | ) |∇ u ε k | dx (5.27) ≤ cR n b ε k (1) + c Z B R B ( ∇ u ε k ) dx + cR n B ( ε k ) ≤ C for some constants c and C independent of k . Thanks to assumption (5.8) and to property (4.25), Lemma 5.1can be applied with a replaced by a ε k . The use of inequality (5.6) of this lemma for the function u ε k , and theequation in (5.9), ensure that k a ε ( |∇ u ε k | ) ∇ u ε k k W , ( B R , R N × n ) (5.28) ≤ C (cid:0) k f k L ( B R , R N ) + ( R − n + R − n − ) k a ε k ( |∇ u ε k | ) ∇ u ε k k L ( B R , R N × n ) (cid:1) , for some constant C = C ( n, N, i a , s a ). Owing to inequalities (5.27) and (5.28), the sequence { a ε k ( |∇ u ε k | ) ∇ u ε k } is bounded in W , ( B R , R N × n ). Thus, there exists a function U ∈ W , ( B R , R N × n ), and a subsequence of { ε k } ,still indexed by k , such that a ε k ( |∇ u ε k | ) ∇ u ε k → U in L ( B R , R N × n )(5.29) and a ε k ( |∇ u ε k | ) ∇ u ε k ⇀ U in W , ( B R , R N × n ).Combining property (4.39) with equations (5.11), (5.25) and (5.29) yields(5.30) a ( |∇ u | ) ∇ u = U ∈ W , ( B R , R N × n ) . On passing to the limit as k → ∞ , from equations (5.28), (5.29) and (5.30) we infer that k a ( |∇ u | ) ∇ u k W , ( B R , R N × n ) ≤ C (cid:0) k f k L ( B R , R N ) + ( R − n + R − n − ) k a ( |∇ u | ) ∇ u k L ( B R , R N × n ) (cid:1) . (5.31)It remains to remove assumption (5.7). Suppose that f ∈ L (Ω , R N ). Let u be an approximable local solutionto equation (2.11), and let f k and u k be as in the definition of this kind of solution. Applying inequality (5.31)to the function u k tells us that a ( |∇ u k | ) ∇ u k ∈ W , ( B R , R N × n ), and k a ( |∇ u k | ) ∇ u k k W , ( B R , R N × n ) (5.32) ≤ C (cid:0) k f k k L ( B R , R N ) + ( R − n + R − n − ) k a ( |∇ u k | ) ∇ u k k L ( B R , R N × n ) (cid:1) , for some constant C independent of k . Hence, by equation (5.4), the sequence { a ( |∇ u k | ) ∇ u k } is bounded in W , ( B R , R N × n ). Thereby, there exist a subsequence, still indexed by k , and a function U ∈ W , ( B R , R N × n ),such that(5.33) a ( |∇ u k | ) ∇ u k → U in L ( B R , R N × n ) and a ( |∇ u k | ) ∇ u k ⇀ U in W , ( B R , R N × n ) . By assumption (5.3), we have that ∇ u k → ∇ u a.e. in Ω. Hence, thanks to properties (5.33),(5.34) a ( |∇ u | ) ∇ u = U ∈ W , ( B R , R N × n ) . Inequality (2.15) follows on passing to the limit as k → ∞ in (5.32), via (5.4), (5.33) and (5.34). (cid:3) Second-order regularity: Dirichlet problems
Generalized solutions, in the approximable sense, to the Dirichlet problem(6.1) ( − div ( a ( |∇ u | ) ∇ u ) = f in Ω 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 ∈ L q (Ω , R N ) for some q ≥
1. An approximatelydifferentiable function u : Ω → R N is called an approximable solution to the Dirichlet problem (6.1) if there POINTWISE DIFFERENTIAL INEQUALITY AND NONLINEAR ELLIPTIC SYSTEMS 25 exists a sequence { f k } ⊂ C ∞ (Ω , R N ) such that f k → f in L q (Ω , R N ), and the sequence { u k } of weak solutionsto the Dirichlet problems(6.2) ( − div ( a ( |∇ u k | ) ∇ u k ) = f k in Ω u k = 0 on ∂ Ωsatisfies(6.3) u k → u and ∇ u k → ap ∇ u a.e. in Ω.As above, in what follows ap ∇ u will simply be denoted by ∇ u .Recall that, under the assumption that f ∈ L (Ω , R N ) ∩ ( W ,B (Ω , R N )) ′ , a function u ∈ W ,B (Ω , R N ) iscalled a weak solution to the Dirichlet problem (6.1) if(6.4) Z Ω a ( |∇ u | ) ∇ u · ∇ ϕ dx = Z Ω f · ϕ dx for every ϕ ∈ W ,B (Ω , R N ). A unique weak solution to problem (6.1) exists whenever | Ω | < ∞ .Before accomplishing the proof of our global estimates, we recall the notions of capacity and of Marcinkiewiczspaces that enter conditions (2.22) and (2.23), respectively, in the statement of Theorem 2.7.The capacity cap Ω ( E ) of a set E ⊂ Ω relative to Ω is defined as(6.5) cap Ω ( E ) = inf (cid:26) Z Ω |∇ v | dx : v ∈ C , (Ω) , v ≥ E (cid:27) . Here, C , (Ω) denotes the space of Lipschitz continuous, compactly supported functions in Ω.The Marcinkiewicz space L q, ∞ ( ∂ Ω) is the Banach function space endowed with the norm defined as(6.6) k ψ k L q, ∞ ( ∂ Ω) = sup s ∈ (0 , H n − ( ∂ Ω)) s q ψ ∗∗ ( s )for a measurable function ψ on ∂ Ω. Here, ψ ∗∗ ( s ) = s ∫ s ψ ∗ ( r ) dr for s >
0, where ψ ∗ denotes the decreasingrearrangement of ψ . The Marcinkiewicz space L , ∞ log L ( ∂ Ω) is equipped with the norm given by(6.7) k ψ k L , ∞ log L ( ∂ Ω) = sup s ∈ (0 , H n − ( ∂ Ω)) s log (cid:0) Cs (cid:1) ψ ∗∗ ( s ) , for any constant C > H n − ( ∂ Ω). Different constants C result in equivalent norms in (6.7).The next lemma stands with respect to Theorems 2.6 and 2.7 that Lemma 5.1 stands to Theorem 2.4. Itfollows from Theorem 2.1 and inequality (2.10), via the same proof of [28, Theorem 3.1, Part (ii)]. Lemma 6.1.
Let n ≥ , N ≥ , and let Ω be a bounded open set in R n with ∂ Ω ∈ C . 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, i a , s a , L Ω , d Ω ) such that, if (6.8) K Ω ( r ) ≤ K ( r ) for r ∈ (0 , ,for some function K : (0 , → [0 , ∞ ) satisfying (6.9) lim r → + K ( r ) < c , then k a ( |∇ u | ) ∇ u k W , (Ω , R N × n ) ≤ C (cid:0) k div ( a ( |∇ u | ) ∇ u ) k L (Ω , R N ) + k a ( |∇ u | ) ∇ u k L (Ω , R N × n ) (cid:1) (6.10) for some constant C = C ( n, N, i a , s a , L Ω , d Ω , K ) , and for every function u ∈ C (Ω , R N ) ∩ C (Ω , R N ) such that (6.11) u = 0 on ∂ Ω .In particular, if Ω is convex, then inequality (6.10) holds whatever K Ω is, and the constant C in (6.10) onlydepends on n, N, i a , s a , L Ω , d Ω . The following gradient bound for solutions to the Dirichlet problem (6.1) is needed to deal with lower-orderterms appearing in our global estimates.
Proposition 6.2.
Assume that n ≥ , N ≥ . Let Ω be an open set in R n such that | Ω | < ∞ . Assume that thefunction a : [0 , ∞ ) → [0 , ∞ ) is continuously differentiable in (0 , ∞ ) and fulfills conditions (2.12) and (2.13) .Let f ∈ L (Ω , R N ) ∩ ( W ,B (Ω , R N )) ′ and let u be the weak solution to the Dirichlet problem (6.1) . Then, thereexists a constant C = C ( n, N, i a , s a , | Ω | ) such that (6.12) k a ( |∇ u | ) ∇ u k L (Ω , R N × n ) ≤ C k f k L (Ω , R N ) . The same conclusion holds if f ∈ L (Ω , R N ) and u is an approximable solution to the Dirichlet problem (6.1) .Proof. Assume that f ∈ L (Ω , R N ) ∩ ( W ,B (Ω , R N )) ′ and that u is the weak solution to the Dirichlet problem(6.1). Given t >
0, let T t ( u ) : Ω → R N be the function defined by(6.13) T t ( u ) = u in {| u | ≤ t } t u | u | in {| u | > t } .Then T t ( u ) ∈ W ,B (Ω , R N ), and(6.14) ∇ T t ( u ) = ∇ u a.e. in {| u | ≤ t } t | u | (cid:16) I − u | u | ⊗ u | u | (cid:17) ∇ u a.e. in {| u | > t } Observe that a ( | P | ) P · ( I − ω ⊗ ω ) P ≥ P ∈ R N × n and any vector ω ∈ R N such that | ω | ≤
1. Thus, on making use of T t ( u ) as a testfunction ϕ in equation (6.4), one deduces that Z {| u |≤ t } a ( |∇ u | ) |∇ u | dx ≤ Z Ω a ( |∇ u | ) ∇ u · ∇ T t ( u ) dx = Z Ω f · T t ( u ) dx (6.15) = Z {| u |≤ t } f · u dx + Z {| u >t } f · t u | u | dx ≤ t k f k L (Ω , R N ) . Hence, by the first inequality in (4.21), Z {| u |≤ t } B ( |∇ u | ) dx ≤ t k f k L (Ω , R N ) . (6.16)On the other hand, the chain rule for vector-valued functions ensures that the function | u | ∈ W ,B (Ω), and |∇ u | ≥ |∇| u || a.e. in Ω. Inequality (6.16) thus implies that Z {| u |
0. Hence, by (6.17), |{| u | ≥ t }| B σ (cid:18) tC ( t k f k L (Ω , R N ) ) σ (cid:19) ≤ t k f k L (Ω , R N ) for t > |{ B ( |∇ u | ) > s, | u | ≤ t }| ≤ s Z { B ( |∇ u | ) >s, | u |≤ t } B ( |∇ u | ) dx ≤ t k f k L (Ω , R N ) s for t > s > |{ B ( |∇ u | ) > s }| ≤ |{| u | > t }| + |{ B ( |∇ u | ) > s, | u | ≤ t }| (6.25) ≤ t k f k L (Ω , R N ) B σ ( Ct σ ′ / ( t k f k L (Ω , R N ) ) σ ) + t k f k L (Ω , R N ) s for t > s > t = (cid:0) C k f k /σL (Ω , R N ) B − σ ( s ) (cid:1) σ ′ in inequality (6.25) results in(6.26) |{ B ( |∇ u | ) > s }| ≤ k f k σ ′ L (Ω , R N ) C σ ′ B − σ ( s ) σ ′ s for s > s = B ( b − ( τ )) in (6.26) and make use of (4.8) to obtain that(6.27) |{ b ( |∇ u | ) > τ }| ≤ k f k σ ′ L (Ω , R N ) C σ ′ H σ ( b − ( τ )) σ ′ B ( b − ( τ )) for τ > H σ is defined as in (4.7), with A replaced by B . Thanks to inequality (6.27), Z Ω b ( |∇ u | ) dx = Z ∞ |{ b ( |∇ u | ) > τ }| dτ ≤ λb ( | Ω | ) + 2 C − σ ′ k f k σ ′ L (Ω , R N ) Z ∞ λ H σ ( b − ( τ )) σ ′ B ( b − ( τ )) dτ (6.28)for λ >
0. Owing to inequalities (4.20) and (4.21), and to Fubinis’s theorem, the following chain holds: Z ∞ λ H σ ( b − ( τ )) σ ′ B ( b − ( τ )) dτ ≤ ( s a + 1) Z ∞ b − ( λ ) H σ ( s ) σ ′ sB ( s ) b ( s ) ds (6.29) = ( s a + 1) Z ∞ b − ( λ ) b ( s ) sB ( s ) Z s (cid:18) tB ( t ) (cid:19) σ − dt ds ≤ ( s a + 1)( s a + 2) Z ∞ b − ( λ ) s Z s (cid:18) tB ( t ) (cid:19) σ − dt ds = ( s a + 1)( s a + 2) (cid:18) Z b − ( λ )0 (cid:18) tB ( t ) (cid:19) σ − Z ∞ b − ( λ ) dss dt + Z ∞ b − ( λ ) (cid:18) tB ( t ) (cid:19) σ − Z ∞ t dss dt (cid:19) = ( s a + 1)( s a + 2) (cid:18) b − ( λ ) Z b − ( λ )0 (cid:18) tB ( t ) (cid:19) σ − dt + Z ∞ b − ( λ ) (cid:18) tB ( t ) (cid:19) σ − dtt (cid:19) ≤ ( s a + 1)( s a + 2) σ ′ (cid:18) b − ( λ ) Z b − ( λ )0 (cid:18) b ( t ) (cid:19) σ − dt + Z ∞ b − ( λ ) (cid:18) b ( t ) (cid:19) σ − dtt (cid:19) for λ >
0. The function t sa +1+ ε b ( t ) is increasing for every ε >
0. Hence, if 0 < ε < σ − s a −
2, then1 b − ( λ ) Z b − ( λ )0 (cid:18) b ( t ) (cid:19) σ − dt = 1 b − ( λ ) Z b − ( λ )0 (cid:18) t s a +1+ ε b ( t ) (cid:19) σ − t − sa +1+ εσ − dt (6.30) ≤ b − ( λ ) (cid:18) b − ( λ ) s a +1+ ε λ (cid:19) σ − Z b − ( λ )0 t − sa +1+ εσ − dt = σ − σ − s a − − ε λ − σ − for λ > < ε < i a + 1, then the function t ε b ( t ) is decreasing. Hence, Z ∞ b − ( λ ) (cid:18) b ( t ) (cid:19) σ − dtt = Z ∞ b − ( λ ) (cid:18) t ε b ( t ) (cid:19) σ − t − εσ − − dt ≤ (cid:18) b − ( λ ) ε λ (cid:19) σ − Z ∞ b − ( λ ) t − εσ − − dt (6.31) = σ − ε λ − σ − for λ > c = c ( σ, i a , s a ) such that Z ∞ λ H σ ( b − ( τ )) σ ′ B ( b − ( τ )) dτ ≤ cλ − σ − for λ > λ = k f k L (Ω , R N ) .The assertion about the case when f ∈ L (Ω , R N ) 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 f k and u k appearing in thedefinition of approximable solutions, and passing to the limit as k → ∞ in the resultant inequality. Fatou’slemma plays a role here. (cid:3) A last preliminary result, proved in [27, Lemma 5.2], is needed in an approximation argument for the domainΩ in our proof of Theorem 2.7.
Lemma 6.3.
Let Ω be a bounded Lipschitz domain in R n , n ≥ such that ∂ Ω ∈ W , . Assume that the function K Ω ( r ) , defined as in (2.20) , is finite-valued for r ∈ (0 , . Then there exist positive constants r and C and asequence of bounded open sets { Ω m } , such that ∂ Ω m ∈ C ∞ , Ω ⊂ Ω m , lim m →∞ | Ω m \ Ω | = 0 , the Hausdorffdistance between Ω m and Ω tends to as m → ∞ , (6.33) L Ω m ≤ CL Ω , d Ω m ≤ Cd Ω and (6.34) K Ω m ( r ) ≤ C K Ω ( r ) for r ∈ (0 , r ) and m ∈ N .Proof of Theorem 2.7. It suffices to prove Part (i). Part (ii) will then follow, since, by [27, Lemmas 3.5 and3.7],(6.35) K Ω ( r ) ≤ C sup x ∈ ∂ Ω kBk X ( ∂ Ω ∩ B r ( x )) for r ∈ (0 , r ),for suitable constants r and C depending on n , L Ω and d Ω .We split the proof in three separate steps, where approximation arguments for the differential operator, the 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(6.36) f ∈ C ∞ (Ω , R N ) , and(6.37) ∂ Ω ∈ C ∞ are in force. Given ε ∈ (0 , u ε the weak solution to the system(6.38) ( − div ( a ε ( |∇ u ε | ) ∇ u ε ) = f in Ω u ε = 0 on ∂ Ω , where a ε is the function defined by (4.24). An application of [26, Theorem 2.1] tells us that k∇ u ε k L ∞ (Ω , R N × n ) ≤ C (6.39)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 a ε be either increasing or decreasing; such an additional assump-tion can however be dropped via a slight variant in the proof. Inequality (6.39) implies that, for each ε ∈ (0 , c ≤ a ε ( |∇ u ε | ) ≤ c in Ωfor suitable positive constants c and c .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 ε ∈ W , (Ω , R N ). Consequently, u ε ∈ W , (Ω , R n ) ∩ W , ∞ (Ω , R N ) ∩ W , (Ω , R N ). One can thenfind a sequence { u k } ⊂ C ∞ (Ω , R N ) ∩ C (Ω , R N ) such that u k = 0 on ∂ Ω for k ∈ N , and(6.41) u k → u ε in W , (Ω , R N ), u k → u ε in W , (Ω , R N ), ∇ u k → ∇ u ε a.e. in Ω , as k → ∞ – see e.g. [15, Chapter 2, Corollary 3]. One also has that(6.42) k∇ u k k L ∞ (Ω , R N × n ) ≤ C k∇ u ε k L ∞ (Ω , R N × n ) for some constant C independent of k , and, by the chain rule for vector-valued Sobolev functions [46, Theorem2.1], |∇|∇ u k || ≤ |∇ u k | a.e. in Ω. Moreover, [26, Equation (6.12)] tells us that − div ( a ε ( |∇ u k | ) ∇ u k ) → f in L (Ω , R N ) , (6.43)as k → ∞ . Assumption (2.22) enables us to apply inequality (6.10), with a replaced by a ε and u replaced by u k , to deduce that k a ε ( |∇ u k | ) ∇ u k k W , (Ω , R N × n ) ≤ C (cid:16) k div ( a ε ( |∇ u k | ) ∇ u k ) k L (Ω , R N ) + k a ε ( |∇ u k | ) ∇ u k k L (Ω , R N × n ) (cid:17) (6.44)for k ∈ N , and for some constant C = C ( n, N, i a , s a , L Ω , d Ω , K Ω ). Notice that this constant actually depends onthe function a ε only through i a and s a , and it is hence independent of ε , owing to (4.25). Equations (6.42)–(6.44)ensure that the sequence { a ε ( |∇ u k | ) ∇ u k } is bounded in W , (Ω , R N × n ). Therefore, there exist a subsequenceof { u k } , still denoted by { u k } , and a function U ε ∈ W , (Ω , R N × n ) such that(6.45) a ε ( |∇ u k | ) ∇ u k → U ε in L (Ω , R N × n ), a ε ( |∇ u k | ) ∇ u k ⇀ U ε in W , (Ω , R N × n ) . Equation (6.41) entails that ∇ u k → ∇ u ε a.e. in Ω. As a consequence,(6.46) a ε ( |∇ u k | ) ∇ u k → a ε ( |∇ u ε | ) ∇ u ε a.e. in Ω.From equations (6.46) and (6.45) one infers that(6.47) a ε ( |∇ u ε | ) ∇ u ε = U ε ∈ W , (Ω , R N × n ) . Furthermore, passing to the limit as k → ∞ in (6.44) yields k a ε ( |∇ u ε | ) ∇ u ε k W , (Ω , R N × n ) ≤ C (cid:0) k f k L (Ω , R N ) + k a ε ( |∇ u ε | ) ∇ u ε k L (Ω , R N × n ) (cid:1) . (6.48) 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 k a ε ( |∇ u ε | ) ∇ u ε k W , (Ω , R N × n ) ≤ C (6.49)for some constant C , independent of ε . By the last inequality, the family of functions { a ε ( |∇ u ε | ) ∇ u ε } isuniformly bounded in W , (Ω , R N × n ) for ε ∈ (0 , { ε m } converging to 0 anda function U ∈ W , (Ω , R N × n ) such that(6.50) a ε m ( |∇ u ε m | ) ∇ u ε m → U in L (Ω , R N × n ), a ε m ( |∇ u ε m | ) ∇ u ε m ⇀ U in W , (Ω , R N × n ) . An argument parallel to that of the proof of (5.25) yields ∇ u ε m → ∇ u a.e. in Ω.(6.51)We omit the details, for brevity. Let us just point out that, in this argument, one has to make use of theinequality Z Ω B ( |∇ u ε m | ) dx ≤ C (cid:18) Z Ω e B ( | f | ) dx + B ( ε m ) (cid:19) , (6.52)instead of (5.12). Inequality (6.52) easily follows on choosing u ε m as a test function in the definition of weaksolution to problem (6.38), with ε = ε m . Coupling equations (6.50) and (6.51) implies that(6.53) a ( |∇ u | ) ∇ u = U ∈ W , (Ω , R N × n ) . On the other hand, on exploiting equations (6.51) and (6.39), the dominated convergence theorem for Lebesgueintegrals and inequality (6.12) (applied with a and u replaced by a ε m and u ε m ) one can deduce thatlim m →∞ k a ε m ( |∇ u ε m | ) ∇ u ε m k L (Ω , R N × n ) = k a ( |∇ u | ) ∇ u k L (Ω , R N × n ) ≤ C k f k L (Ω , R N ) (6.54)for some constant C = C ( n, N, i a , s a , | Ω | ). Combining equations (6.48), (6.50), (6.53) and (6.54) yields(6.55) k a ( |∇ u | ) ∇ u k W , (Ω , R N × n ) ≤ C k f k L (Ω , R N ) for some constant C = C ( n, N, i a , s a , L Ω , d Ω , K Ω ). Step 2 . Assume now that the temporary condition (6.36) is still in force, but Ω is just as in the statement. Let { Ω m } be a sequence of open sets approximating Ω in the sense of Lemma 6.3. For each m ∈ N , denote by u m the weak solution to the Dirichlet problem(6.56) ( − div ( a ( |∇ u m | ) ∇ u m ) = f in Ω m u m = 0 on ∂ Ω m , where f is continued by 0 outside Ω. Owing to our assumptions on Ω m , inequality (6.55) holds for u m . Thereby,there exists a constant C ( n, N, i a , s a , L Ω , d Ω , K Ω ) such that k a ( |∇ u m | ) ∇ u m k W , (Ω , R N × n ) ≤ k a ( |∇ u m | ) ∇ u m k W , (Ω m , R N × n ) ≤ C k f k L (Ω m , R N ) = C k f k L (Ω , R N ) . (6.57)Observe that the dependence of the constant C on the specified quantities, and, in particular, its independenceof m , is due to properties (6.33) and (6.34).Thanks to (6.57), the sequence { a ( |∇ u m | ) ∇ u m } is bounded in W , (Ω , R N × n ), and hence there exists a subse-quence, still denoted by { u m } and a function U ∈ W , (Ω , R N × n ), such that a ( |∇ u m | ) ∇ u m → U in L (Ω , R N × n ),(6.58) a ( |∇ u m | ) ∇ u m ⇀ U in W , (Ω , R N × n ) . We now notice that there exists α ∈ (0 , m , such that u m ∈ C ,α loc (Ω , R N ), and for every openset Ω ′ ⊂⊂ Ω with a smooth boundary k u m k C ,α (Ω ′ , R N ) ≤ C, (6.59) 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 k∇ u m k C α (Ω ′ , R N × n ) ≤ C. (6.60)Since the function f satisfies assumption (6.36), a basic energy estimate for weak solutions tells us that Z Ω m B ( |∇ u m | ) dx ≤ C (6.61)for some constant C independent of m . Thus, as a consequence of the Poincar´e inequality (4.4), Z Ω m B ( | u m | ) dx ≤ C , (6.62)where the constant C is independent of m , for | Ω m | is uniformly bounded. Inequalities (6.60) and (6.62), via aSobolev type inequality, tell us that k u m k L ∞ (Ω ′ , R N ) ≤ C (6.63)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 ∈ C (Ω , R N ) such that(6.64) u m → v and ∇ u m → ∇ v in Ω.Hence,(6.65) a ( |∇ u m | ) ∇ u m → a ( |∇ v | ) ∇ v in Ω.Owing to equations (6.65) and (6.58),(6.66) a ( |∇ v | ) ∇ v = U ∈ W , (Ω , R N × n ) . Next, we pick a test function ϕ ∈ C ∞ (Ω , R N ) (continued by 0 outside Ω) in the definition of weak solution toproblem (6.56). Passing to the limit as m → ∞ in the resulting equation yields, via (6.58) and (6.66),(6.67) Z Ω a ( |∇ v | ) ∇ v · ∇ ϕ dx = Z Ω f · ϕ dx . Inequality (6.61) tells us that R Ω B ( |∇ u m | ) dx ≤ C for some constant C independent of m . Therefore, this in-equality is still true if u m is replaced with v . Consequently, thanks to inequality (4.22), R Ω e B ( a ( |∇ v | ) |∇ v | ) dx < ∞ . Thus, since under our assumptions on a the space C ∞ (Ω , R N ) is dense in W ,B (Ω , R N ), equation (6.67)holds for every function ϕ ∈ W ,B (Ω , R N ) 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 aconstant C = C ( n, N, i a , s a , L Ω , d Ω , K Ω ) such that(6.68) k a ( |∇ u | ) ∇ u k W , (Ω , R N × n ) ≤ C k f k L (Ω , R N ) . Step 3 . Finally, assume that both Ω and f are as in the statement. The definition of approximable solutionentails that there exists a sequence { f k } ⊂ C ∞ (Ω , R N ), such that f k → f in L (Ω , R N ) and the sequence of weaksolutions { u k } ⊂ W ,B (Ω , R N ) to problems (6.2), fulfills u k → u and ∇ u k → ∇ u a.e. in Ω. An application ofinequality (6.68) with u and f replaced by u k and f k , tells us that a ( |∇ u k | ) ∇ u k ∈ W , (Ω , R N × n ), and k a ( |∇ u k | ) ∇ u k k W , (Ω , R N × n ) ≤ C k f k k L (Ω , R N ) ≤ C k f k L (Ω , R N ) (6.69)for some constants C and C , depending on N , i a , s a and Ω. Therefore, the sequence { a ( |∇ u k | ) ∇ u k } is boundedin W , (Ω , R N × n ), whence there exists a subsequence, still indexed by k , and a function U ∈ W , (Ω , R N × n )such that(6.70) a ( |∇ u k | ) ∇ u k → U in L (Ω , R N × n ), a ( |∇ u k | ) ∇ u k ⇀ U in W , (Ω , R N × n ) . Inasmuch as ∇ u k → ∇ u a.e. in Ω, one hence deduces that a ( |∇ u | ) ∇ u = U ∈ W , (Ω , R N × n ). Thereby, thesecond inequality in (2.19) follows from equations (6.69) and (6.70). The first inequality in (2.19) holds trivially.The proof is complete. (cid:3) Proof of Theorem 2.6.
The proof parallels that of Theorem 2.7. However,
Step 2 requires a variant. The se-quence { Ω m } of bounded sets, with smooth boundaries, coming into play in this step has to be chosen in such away 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 K Ω if Ω is convex. (cid:3) 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 “Directand inverse problems for partial differential equations: theoretical aspects and applications”, grant number201758MTR2 (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.
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Fakult¨at f¨ur Mathematik, University Bielefeld, Universit¨atsstrasse 25, 33615 Bielefeld, Germany
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Email address : [email protected] Fakult¨at f¨ur Mathematik, University Bielefeld, Universit¨atsstrasse 25, 33615 Bielefeld, Germany
Email address : [email protected] Department of Mathematics, Link¨oping University,E-581 83 Link¨oping, Sweden and Peoples Friendship Univer-sity of Russia (RUDN University), 6 Miklukho-Maklay St, Moscow, 117198, Russian Federation
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