Global gradient estimates in elliptic problems under minimal data and domain regularity

Transcription

1 Global gradient etimate in elliptic problem under minimal data and domain regularity Andrea Cianchi Dipartimento di Matematica e Informatica U.Dini, Univerità di Firenze Piazza Ghiberti 27, 522 Firenze, Italy Vladimir Maz ya Department of Mathematic, Linköping Univerity, SE Linköping, Sweden and Department of Mathematical Science, M&O Building Univerity of Liverpool, Liverpool L69 3BX, UK Introduction The preent paper i mainly devoted to report on ome contribution by the author on integrability propertie of the gradient of olution to boundary value problem for nonlinear elliptic equation in divergence form. Thi i a very claical iue in the theory of partial differential equation, whoe modern development can be traced back to the work of everal author in the late fiftie and early ixtie of the lat century. We do not attempt to provide an even partial account of the vat bibliography on thi topic. We jut refer to the the monograph [BF, Gia, GT, Gi, LU, Mo] and to the recent urvey paper [Mi]. The leitmotif of our invetigation i the aim at aumption on the regularity of the ground domain and of the precribed data, which are minimal, in a ene, for a certain gradient bound to hold. A ditinctive feature of our approach i in the derivation of etimate which are flexible enough to be applied in the proof of gradient bound for a wide choice of norm. Mot of the relevant etimate are formulated in term of pointwie inequalitie for the ditribution function of the length of the gradient, or, equivalently, for it decreaing rearrangement. With thi tool at dipoal, global bound for any rearrangement invariant norm of the gradient of olution to either Dirichlet or Neumann boundary value problem are imply reduced to one-dimenional inequalitie for Hardy type operator. The latter depend both on the cla of elliptic differential operator under conideration, and on the regularity of the domain. The firt et of reult to be preented i focued on boundary value problem involving nonlinear differential operator with a quite general tructure, and need not enjoy additional moothne propertie. When dealing with thi kind of problem, no (uniform) gradient integrability can be expected beyond the natural energy level dictated by the nonlinearity of the problem. Moreover, domain regularity only play a role when Neumann boundary condition are impoed, and can effectively be precribed in term of either of two function - the ioperimetric Mathematic Subject Claification: 35J25, 35B45. Keyword: Nonlinear elliptic equation, boundary value problem, gradient etimate, capacity, perimeter, rearrangement.

2 2 function and the iocapacitary function - which are aociated with the domain and reflect ome of it geometric-functional propertie. Thee function were introduced in [Ma, Ma3] to provide neceary and ufficient condition for the validity of Sobolev type embedding. The ue of ioperimetric inequalitie in the tudy of Dirichlet and Neumann problem for (linear) elliptic equation wa initiated in [Ma2, Ma3], where variou a priori etimate for olution and unique olvability reult were obtained under weak aumption on coefficient, data and domain. The tandard ioperimetric inequality in R n wa exploited in [Ta] and [Ta2] to etablih ymmetrization comparion principle for linear and nonlinear, repectively, elliptic Dirichlet problem. Thi line of reearch ha ubequently been developed in a rich literature; reult and reference an thi topic can be found e.g. in the book [Ka, Ke], and in the urvey paper [Tr]. After recalling and rephraing ome fundamental reult along thi direction, we decribe an alternative approach in the analyi of Neumann problem, which exploit the iocapacitary function. Such an approach can actually lead to tronger reult than thoe obtained via the ioperimetric function when domain with complicated geometric configuration are taken into account. The econd part of our urvey deal with boundary value problem for differential operator having a pecial tructure and ome moothne, which allow for higher gradient integrability. Under harp integrability aumption on the curvature of the boundary of the domain, a rearrangement etimate for the gradient of olution, both to Dirichlet and Neumann problem, i exhibited. Noticeably, thi etimate provide a pointwie bound for a power of the rearrangement of the gradient which depend linearly on the rearrangement of the datum. A a conequence, it tranlate verbatim the linear theory of integrability of the gradient of olution to boundary value problem for the Laplace equation to a parallel theory for nonlinear problem. A an intermediate tep, of independent interet, we how an L bound for the gradient under minimal integrability aumption on the right-hand ide of the equation. We emphaize that the bound in quetion turn out to hold alo for ytem. 2 Part I: equation with general nonlinearitie Thi ection i devoted to equation ubject to cutomary ellipticity condition, without any extra moothne aumption. Specifically, we deal with equation of the form (2.) div(a(x, u)) = f(x) in Ω, where Ω i a domain, namely a connected open et in R n, n 2, having finite Lebegue meaure Ω, f i an integrable function, and a : Ω R n R n i a Carathéodory function uch that, for a.e. x Ω, (2.2) a(x, ξ) ξ ξ p for ξ R n, for ome exponent p >. Although no additional hypothei on the function a(x, ξ) i really needed for our etimate, in order to enure that uitable notion of olution are meaningful we alo aume that a function h L p (Ω) and a contant C exit uch that, for a.e. x Ω, (2.3) a(x, ξ) C ξ p + h(x) for ξ R n. p Here, p = p, the Hölder conjugate of p. The concluion that can be derived on gradient ummability propertie of olution to boundary value problem for equation (2.) depend on whether Dirichlet or Neumann condition are impoed. In the former cae, only the meaure of Ω i relevant, wherea in the latter cae it regularity play a role a well.

3 3 2. Dirichlet problem Here, we are concerned with Dirichlet problem obtained by coupling equation (2.) with homogeneou boundary condition, namely { div(a(x, u)) = f(x) in Ω (2.4) u = on Ω. If f ha a ufficiently trong degree of integrability to belong to W,p (Ω), the dual of the Sobolev pace W,p (Ω), then a weak olution u W,p (Ω) to problem (2.4) i well defined by requiring that (2.5) a(x, u) φ dx = fφ dx Ω for every φ W,p (Ω). For intance, by the tandard Sobolev embedding theorem for W,p (Ω), thi i certainly the cae when f L q (Ω) with q np np n+p if p < n, q > if p = n, and q if p > n. Under the additional trict monotonicity aumption that (2.6) [a(x, ξ) a(x, η)] (ξ η) > for ξ, η R n with ξ η, the exitence and uniquene of a weak olution to problem (2.4) i well known, and follow via the Browder-Minthy theory of monotone operator. On the other hand, if f / W,p (Ω), then olution to (2.4) can till be conidered, but have to be interpreted in a uitable generalized ene. A hown by claical example [Se], allowing olution jut in the ditributional ene may lead to pathological, non-uniquene phenomena. The notion of entropy olution, introduced in [BBGGPV], turn out to be well uited for our framework. It definition i briefly recalled below. Let u mention that, a poteriori, uch a definition turn out to be equivalent to other definition (uch a renormalized olution [M, M2], and olution obtained a limit of approximation [DaA]) available in the literature. Given any t, t 2 R, with t < t 2, let T t,t 2 : R R be the function defined a t if < t (2.7) T t,t 2 () = if t t 2 t 2 if t 2 <. For p, et (2.8) W,p,T (Ω) = {u i a meaurable function in Ω: T t,t(u) W,p (Ω) for t > }. Then [BBGGPV, Lemma 2.] enure that, for every u W,p,T (Ω), a unique meaurable function V u : Ω R n exit uch that (2.9) ( T t,t (u) ) = V u χ { u <t} a.e. in Ω for every t >, where χ E denote the characteritic function of the et E. Furthermore, a function u W,p,p,T (Ω) belong to W (Ω) if and only if u L p (Ω) and V u L p (Ω). In thi cae, V u = u, the weak gradient of u. In what follow, with abue of notation, for every u W,p,T (Ω) we denote V u by u. Ω

4 4 A function u W,p,T (Ω) i called an entropy olution to the Dirichlet problem (2.4) if (2.) a(x, u, u) ( T t,t 2 (u φ) ) dx f(x)t t,t 2 (u φ) dx Ω for every φ W,p (Ω) L (Ω) and every t < t 2. For any f L (Ω) there exit a unique (entropy) olution u to the Dirichlet problem (2.4) [BBGGPV, Theorem 6.]. Let u warn that uch a olution need not be a weakly differentiable function in general, and, even if it i o, it weak gradient need to belong to the energy pace L p (Ω) if f ha not a ufficiently high integrability degree. Uniform etimate for norm of u tronger than L p (Ω) cannot hold without further aumption on the function a(x, ξ) (in addition to (2.2)), and on Ω. Thi fact i well known even in the cae of linear equation [Me]. Thu, when f W,p (Ω), the olution u to (2.4) i in fact a weak olution, and hence u L p (Ω); thu, there i not much to add a far a the integrability of u i concerned. On the other hand, if f / W,p (Ω), the quetion arie of how the integrability degree of f i reflected by that of u. Thi quetion can be poed in the framework of all rearrangement-invariant (quai-)norm, in a ene the widet cla of (quai-)norm which only depend on integrability propertie of function. Let u briefly recall a few baic fact on thi function pace etting. A quai-normed function pace X(Ω) on Ω i a linear pace of meaurable function on Ω equipped with a functional X(Ω), a quai-norm, enjoying the following propertie: (i) g X(Ω) > if g ; λg X(Ω) = λ g X(Ω) for every λ R and g X(Ω); g +g 2 X(Ω) c( g X(Ω) + g 2 X(Ω) ) for ome contant c and for every g, g 2 X(Ω); (ii) g g 2 a.e. in Ω implie g X(Ω) g 2 X(Ω) ; (iii) g k g a.e. implie g k X(Ω) g X(Ω) a k ; (iv) if G i a meaurable ubet of Ω and G <, then χ G X(Ω) < ; (v) for every meaurable ubet G of Ω with G <, there exit a contant C uch that G g dx C g X(Ω) for every g X(Ω). The pace X(Ω) i called a Banach function pace if (i) hold with c =. In thi cae, the functional X(Ω) i actually a norm which render X(Ω) a Banach pace. The decreaing rearrangement g : [, ) [, ] of a meaurable function g on Ω i defined a g () = up{t : {x Ω : g(x) > t} > } for. In other word, g i the (unique) non increaing, right-continuou function in [, ) which i equimeaurable with g. We alo et g () = g (r)dr for >, and oberve that g g, ince g i non-increaing. A quai-normed function pace (in particular, a Banach function pace) X(Ω) i aid to be rearrangement-invariant if there exit a quai-normed function pace X(, Ω ) on the interval (, Ω ), called the repreentation pace of X(Ω), having the property that (2.) g X(Ω) = g X(, Ω ) for every g X(Ω). Obviouly, if X(Ω) i a rearrangement-invariant quai-normed pace, then (2.2) g X(Ω) = g 2 X(Ω) if g = g 2. Ω

5 5 Let u mention that, if X(Ω) i a Banach function pace, property (2.2) i in fact equivalent to the exitence of a repreentation pace X(, Ω ) fulfilling (2.). For cutomary pace X(Ω), an expreion for the quai-norm X(, Ω ) i immediately derived from that of X(Ω), via elementary propertie of rearrangement. Cutomary intance of rearrangement-invariant quainormed pace are Lebegue pace, Lorentz pace, Lorentz-Zygmund pace, Orlicz pace. Their definition will be recalled below. We refer the reader to [BS] for a comprehenive treatment of rearrangement-invariant pace. A paradigmatic reult which can give the flavor of the material collected in thi paper i the following theorem from [Ta2]. Theorem 2. Let u be the olution to the Dirichlet problem (2.4). If < r p, then ( Ω (2.3) u L r (Ω) (nωn /n ) p r n (p ) ( Here, ω n = π n/2 /Γ( + n 2 ), the meaure of the unit ball in Rn. ) r ) f p r (ρ)dρ d. Remark 2.2 Theorem 2. can be interpreted a a comparion principle between gradient norm of the olution u to problem (2.4), and the (radially ymmetric) olution to a ymmetrized problem. Indeed, the right-hand ide of inequality (2.3) agree with the L r norm of the gradient of the (radially decreaing) olution to the p-laplace equation in a ball, with the ame meaure a Ω, and with a right-hand ide which i radially decreaing and whoe decreaing rearrangement equal f. Owing to Theorem 2., bound for any norm u L r (Ω), with r p, in term of another rearrangement invariant quai-norm of the datum f, are reduced to one-dimenional Hardytype inequalitie. Corollary 2.3 Let < r p and let X(Ω) be a rearrangement invariant quai-normed pace uch that ( ) (2.4) n p (p ) p ϕ(ρ)dρ C ϕ L r (, Ω ) X(, Ω ), for ome contant C and every nonnegative and non-increaing function ϕ X(, Ω ). If f X(Ω) and u i the olution to the Dirichlet problem (2.4), then (2.5) u L r (Ω) C(nω /n n ) p p f X(Ω). Corollary 2.3 can be exploited, for intance, to derive etimate for Lebegue norm of the gradient in term of Lorentz norm of the datum. Recall that, if either q (, ] and k (, ], or q = and k (, ], the Lorentz pace L q,k (Ω) i defined a the et of all meaurable function g on Ω for which the expreion (2.6) g L q,k (Ω) = q k g () L k (, Ω ) i finite. On ha that L q,q (Ω) = L q (Ω)

6 6 for every q [, ]. The pace L q, (Ω) i called Marcinkiewicz pace, or weak Lebegue pace. Moreover, L q,k (Ω) L q,k 2 (Ω) if k < k 2, and, L q,k loc (Ω) L q 2,k 2 loc (Ω) if q > q 2 and k, k 2 are admiible exponent in (, ]. If q >, then (2.7) q k g () L k (, Ω ) q k g () L k (, Ω ), up to multiplicative contant depending on q and k. Furthermore, if either q > and k [, ], or q = k =, then L q,k (Ω) i in fact a Banach function pace, up to equivalent norm. The following reult appear in [Ta] in the cae of linear equation, and in [AFT] in the nonlinear cae. np np n+p Theorem 2.4 Let p (, n), and let < q. Let u be the olution to the Dirichlet nq q, problem (2.4), with f L n q (Ω). Then there exit a contant C uch that In particular, u nq(p ) L n q (Ω) u nq(p ) L n q (Ω) C f C f p nq q, L n q (Ω) p L q (Ω).. Gradient regularity in Marcinkiewicz pace, for q =, i etablihed in [BBGGPV]. It alo follow from a counterpart for Dirichlet problem of Propoition 2.3, Subection 2.2, which deal with Neumann problem. The cae when p = n and q = can be found in [DHM]. Bound for more general rearrangement-invariant norm of u, till weaker than L p (Ω), can be etablihed through the following pointwie etimate for u. The proof of uch etimate exploit an argument imilar to that ued for [ACMM, Theorem 3.3], and omewhat enhance a reult in the ame direction from [AFT]. Theorem 2.5 Let u be the olution to the Dirichlet problem (2.4). Then there exit a contant C = C(n, p) uch that (2.8) u () C(n, p) ( Ω 2 τ p n ( τ ) ) p p f (ρ) dρ dτ for (, Ω ). Theorem 2.5 ha an obviou corollary, by the very definition of rearrangement-invariant quai-norm. Corollary 2.6 Let X(Ω) be a quai-normed rearrangement-invariant pace and let f X(Ω). Let u be the olution to the Dirichlet problem (2.4). Aume that Y (Ω) i another quai-normed rearrangement-invariant pace uch that (2.9) ( Ω 2 τ p n ( τ ) p ϕ(ρ) dρ dτ ) p C ϕ Y (, Ω ) p, X(, Ω ) for ome contant C and every nonnegative and non-increaing function ϕ X(, Ω ). Then there exit a contant C = C (C, n, p) uch that (2.2) u Y (Ω) C f p X(Ω).

7 7 A an application of Corollary 2.6, let u mention a gradient bound for Lorentz norm ee [Ta] (linear equation) and [AFT] (nonlinear equation). Theorem 2.7 Let < p < n, max{, n/(np n+k)} < q < np/(np n+p) and < k <. Let u be the olution to the Dirichlet problem (2.4), with f L q,k (Ω). Then there exit a contant C uch that u L nq(p ) n q,(p )k (Ω) C f p L q,k (Ω). Let u mention that gradient etimate in Lorentz pace for local olution to nonlinear elliptic problem are proved in [Mi]. 2.2 Neumann problem We focu here on the Neumann problem { div(a(x, u)) = f(x) in Ω (2.2) a(x, u) n = on Ω, where n denote the outward unit normal on Ω. Given p [, ], we define the Sobolev type pace { } V,p (Ω) = u W, loc (Ω) : u Lp (Ω). If Ω i connected, and B i any ball uch that B Ω, then V,p (Ω) i a Banach pace equipped with the norm u V,p (Ω) = u L p (B) + u L p (Ω). Note that replacing B by another ball reult in an equivalent norm. Moreover, if Ω i regular enough, for intance a Lipchitz domain, then, by a Poincaré type inequality, V,p (Ω) L p (Ω), and hence V,p (Ω) = W,p (Ω), the uual Sobolev pace. When a atifie aumption (2.2) (2.3), and f belong to the topological dual V,p (Ω) of V,p (Ω), a weak olution to problem (2.2) i well defined a a function u V,p (Ω) uch that (2.22) a(x, u) φ dx = fφ dx Ω for every φ V,p (Ω). Under the additional monotonicity aumption (2.6), and the compatibility condition (2.23) f(x)dx =, Ω the exitence and uniquene (up to additive contant) of a weak olution to problem (2.2) can be derived via a rather tandard application of the Browder-Minthy theory of monotone operator. If f / V,p (Ω), a generalized notion of olution to the Neumann problem (2.2) ha to be adopted in the pirit, for intance, of the entropy olution recalled in Section 2.. Let u et (2.24) V,p T (Ω) = {u i a meaurable function in Ω : T t,t(u) V,p (Ω) for t > }. An extended notion of gradient V u, again imply denoted by u, for function u V,p T (Ω) can be introduced via (2.9). One ha that a function u V,p T (Ω) belong to V,p (Ω) if and only if V u L p (Ω), and, in thi cae, V u = u, the weak gradient of u. Ω

8 8 In analogy with (2.), a function u V,p T (Ω) i called an entropy olution to the Neumann problem (2.2) if (2.25) a(x, u, u) ( T t,t 2 (u φ) ) dx f(x)t t,t 2 (u φ) dx Ω for every φ V,p (Ω) L (Ω), and every t < t 2. If Ω i a regular Lipchitz, ay domain, a unique (up to additive contant) entropy olution to the Neumann problem (2.2) can be hown to exit for every f L (Ω). On the other hand, unlike the cae of Dirichlet problem, in general a balance between the regularity of Ω and the degree of integrability of f i needed in order to guarantee the exitence of a olution to the Neumann problem (2.2). We are not going to dicu thi iue here. We refer to the paper [ACMM], where the exitence of generalized olution to Neumann problem in poibly irregular domain i dicued. We limit ourelve to auming that a olution doe exit, and to decribing the a priori gradient etimate which can be derived depending on the (ir)regularity of Ω and of f. In our approach, the regularity of Ω i precribed in term of either the ioperimetric function, or the iocapacitary function of Ω. Thee function are the optimal one in inequalitie between the meaure of ubet of Ω and either their relative perimeter, or their condener capacity, repectively. The ue of each one of thee two function ha it own advantage. The ioperimetric function ha a tranparent geometric character, and it i uually eaier to invetigate. The iocapacitary function can be le imple to tudy, but it i in a ene more appropriate, ince it lead to finer concluion in general. Let u incidentally mention that uitable verion of thee function have alo recently been employed in the analyi of eigenvalue problem on noncompact Riemannian manifold [CM3, CM5]. The ioperimetric function λ Ω : [, Ω /2] [, ) of Ω i defined a (2.26) λ Ω () = inf{p (E, Ω) : E Ω /2} for [, Ω /2]. Here, P (E; Ω) i the perimeter of E relative to Ω, which agree with H n ( M E Ω), where H n denote the (n )-dimenional Haudorff meaure, and M E tand for the eential boundary of E in the ene of geometric meaure theory. The relative ioperimetric inequality tell u that (2.27) λ Ω ( E ) P (E; Ω) for every meaurable et E Ω with E Ω /2, and i a traightforward conequence of definition (2.26). The ioperimetric function λ Ω i explicitly known only for very pecial domain, uch a ball [Ma5, BuZa] and convex cone [LP]. However, variou qualitative and quantitative propertie of λ Ω have been invetigated (in the even more general Riemannian framework), in connection, for intance, with Sobolev inequalitie [HK, Ma, Ma5, MP] and eigenvalue etimate [Ch, Ci2, Ga]. The function λ Ω i known to be trictly poitive in (, Ω /2] when Ω i connected [Ma5, Lemma 3.2.4]. The only piece of information on λ Ω which i relevant in view of our application i it aymptotic behavior a +, which depend on the regularity of Ω. Heuritically peaking, a fater decay of λ Ω to reult in a et Ω with a more irregular geometry. The decay of λ Ω for pecific open et, and cutomary clae of et, i exhibited in the example at the end of thi Subection. A counterpart of Theorem 2. for Neumann problem, which make ue of the ioperimetric function, i the content of Theorem 2.8 below, and can be proved by the technique of [Ma3] in the linear cae. The nonlinear cae for regular domain i contained in [Ma5]; general domain and nonlinearitie are treated in [Ci2]. Ω

9 9 Theorem 2.8 Let u be a olution to the Neumann problem (2.2). If < r p, then there exit a contant C = C(p, r) uch that ( Ω (2.28) u L r (Ω) C λ Ω () r p ( ) r ) f p r (ρ)dρ d. A a conequence of Theorem 2.8, bound for L r norm of the gradient of olution to the Neumann problem (2.2) are reduced to one-dimenional inequalitie for a Hardy type operator depending on Ω jut through λ Ω. Corollary 2.9 Let < r p and let X(Ω) be a rearrangement invariant quai-normed pace uch that ( ) (2.29) λ Ω() p p p ϕ(ρ)dρ C ϕ, X(, Ω ) L r (, Ω ) for ome contant C and every nonnegative and non-increaing function ϕ X(, Ω ). If f X(Ω) and u i the olution to the Neumann problem (2.2), then (2.3) u L r (Ω) C f where C = C (C, p, r). p X(Ω), Corollary 2.9 can be exploited, in combination with Hardy type inequalitie for non-increaing function (ee e.g. [CPSS]), to etablih bound for Lebegue norm of u in term of Lebegue, or Lorentz norm of f, depending on λ Ω. Some pecial cae are dealt with in the example below. An analogue, for Neumann problem, of the rearrangement pointwie etimate of Theorem 2.5 i the content of the next reult. In what follow, we et (2.3) med(u) = up{t R : {u > t} Ω /2}, a median of u. Furthermore, we define u + = u +u 2 and u = u u 2, the poitive and the negative part of u, repectively. Theorem 2. Let u be the olution to the Neumann problem (2.2) atifying med(u) =. Then ( (2.32) u ± 2 Ω /2 ( τ ) ) p p () λ Ω (τ) p f ±(ρ) dρ dτ for (, Ω ). 2 Theorem 2. follow via a direct argument, which parallel the proof of Theorem 2.5, but with the tandard ioperimetric inequality in R n replaced with the relative ioperimetric inequality in Ω. Alternatively, it can be derived from Theorem 2.4 and inequality (2.52) below. Corollary 2. Let X(Ω) be a rearrangement-invariant pace and let f X(Ω). Let u be a olution to the Neumann problem (2.2). Aume that Y (Ω) i a rearrangement-invariant pace uch that (2.33) ( Ω 2 λ Ω (τ) p ( τ ) p ϕ(ρ) dρ dτ ) p C ϕ Y (, Ω ) p, X(, Ω )

10 for ome contant C and every nonnegative and non-increaing function ϕ X(, Ω ). Then there exit a contant C = C (C) uch that (2.34) u Y (Ω) C f p X(Ω). Let u now turn to gradient etimate obtained in term of the iocapacitary function, which are etablihed in [CM] and [ACMM]. The definition of iocapacitary function of Ω i analogou to that of ioperimetric function, provided that the relative perimeter of ubet of Ω i replaced with their condener capacity. Recall that the tandard p-capacity of a et E Ω can be defined, for p, a { (2.35) C p (E) = inf u p dx : u W,p (Ω), u a.e. in ome neighbourhood of E Ω A property i aid to hold C p -quai everywhere in Ω, C p -q.e. for hort, if it i fulfilled outide a et of p-capacity zero. Each function u W,p (Ω) ha a repreentative ũ, called the precie repreentative, which i C p -quai continuou, in the ene that for every ε >, there exit a et A Ω, with C p (A) < ε, uch that f Ω\A i continuou in Ω \ A. The function ũ i unique, up to ubet of p-capacity zero. A tandard reult in the theory of capacity tell u that, for every et E Ω, { (2.36) C p (E) = inf u p dx : u W,p (Ω), u C p -q.e. in E Ω ee e.g. [MZ, Corollary 2.25]. Conitently, the p-capacity C p (E, G) of the condener (E, G), where E G Ω, can be defined a (2.37) { } C p (E, G) = inf u p dx : u W,p (Ω), u C p -q.e. in E and u C p -q.e. in Ω \ G. Ω The p-iocapacitary function ν Ω,p : [, Ω /2) [, ) of Ω i given by (2.38) ν Ω,p () = inf {C p (E, G) : E and G are meaurable ubet of Ω uch that E G Ω, E and G Ω /2} }, }. for [, Ω /2). Clearly, the function ν Ω,p i non-decreaing. Definition 2.38 immediately yield the iocapacitary inequality, which tell u that (2.39) ν Ω,p ( E ) C p (E, G) for every meaurable et E G Ω with G Ω /2. The decay to of ν Ω,p () a + i related to the regularity of Ω, and play a role in our reult. Example of uch decay for ome familie of domain are preented below. Condition, in term of ν Ω,p, for gradient bound, in Lebegue pace, for the olution to the Neumann problem 2.2 read a follow. Theorem 2.2 Aume that f L q (Ω) for ome q [, ]. Let u be a olution to the Neumann problem (2.2). Let < r p. Then there exit a contant C uch that (2.4) u L r (Ω) C f p L q (Ω),

11 if either (i) q >, q(p ) r and (2.4) up << Ω 2 or (ii) < q <, < r < q(p ) and p(p ) + p r q ν Ω,p () <, (2.42) or (iii) q = and (2.43) or (iv) q = and (2.44) Ω /2 Ω /2 Ω /2 ( ) rq p[q(p ) r] d <, ν Ω,p () ( ( ) r p(p ) d <, ν Ω,p () ν Ω,p () ) r p(p ) d r p <. Moreover the contant C in (2.4) depend only on p, q, r and on the left-hand ide either of (2.4), or (2.42), or (2.43) or (2.44), repectively. In the borderline ituation when q =, cae (iv) of Theorem 2.2 can be omewhat improved on calling into play Marcinkiewicz norm, which extend L q, (Ω). Recall that, given a bounded non-decreaing function ω : (, Ω ) (, ), the Marcinkiewicz pace M ω (Ω) aociated with ω i the et of all meaurable function u in Ω uch that the quantity (2.45) u Mω(Ω) = up ω()u () (, Ω ) i finite. The expreion (2.45) i equivalent to a norm, which make M ω (Ω) a rearrangementinvariant pace, if and only if up (, Ω ) ω(ρ) <. Clearly, M ω = L r, (Ω) if ω() = r ω() dρ for ome r. Propoition 2.3 Aume that f L (Ω). Let u be a olution to the Neumann problem (2.2). Let ω Ω,p : (, Ω ) [, ) be the function defined by (2.46) ω Ω,p () = (ν Ω,p p (/2)) p for (, Ω ). Then there exit a contant C = C(p, n) uch that (2.47) u MωΩ,p (Ω) C f p L (Ω). Verion of Theorem 2. and the enuing Corollary 2., which involve the iocapacitary function intead of the ioperimetric function of Ω, have the following form.

12 2 Theorem 2.4 Let u be the weak olution to the Neumann problem (2.2) atifying med(u) =. Then (2.48) u ± () ( 2 Ω /2 2 ( τ ) p f±(ρ) dρ p d( DνΩ,p )(τ) ) p for (, Ω ). Corollary 2.5 Let X(Ω) be a rearrangement-invariant pace and let f X(Ω). Let u be a olution to the Neumann problem (2.2). Aume that Y (Ω) i a rearrangement-invariant pace uch that (2.49) ( Ω ( r ) p ϕ(ρ) dρ d( Dν p p )(τ) ) p C ϕ Y (, Ω ) p, X(, Ω ) for ome contant C and every nonnegative and non-increaing function ϕ X(, Ω ). Then there exit a contant C = C (C) uch that (2.5) u Y (Ω) C f p X(Ω). For any connected open et Ω with finite meaure, we have that (2.5) ν () λ Ω () a +, a hown by an eay variant of [Ma5, Lemma 2.2.5], and, if p >, ( Ω /2 ) dρ p (2.52) ν Ω,p () for (, Ω /2) λ Ω (ρ) p [Ma5, Propoition 4.3.4/]. Owing to inequality (2.52), one can how that Ω /2 2 ( τ ) p f±(ρ) dρ Ω /2 ( τ p d( DνΩ,p )(τ) 2 ) p f±(ρ) dρ dτ λ Ω (τ) p for (, Ω /2), for every f L (Ω). Thu, Theorem 2.4 and Corollary 2.5 are alway at leat a harp a their counterpart, Theorem 2. and Corollary 2., involving the ioperimetric function. A revere inequality in (2.52) doe not hold in general, even up to a multiplicative contant. Thi account for the fact that the reult on the Neumann problem (2.2) which can be derived in term of ν Ω,p can be tronger than thoe reting upon λ Ω ee Example 4 and 5, below. However, the two ide of (2.52) are equivalent when Ω i ufficiently regular, a in Example -3. In thi cae, the gradient bound which follow by exploiting the ioperimetric and the iocapacitary function coincide. Example. (John domain). A bounded open et Ω in R n i called a John domain if there exit a contant c (, ) and a point x Ω having the property that, for every x Ω, there exit a rectifiable curve ϖ : [, l] Ω, parametrized by arclenght, uch that ϖ() = x, ϖ(l) = x, and dit (ϖ(), Ω) c for [, l]. Note that, in particular, any Lipchitz domain i a John domain. The notion of John domain arie in connection with the analyi of holomorphic dynamical ytem and quaiconformal

13 3 mapping, and ha been ued in recent year in the tudy of Sobolev inequalitie. In particular, a reult from [KM] (complementing [HK]) implie that, if Ω i a John domain in R n, then Moreover, if p < n, then λ Ω () n n for (, Ω /2). ν Ω,p () n p n for (, Ω /2), and, if p n, then, for every γ >, ν Ω,p () γ for (, Ω /2), where the notation mean that the left-hand ide i bounded from below by a poitive contant time the right-hand ide. In fact, in the borderline cae when p = n, one can how that ν Ω,p () log n ( ) for (, Ω /2). When < p < n, Theorem 2.2, cae (i) and (iv), yield the etimate (2.53) u q(p )n L n q (Ω) if < q np np+p n, and C f (2.54) u L r (Ω) C f p L q (Ω), p L (Ω), if < r < (p )n, for a olution u to the Neumann problem 2.2. Moreover, from Propoition 2.3, one obtain that u L (p )n, (Ω) C f p L (Ω). Note that inequalitie (2.53) and (2.54) can alo be proved via Corollary 2.9. Example 2. (Hölder domain). Let Ω be a bounded domain in R n with a Hölder boundary with exponent α (, ), and let < p < α (n ) +. By a Sobolev embedding theorem of [La], and [Ma5, Theorem 6.3.3/], we have that αp (2.55) ν Ω,p () n +α for (, Ω /2). Thu, owing to Theorem 2.2, if u i a olution to the Neumann problem 2.2, then (2.56) u L r (Ω) C f if q >, where r = min { q(p )(n +α) n α(q ), p }, and (2.57) u L r (Ω) C f if < r < (p ) ( + p L q (Ω), p L (Ω), α (n )). In fact, by Propoition 2.3, one alo ha that u (p )(+ α C f L (n ) ), (Ω) p L (Ω).

14 4 On the other hand, by [La] again and [Ma5, Corollary 5.2.3] (ee alo [Ci, Theorem ] for the cae n = 2), λ Ω () n n +α for (, Ω /2). Thu, making ue of Corollary 2.9 lead to the ame concluion (2.56) and (2.57). Example 3 (An unbounded funnel). Let ζ : [, ) (, ) be a differentiable convex function uch that lim ρ ζ(ρ) =. Conider the unbounded et Ω = {x R n : x n >, x < ζ(x n )} (ee Figure ), where x = (x, x n ) and x = (x,..., x n ) R n. Aume that (2.58) ζ(ρ) n dρ <, whence Ω <. Let Υ : [, ) [, ) be the function given by By [Ma5, Example 5.3.3/2], Υ(ρ) = nω n ζ(τ) n dτ for ρ >. ρ x n ζ( x n ) Figure : an unbounded funnel λ Ω () ( ζ(υ ()) ) n a +, and, by [Ma5, Example 6.3.6/2], if p >, ( Υ () ν Ω,p () ζ(τ) n Υ ( Ω /2) p dτ) p a +. Aume, for intance, that ζ(ρ) = (+ρ) β for ome β > n. Then λ Ω () β(n ) β(n ) a +,

15 5 and ν Ω,p () β(n )+p β(n ) a +. An application of Theorem 2.2 tell u that, if u i a olution to the Neumann problem (??), then u L r (Ω) C f p L q (Ω), provided that r p and r < q(p )[β(n ) ] q+β(n ). The ame concluion can be derived via Propoition 2.3. Example 4 (Courant-Hilbert) Let u conider the Neumann problem (2.2) in the domain Ω R 2 diplayed in Figure 2. Thi et i exhibited in [CH] a an example of a domain in which the Poincaré inequality fail. In L l Figure 2: Courant-Hilbert domain the figure, L = 2 k and l = δ(2 k ), where k N and δ : [, ) [, ) i any function uch that: δ(2) cδ() for ome c > and for every > ; the function p+ δ() i non-decreaing; the function +ε δ() i non-increaing for ome ε >. One can how that, if p 2, then (2.59) λ Ω () δ ( /2) a +, and (2.6) ν Ω,p () δ ( /2) p 2 a + [CM4]. Aume, for intance, that δ() γ a +, for ome γ (, p + ). Then, from Theorem 2.2 on can infer that (2.6) u L r (Ω) C f p L q (Ω),

16 6 if q >, where r = min { 2pq(p ) The ue of Corollary 2.9 only yield 2p q(p+ γ), p}, and, by Propoition 2.3, u 2p(p ) L p +γ, (Ω) C f u L r (Ω) C f p L q (Ω), p L (Ω). if q >, where r = min { 2q(p ) γq 2q+2, p}, a weaker concluion then (2.6), ince 2pq(p ) 2p q(p+ γ). 2q(p ) γq 2q+2 < Example 5 (Nikodým) We conclude with the highly irregular domain Ω R 2 illutrated in Figure 3, which wa introduced by Nikodým in hi tudy of Sobolev embedding. L l Figure 3: Nikodým domain Here, L = 2 k and l = 2 βk, where β > and k N. One ha that (2.62) λ Ω () β a +, and, if < p < 2, (2.63) ν Ω,p () β a + [Ma5, Section 6.5]. Thu, ince (2.64) ( Ω /2 ) dρ p p(β )+ a +, λ Ω (ρ) p the iocapacitary function ν Ω,p () i not equivalent to ( Ω /2 ) dr p λ Ω (r) for uch a domain Ω. p In fact, the etimate for the gradient of olution to the Neumann problem (2.2), which can

17 7 be derived via the iocapacitary function, are tronger than thoe obtained by the ioperimetric function. To ee thi, note that Theorem 2.2 implie that, if q and r p, then (2.65) u L r (Ω) C f provided that (2.66) r < pq(p ) q(β ) + p. p L q (Ω), Such a concluion i tronger than what follow via Corollary 2.9, which yield (2.65) only for r < q(p ) q(β ) +. 3 Part II: equation with pecial nonlinearitie We focu here on either Dirichlet problem of the form { div(a( u ) u) = f(x) in Ω (3.) u = on Ω, or Neumann problem of the form div(a( u ) u) = f(x) in Ω (3.2) u = on Ω. n We aume that a : [, ) [, ) i of cla C (, ), ta (t) (3.3) < inf t> a(t) up ta (t) t> a(t) <, and there exit p (, ) and c, C > uch that (3.4) ct p ta(t) C(t p + ) for t >. In particular, the tandard p-laplace operator, correponding to the choice a(t) = t p 2, with p >, i included in thi framework, ince i a = a = p 2 for thi choice of a. We hall preent a rearrangement etimate for the gradient of olution to problem (3.) and (3.2), in the pirit of thoe exhibited in Section 2 for problem with general operator. Of coure, the relevant etimate will be eentially tronger, and apt to etablih any integrability property of the gradient, depending on the integrability of the right-hand ide f. Our gradient rearrangement etimate i tated in Subection 3.2. We preliminarily dicu the problem of the global boundednee of the gradient in Subection 3.. With thi regard, let u point out that aumption (3.4) i irrelevant for the reult of Subection 3. to hold. We keep it in force jut for implicity of expoition, ince dropping (3.4) entail the ue of Orlicz-Sobolev pace, intead of the tandard Sobolev pace, a a functional framework - ee [CM2, CM6]. We purue minimal regularity aumption on Ω enuring gradient regularity. Bounded domain Ω whoe boundary Ω W 2 L n, will be allowed. Thi mean that Ω i locally the

18 8 ubgraph of a function of n variable whoe econd-order weak derivative belong to the Lorentz pace L n,. Thi i the weaket poible integrability aumption on econd-order derivative for the firt-order derivative to be continuou, and hence for Ω C, [CP]. Note that, by contrat, more tandard reult on global boundedene of the gradient of olution require that the function whoe ubgraph locally agree with Ω have a modulu of continuity atifying a Dini condition ee [Li, Section 3], [Li2, Theorem 5.], and alo [An, Remark on Lemma A3.]. The cae of arbitrary convex domain i alo covered by our reult. Thi cae i of pecial interet for the Neumann problem (3.2), for which, unlike the cae of Dirichlet problem, even for bounded f an approach via tandard barrier argument doe not apply. Partial contribution in thi connection can be found in [Li4, Example, page 58, and Remark, page 62]. 3. Global boundedne of the gradient In thi Subection we exhibit a minimal integrability aumption on f enuring the global boundedne of the gradient of olution to (3.) and (3.2), and hence the Lipchitz continuity of olution [CM2, CM6]. The relevant aumption amount to requiring that f belong to the Lorentz pace L n, (Ω), which i borderline for the family of Lebegue pace L q (Ω) with q > n, inamuch a L q (Ω) L n, (Ω) L n (Ω) for any uch q. Let u notice that, if f L n, (Ω) and Ω W 2 L n,, the exitence and uniquene (up to additive contant in the Neumann cae) of a weak olution to problem (3.) and (3.2) follow via claical minimization argument for trictly convex functional of which thee problem are the Euler equation. We emphaize that our reult harpen more tandard reult available in the literature even for the Lapalce operator. Theorem 3. Let Ω be a bounded domain in R n, n 3, uch that either Ω W 2 L n,, or Ω i convex. Aume that the function a C (, ) and fulfill (3.3)-(3.4). Let f L n, (Ω), and let u be either the olution to the Dirichlet problem (3.) or to the Neumann problem (3.2). Then (3.5) u L (Ω) C f for ome contant C = C(p, Ω). Some remark about Theorem 3. are in order. p L n, (Ω) Remark 3.2 Theorem 3. hold, at leat if a i monotone (either increaing or decreaing), even in the vector-valued cae, namely if the equation in problem (3.) or (3.2) are replaced by ytem. Thi i poible, owing to the Uhlenbeck tructure of the differential operator, which only depend on the length of the gradient. Recall that, in contrat with the calar cae, olution to nonlinear elliptic ytem with a more general tructure can be irregular. Example are produced in [SY], where the exitence of nonlinear elliptic ytem, with regular differential operator depending only on the gradient, but endowed with olution which are not even bounded, i proved. Earlier example of irregular olution to elliptic ytem are rooted in the paper [DeG], and include [GM] and [Ne]. Remark 3.3 A verion of Theorem 3. alo hold in the cae when n = 2, under the lightly tronger aumption that f L q (Ω) for ome q > n.

19 9 Remark 3.4 The harpne of aumption f L n, (Ω) for the boundedne of the gradient can be demontrated by conidering the Dirichlet problem for the Poion equation { u = f in Ω when Ω i a ball in R n [Ci3]. u = on Ω, Remark 3.5 The aumption Ω W 2 L n, i optimal for the boundedne of the gradient, a long a the regularity of Ω i precribed in term of integrability propertie of the econd-order derivative of the function which locally repreent it boundary. Thi can be hown by example of Dirichlet and Neumann problem for the p-laplace equation in domain whoe boundarie have conical ingularitie [CM7]. Example of the ame nature alo how that the concluion of Theorem 3. may fail under light local non-mooth perturbation of convex domain. Both example involve a domain Ω whoe boundary contain, i mooth outide a neighborhood of, and in uch neighborhood Ω agree with {x = (x, x n ) : x n < L x } for ome number L >. In other word, in a neighborhood of the domain Ω i bounded by an inward cone, whoe aperture i arctan(/l). The hape of Ω far from i immaterial. Aume firt that 2 p n and that the cone in the definition of Ω i very harp, namely that L i very large. Conider the Dirichlet problem { div( u p 2 u) = f(x) in Ω (3.6) u = on Ω. One ha that Ω W 2 L q for every q < n, and, in fact, Ω W 2 L q, for every q < n, but Ω / W 2 L n,. The function f can be choen in uch a way that it i mooth, vanihe in a neighborhood of, and the olution u to (3.6) atifie (3.7) u(x) x α(l) F ( x n / x ) a x, for ome mooth function F : R R, and ome exponent α(l) > uch that lim α(l), L ee [KM]. Thu, given any q > n, we have that u / L q (Ω) for ufficiently large L, even if f i very mooth. If, intead, Ω W 2 L n,, then Theorem 3. enure that u L (Ω) provided that f L n, (Ω), and hence, in particular, if f L (Ω). Suppoe next that the cone in the definition of Ω i almot flat, namely that L i very mall. Clearly, Ω can be contructed in uch a way that it i convex when L =. Conider the Neumann problem for the Laplace equation { u = f(x) in Ω (3.8) u ν = on Ω. One can how that there exit function f which are mooth, vanih in a neighborhood of, and uch that olution u to (3.8) atifie u(x) x β(l) F ( x n / x ) a x,

20 2 up to multiplicative contant independent of x, for ome mooth function F : R R. Here, β(l) i a poitive exponent uch that β(l) < if L i ufficiently cloe to, ee e.g. [KMR, Section 2.3.2]. Thu, if L i ufficiently mall, there exit q < uch that u / L q (Ω). An analogou concluion hold if the Neumann condition in (3.8) i replaced with the Dirichlet condition u = on Ω. Thi i another example howing that the regularity aumption on Ω in Theorem 3. cannot be eentially relaxed. Indeed, boundedne, and high integrability, of u need not be guaranteed, yet for the Laplace equation with a mooth right-hand ide, even if Ω i mooth everywhere, except at a ingle point, in a neighborhood of which Ω i almot flat, and the regularity aumption Ω W 2 L n, i jut lightly relaxed. The ame example alo demontrate that even a mild local perturbation of convexity may affect the concluion of Theorem 3.. Remark 3.6 In [DM], the aumption f L n, loc (Ω) ha independently, and by different technique, been hown to enure the local boundedne of the gradient of local olution to nonlinear equation, and ytem with pecial tructure. The ame aumption alo yield the continuity of the gradient in Ω [DM2, KM2]. The quetion could thu be raied of whether the tronger concluion u C (Ω) hold in Theorem 3.. The anwer i however negative in general, at leat for convex domain. A counterexample can be produced a follow. Conider a dik D R 2, fix a point x D, and conider a equence of point x k D uch that x k x. Let Ω R 2 be the convex domain obtained a the union of D and of the equence of et bounded by D and by the tangent traight-line to D at the point x k. Let f be a nonnegative, mooth, radially ymmetric function about the center of D, which ha compact upport in D. Conider the olution u to the (calar) equation u = f(x) in Ω, ubject to the Dirichlet condition u = on Ω. If u were continuou on Ω, then it hould vanih at the interection of the tangent traight-line to D at the point x k, and hence one would have u(x) = u ν (x) = a well. Thi contradict the fact that u ν (x) v ν (x) >, where v i the olution to the problem v = f(x) in D, with v = on D. Variant of thi example could be exhibited for Neumann problem. Alo, the domain Ω can be modified into another domain whoe boundary i of cla C. Analogue in dimenion n 3 can be produced on replacing the dik with a ball, and the tangent traight-line with the tangent hyperplane. Remark 3.7 A local etimate at the boundary for olution to the Neumann problem (2.2), in convex domain, in the pecial cae when a(t) = t p 2 and f =, ha recently been etablihed in [BL]. 3.2 A pointwie rearrangement etimate for the gradient In order to grap the pirit of the reult of thi ection, baed on [CM7], conider a prototypal problem given by the Poion equation in the whole of R n. If n 3, then the unique olution decaying to at infinity to the equation (3.9) u = f in R n admit a repreentation formula in term of a Riez potential operator, namely f(y) (3.) u(x) = n(n 2)ω n x y n 2 dy for x Rn. R n

21 2 Hence, (3.) u(x) nω n R n f(y) x y n dy for a.e. x Rn. A rearrangement inequality for convolution [On] then implie that (3.2) u () C f (ρ)ρ n dρ for >, where C i a contant depending only on n. Our pointwie gradient rearrangement etimate i a global nonlinear analogue of (3.2) for the olution to (3.) and (3.2), with p [2, n). Theorem 3.8 Let Ω be a domain in R n, n 3, uch that either Ω W 2 L n,, or Ω i convex. Aume that the function a C (, ) and fulfill (3.3)-(3.4) for ome p [2, n). Let f L (Ω), and let u be either the olution to the Dirichlet problem (3.) or to the Neumann problem (3.2). Then there exit a contant C = C(Ω, p) uch that (3.3) u () p C Ω f (ρ)ρ n dρ for (, Ω ). Remark 3.9 A local etimate for the gradient of local olution to nonlinear elliptic equation in term of the Riez potential of the right-hand ide, which extend (3.) in the ame direction a (3.3) extend (3.2) for global olution to nonlinear boundary value problem, i etablihed in [KM]. We preent hereafter ome gradient norm etimate which can be deduced thank to Theorem 3.8. We begin with a general criterion which hold for arbitrary rearrangement-invariant quainorm, which i a traightforward conequence of Theorem 3.8. Corollary 3. Let Ω and u be a in Theorem 3.8. Let X(Ω) and Y (Ω) be rearrangement invariant quai-normed pace on Ω, and let X(, Ω ) and Y (, Ω ), repectively, be their repreentation pace. Aume that there exit a contant C uch that Ω τ (3.4) τ n ϕ(ρ) dρ dτ C ϕ X(, Ω ) Y (, Ω ) for every non-decreaing function ϕ X(, Ω ). If f X(Ω), then there exit a contant C = C (C, Ω, p) uch that (3.5) u p Y (Ω) C f X(Ω). (3.6) Let u notice that inequality (3.4) i equivalent to the pair of inequalitie Ω ϕ(ρ)ρ n dρ C ϕ X(, Ω ) Y (, Ω ) and (3.7) n ϕ(ρ)dρ C ϕ X(, Ω ) Y (, Ω )

22 22 for every non-decreaing function ϕ X(, Ω ). Thi follow from Fubini theorem applied to the double integral appearing in (3.4). Inequality (3.4) i tronger, in general, than jut (3.6), ince, if ϕ : (, Ω ) [, ) i nonincreaing, then (3.8) ϕ() ϕ(ρ)dρ for >. However, inequalitie (3.4) and (3.6) are equivalent in the cae when the quai-norm in X(Ω) fulfil (3.9) ϕ(ρ)dρ C ϕ X(, Ω ) X(, Ω ) for ome contant C and for every ϕ X(, Ω ). Thu, if X(Ω) atifie (3.9), then the ole inequality (3.6) implie the gradient etimate (3.5). The rearrangement-invariant Banach function pace X(Ω) making inequality (3.9) true can be characterized in term of their upper Boyd index I(X). The definition of I(X) relie upon that of dilation operator. The dilation operator D δ : X(, Ω ) X(, Ω ) i defined for δ > and ϕ X(, Ω ) a { ϕ(δ) if δ (, Ω ) D δ ϕ() = otherwie, and i bounded whenever X(Ω) i a rearrangement-invariant Banach function pace [BS, Chapter 3, Prop. 5.]. It norm i denoted by D δ. The Boyd index I(X) of X(Ω) i given by I(X) = lim δ log D δ log(/δ). One ha that I(X) [, ] for every rearrangement-invariant Banach function pace X(Ω). Moreover, inequality (3.9) hold if and only if I(X) < [BS, Theorem 5.5]. We conclude by tating explicit gradient bound for Lebegue, Lorentz, Lorentz-Zygmund, and Orlicz norm for olution to either the Dirichlet problem (3.) or the Neumann problem (3.2). Our firt reult concern gradient etimate in claical Lebegue pace. In the tatement below, C denote a contant independent of u and f. Theorem 3. Let Ω, p and u be a in Theorem 3.8. Aume that f L q (Ω). (i) If q =, then, for every r < n(p ) n, (3.2) u L r (Ω) C f (ii) If < q < n, then (3.2) u qn(p ) L n q (Ω) (iii) If q = n, then,for every r <, C f (3.22) u L r (Ω) C f (v) If q > n, then (3.23) u L (Ω) C f p L (Ω). p L q (Ω). p L. n p L q (Ω).

23 23 Theorem 3. overlap with variou contribution, including [AM, BBGGPV, BG, DMOP, Ma2, Ma3, Li3, Ta, Ta2]. More general and harper etimate in Lorentz and Lorentz-Zygmund pace are contained in the next theorem. The Lorentz-Zygmund pace extend the Lorentz pace, and come into play in certain borderline ituation. If either q (, ], k (, ], β R, or q =, k (, ], β [, ), the Lorentz-Zygmund pace L q,k;β (Ω) i defined a the et of all meaurable function g on Ω making the expreion (3.24) g L q,k;β (Ω) = q k ( + log( Ω /)) β g () L k (, Ω ) finite. If k and the weight multiplying g () on the right-hand ide of (3.24) i non-increaing, then the functional g L q,k;β (Ω) i actually a norm, and L q,k;β (Ω) i a rearrangement-invariant Banach function pace equipped with thi norm. Otherwie, thi functional i only a quai-norm. For certain value of the parameter q, k and β, it i however equivalent to a rearrangementinvariant norm obtained on replacing g by g in the definition. A comprehenive analyi of Lorentz-Zygmund pace i the ubject of [OP]. Theorem 3.2 Let Ω, p and u be a in Theorem 3.8. Aume that f L q,k (Ω). (i) If q = and < k, then (ii) If < q < n and < k, then (iii) If q = n and k >, then u L n(p ) n, (Ω) u L qn(p ) n q C f,k(p ) (Ω) C f p L,k (Ω). u,k(p ); C f L p (Ω) (iv) If either q = n and k, or q > n and < k, then u L (Ω) C f p L q,k (Ω). p L q,k (Ω). p L n,k (Ω). Variou cae of Theorem 3.2 are known, poibly under tronger aumption on Ω ee e.g. [ACMM, AFT, AM, BBGGPV]. Our lat application concern gradient etimate in Orlicz pace. Let A : [, ) [, ] be a Young function, namely a convex function, vanihing at, which i neither identically equal to, nor to. The Orlicz pace L A (Ω) aociated with A i the rearrangement-invariant pace of thoe meaurable function g on Ω uch that the Luxemburg norm g L A (Ω) = inf { λ > : Ω ( g(x) ) } A dx λ i finite. Since we are auming that Ω <, the Orlicz pace L A (Ω) and L B (Ω) agree, up to equivalent norm, if and only if the Young function A and B are equivalent near infinity, in the ene that there exit poitive contant c and t uch that B(t/c) A(t) B(ct) for t t.

24 24 The Young conjugate of A i the Young function à given by Ã(t) = up{t A() : } for t. The Sobolev conjugate, introduced in [C4, C5], of a Young function A uch that (3.25) i the Young function A n defined a ( ) t n dt <, A(t) (3.26) A n (t) = A ( H (t) ) for t, where H : [, ) [, ) i given by ( (3.27) H() = ( ) ) t n /n dt A(t) for, and H i the generalized left-continuou invere of H. Accordingly, given a Young function B uch that (3.28) ( ) t n dt <, B(t) we denote by ( B)n the Sobolev conjugate of B, obtained a in (3.26) (3.27), on replacing A with B. Theorem 3.3 Let Ω, p and u be a in Theorem 3.8. Let A and B be Young function fulfilling (3.25) and (3.28), repectively. Aume that f L A (Ω), and that there exit c > and t > uch that (3.29) B(t) A n (ct) and Ã(t) ( B)n (ct) for t t. Let E be the Young function given by Then E(t) = B(t p ) for t. (3.3) u L E (Ω) C f p L A (Ω). Remark 3.4 Aumption (3.25) and (3.28) are, in fact, irrelevant in the tatement of Theorem 3.3. Indeed, the function A and B can be replaced, if neceary, by Young function equivalent near infinity, which fulfil (3.25) and (3.28). Such a replacement leave the pace L A (Ω) and L B (Ω) unchanged, up to equivalent norm. Theorem 3.3 can be eaily pecialized to the cae when L A (Ω) i a Zygmund pace, namely A(t) i equivalent to t q log α ( + t) near infinity,

25 25 where either q > and α R, or q = and α. In thi cae, the pace L A (Ω) i denoted by L q (logl) α (Ω). Let u notice that the Zugmund pace are indeed a ubcla of the Lorentz- Zygmund pace, ince L q (logl) α (Ω) = L q,q; α q (Ω), up to equivalent norm. If A(t) i equivalent to e tβ near infinity, for ome β >, we denote L A (Ω) by expl β (Ω). Similarly, we ue the notation exp ( expl β) (Ω) for the Orlicz pace aociated with a Young function A(t) equivalent to e etβ e near infinity. Theorem 3.5 Let Ω, p and u be a in Theorem 3.8. Let f L q (logl) α (Ω). (i) If q = and α >, then (3.3) u L n(p ) n (logl) nα n (Ω) (ii) If < q < n and α R, then (3.32) u nq(p ) L n q (logl) n q nα (Ω) (iii) If q = n and α < n, then (3.33) u expl n(p ) n α (Ω) (iv) If q = n and α = n, then (3.34) u exp (expl n(p ) n ) (Ω) C f C f C f C f (v) If either q = n and α > n, or q > n and α R, then (3.35) u L (Ω) C f p L(logL) α (Ω). p L q (logl) α (Ω). p L n (logl) α (Ω). p L n (logl) n (Ω). p L q (logl) α (Ω). Special cae of Theorem 3.5 are known. In particular, ome intance of cae (i) can be found in [BBGGPV, De]. Reference [AFT] A.Alvino, V.Ferone, & G.Trombetti, Etimate for the gradient of olution of nonlinear elliptic equation with L data, Ann. Mat. Pura Appl. (4) 78 (2), [ACMM] A.Alvino, A.Cianchi, V.Maz ya & A.Mercaldo, Well-poed elliptic Neumann problem involving irregular data and domain, Ann. Int. H. Poincaré Anal. Non Linéaire 27 (2), [AM] A.Alvino & A.Mercaldo, Nonlinear elliptic problem with L data: an approach via ymmetrization method, Mediter. J. Math. 5 (28),