Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data
aa r X i v : . [ m a t h . A P ] F e b Lorentz-Morrey global bounds for singular quasilinearelliptic equations with measure data
M.-P. Tran ∗ , T.-N. Nguyen † Abstract
The aim of this paper is to present the global estimate for gradient ofrenormalized solutions to the following quasilinear elliptic problem: ( − div( A ( x, ∇ u )) = µ in Ω ,u = 0 on ∂ Ω , in Lorentz-Morrey spaces, where Ω ⊂ R n ( n ≥ µ is a finite Radonmeasure, A is a monotone Carath´eodory vector valued function defined on W ,p (Ω) and the p -capacity uniform thickness condition is imposed on thecomplement of our domain Ω. It is remarkable that the local gradient es-timates has been proved firstly by G. Mingione in [34] at least for the case2 ≤ p ≤ n , where the idea for extending such result to global ones was alsoproposed in the same paper. Later, the global Lorentz-Morrey and Morreyregularities were obtained by N.C.Phuc in [38] for regular case p > − n .Here in this study, we particularly restrict ourselves to the singular case n − n − < p ≤ − n . The results are central to generalize our techniqueof good- λ type bounds in previous work [40], where the local gradient esti-mates of solution to this type of equation was obtained in the Lorentz spaces.Moreover, the proofs of most results in this paper are formulated globallyup to the boundary results. Keywords: quasilinear elliptic equation; measure data; Lorentz-Morreyspace; capacity uniformly thickness; global bounds, gradient estimates.
Our main purpose in this paper is to establish a global gradient estimate inLorentz-Morrey spaces of solutions (the renormalized solutions) to the followingquasilinear elliptic equations with respect to the given measure datum µ : ( − div( A ( x, ∇ u )) = µ in Ω ,u = 0 on ∂ Ω . (1.1) ∗ Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc ThangUniversity, Ho Chi Minh city, Vietnam; [email protected] † Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh city,Vietnam
1n our study, the given domain Ω is a bounded open subset of R n , n ≥ µ stands for a finite signed Radon measure in Ω. The nonlinear operator A : R n × R n → R is a Carath´eodory vector valued function (that is, A ( ., ξ ) ismeasurable on Ω for every ξ in R n , and A ( x, . ) is continuous on R n for almostevery x in Ω) which satisfies the following growth and monotonicity conditions:for some 1 < p ≤ n : | A ( x, ξ ) | ≤ β | ξ | p − , (1.2) h A ( x, ξ ) − A ( x, η ) , ξ − η i ≥ α (cid:0) | ξ | + | η | (cid:1) p − | ξ − η | , (1.3)for every ( ξ, η ) ∈ R n × R n \ { (0 , } and a.e. x ∈ R n , α and β are positiveconstants.In addition, in order to obtain the global bounds of solution in Lorentz-Morreyspaces, the domain Ω ⊂ R n is under the assumption that its complement R n \ Ω isuniformly p -capacity thick. More precise, we say that the domain R n \ Ω satisfiesthe p -capacity uniform thickness condition if there exist two constants c , r > p (( R n \ Ω) ∩ B r ( x ) , B r ( x )) ≥ c cap p ( B r ( x ) , B r ( x )) , (1.4)for every x ∈ R n \ Ω and 0 < r ≤ r . Here, the p -capacity of any compact set K ⊂ Ω is defined as:cap p ( K, Ω) = inf (cid:26) ˆ Ω |∇ ϕ | p dx : ϕ ∈ C ∞ c , ϕ ≥ χ K (cid:27) , where χ K is the characteristic function of K . This p -capacity density conditionis stronger than the Weiner criterion in [22]: ˆ cap p (( R n \ Ω) ∩ B r ( x ) , B r ( x ))cap p ( B r ( x ) , B r ( x )) ! p − drr = ∞ which characterizes regular boundary points for the Dirichlet problem for the p -Laplace equation. Otherwise, it is weaker than the Reifenberg flatness conditionthat was discussed in various studies [12, 13, 14, 15, 31, 32, 39]. The class of do-mains whose complement satisfies the uniformly p -capacity condition is relativelylarge (including those with Lipschitz boundaries or satisfy a uniform corkscrewcondition). The condition (1.4) is still valid for balls centered outside a uniformly p -thick domain and furthermore, this condition is nontrivial when p ≤ n . Thedefinition and properties of variational capacity can be found in [30].Throughout this paper, the solution to the problem (1.1) is considered inthe sense of renormalized solution , whose definition was presented in [7, 9, 16]and many references therein. More specifically, the datum measure µ is defined in M b (Ω), the space of all Radon measures on Ω with bounded total variation. Notethat, if µ ∈ M b (Ω), then the total variation of µ is bounded positive measure on Ω.It is also remarkable that for every measure µ in M b (Ω) there exists a unique pair2f measures ( µ , µ s ), with µ in M b (Ω) and µ s in M s (Ω), such that µ = µ + µ s ,is µ is nonnegative, so are µ and µ s . Therefore, the measures µ and µ s will becalled the absolutely continuous and the singular part of µ with respect to the p -capacity.The quasilinear elliptic equations with measure data (1.1) and solution regu-larity estimates have been widely studied in several papers in recent years. Forinstance, firstly by L. Boccardo et al. in [9], and later in different works by G.Mingione et al. [23, 24, 33, 34, 35] and N.C. Phuc et al. [2, 3, 27, 36, 37, 38].In [34], G. Mingione firstly proposed the local estimates of solution at least forthe case 2 ≤ p ≤ n , and the extension to global estimates has also been men-tioned by using maximal function. Later, some of other researching approacheshave been studied for different hypotheses of domain Ω, the nonlinear operator A and the case of p . In [37], N.C. Phuc gave the global gradient estimates in theLorentz spaces and later in [38], author also presented his study on the Lorentz-Morrey and Morrey global bounds to this type of equation, for the regular case of2 − n < p ≤ n and Ω is subject to the p -capacity complement thickness condition.There have been further discussions on the global gradient estimates of solutionto this equation, with different possible assumptions. For instance, authors in [27]studied the gradient estimate of solution in Lorentz space under the hypothesesof Ω-Reifenberg domain, for n − n − < p ≤ − n and the nonlinearity A is requiredto satisfy the smallness condition of BMO type. Otherwise, without the assump-tion of Reifenberg flat domain, the gradient estimates were presented under theweaker condition on complement domain of Ω, that is the p -capacity uniformthickness ( p -fat). Later, in our present work, for singular case n − n − < p ≤ − n ,the solution regularity to this quasilinear elliptic equation (1.1) in Lorentz spaces L q,s (Ω) ( q > , < s ≤ ∞ ) were given in [40].In the present paper, our work is studied following the series of works by G.Mingione (in [17], [18], [23, 24], [33, 34]), N.C. Phuc (in [2, 36, 37, 38]), where theglobal bounds of solution to (1.1) were obtained under different hypotheses andassumptions. Herein, our main advantage here is to provide a new continuationresult of solution global bounds in Lorentz-Morrey spaces for singular p , where theproof techniques may be generalized in the same way as our previous work in [40].However, the main new contribution in this paper is that all gradient estimates are global up to the boundary. This research paper gives us a motivation to studyglobal W α,p estimates (0 < α < Let us firstly recall the definition of the
Lorentz space L q,t (Ω) for 0 < q < ∞ and0 < t ≤ ∞ (see in [21]). It is the set of all Lebesgue measurable functions g on Ω3uch that: k g k L q,t (Ω) = (cid:20) q ˆ ∞ λ q L n ( { x ∈ Ω : | g ( x ) | > λ } ) tq dλλ (cid:21) t < + ∞ , (2.1)as t = ∞ . If t = ∞ , the space L q,t (Ω) is the usual weak- L q or Marcinkiewiczspace with the following quasinorm: k g k L q, ∞ (Ω) = sup λ> λ L n ( { x ∈ Ω : | g ( x ) | > λ } ) q . (2.2)In the definition above, the notation L n ( E ) stands for the n -dimensional Lebesguemeasure of a set E ⊂ R n . When t = q , the Lorentz space L q,q (Ω) becomes theLebesgue space L q (Ω).Otherwise, we also give the definition of Lorentz-Morrey spaces . A function g ∈ L q,t (Ω) for 0 < q < ∞ , 0 < t ≤ ∞ is said to belong to the Lorentz-Morreyfunctional spaces L q,t ; κ (Ω) for some 0 < κ ≤ n if k g k L q,t ; κ (Ω) := sup <ρ Let µ = µ + µ s ∈ M b (Ω) , where µ ∈ M (Ω) and µ s ∈ M s (Ω) .A measurable function u defined in Ω and finite almost everywhere is called arenormalized solution of (1.1) if T k ( u ) ∈ W ,p (Ω) for any k > , |∇ u | p − ∈ L r (Ω) for any < r < nn − , and u has the following additional property. For any k > there exist nonnegative Radon measures λ + k , λ − k ∈ M (Ω) concentrated on the sets u = k and u = − k , respectively, such that µ + k → µ + s , µ − k → µ − s in the narrowtopology of measures and that ˆ {| u | 0, which satisfies − div A ( x, ∇ T k ( u )) = µ k in the sense of distribution in Ω for a finite measure µ k in Ω. Definition 2.2 Let u be a measurable function defined on Ω which is finite al-most everywhere, and satisfies T k ( u ) ∈ W , (Ω) for every k > . Then, thereexists a unique measurable function v : Ω → R n such that: ∇ T k ( u ) = χ {| u |≤ k } v, almost everywhere in Ω , for every k > . (2.5) Moreover, the function v is so-called “distributional gradient ∇ u ” of u . u to(1.1). For the proof, we refer the reader to [16, Theorem 4.1]. Remark 2.3 Let Ω is an open bounded domain in R n . Then, there exists C = C ( n, p, α, β ) such that for any the renormalized solution u to (1.1) with a givenfinite measure data µ there holds: k∇ u k L ( p − nn − p , ∞ (Ω) ≤ C [ | µ | (Ω)] p − . (2.6)Let us now state main results of boundedness property of maximal functionand gradient estimates of solution to (1.1) on Lorentz-Morrey spaces, where theproofs would be found in Section 4, respectively. Theorem 2.4 Let n − n − < p ≤ − n and suppose that Ω ⊂ R n is a boundeddomain whose complement satisfies a p -capacity uniform thickness condition withconstants c , r > . Let µ ∈ M b (Ω) , < R < diam (Ω) and the balls D = B R ( x ) , D = B R ( x ) be such that D ∩ Ω = ∅ , where x is fixed in Ω . Then,for any γ ∈ h − p , ( p − nn − (cid:17) and for any renormalized solution u to (1.1) withgiven measure data µ , there exist Θ = Θ( n, p, α, β, c ) > p and constant C > depending on n, p, α, β, c , diam (Ω) /r such that the following estimate L n (cid:16)n ( M ( χ D |∇ u | γ )) /γ > ε − λ, ( M ( χ D µ )) p − ≤ ε p − γ λ o ∩ D (cid:17) ≤ Cε L n (cid:16)n ( M ( χ D |∇ u | γ )) /γ > λ o ∩ D (cid:17) (2.7) holds for any λ > ε − p − γ k∇ u k L γ ( D ) R − nγ , ε ∈ (0 , . Theorem 2.5 Let n − n − < p ≤ − n and suppose that Ω ⊂ R n is a boundeddomain whose complement satisfies a p -capacity uniform thickness condition withconstants c , r > . Then, there exist Θ = Θ( n, p, α, β, c ) > p , β = β ( n, p, α, β ) ∈ (0 , ] and a constant C = C ( n, p, α, β, c , diam (Ω) /r ) > such that for any < q < Θ , < s ≤ ∞ , p − − β ) < θ ≤ n and for any solution u to (1.1) with a finite measure µ ∈ M b (Ω) , there holds sup ρ ∈ (0 ,T ) ,x ∈ Ω ρ − nq + θ − p − k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) + sup ρ ∈ (0 ,T ) ,x ∈ Ω Cρ − nq + θ − p − k ( M ( χ B ρ µ )) p − k L q,s ( B ρ ( x )) , for any γ ∈ h − p , ( p − nn − (cid:17) , where T = diam (Ω) . It can be noticed that in this theorem and in what follows, for simplicity, theset { x ∈ Ω : | g ( x ) | > Λ } is denoted by {| g | > Λ } (in order to avoid the confusionthat may arise). And the fractional maximal function M α of each locally finitemeasure µ by: M α ( µ )( x ) = sup ρ> | µ | ( B ρ ( x )) ρ n − α , ∀ x ∈ R n , < α < n. (2.8)5or the case α = 0, the definition of M α becomes M is essentially the Hardy-Littlewood maximal function M defined for each locally integrable function f in R n by: M ( f )( x ) = sup ρ> B ρ ( x ) | f ( y ) | dy, ∀ x ∈ R n . (2.9)Otherwise, the notion of fractional maximal function M Tα is also defined as: M Tα ( | µ | )( y ) = sup <ρ It refers to [21] that the operator M is bounded from L s ( R n ) to L s, ∞ ( R n ) , for s ≥ , this means, L n ( { M ( g ) > λ } ) ≤ Cλ s ˆ R n | g | s dx, for all λ > . (2.11) Remark 2.7 In [21], it allows us to present a boundedness property of maximalfunction M in the Lorentz space L q,s ( R n ) , for q > as follows: k M ( f ) k L q,s (Ω) ≤ C k f k L q,s (Ω) . (2.12)The following theorem provides our main result of gradient estimate of solu-tion in the Lorentz-Morrey spaces. Theorem 2.8 Let n − n − < p ≤ − n , µ ∈ M b (Ω) and suppose that Ω ⊂ R n is a bounded domain whose complement satisfies a p -capacity uniform thicknesscondition with constants c , r > . Then, there exist Θ = Θ( n, p, α, β, c ) > p , β = β ( n, p, α, β ) ∈ (0 , / and C = C ( n, p, α, β, c , diam (Ω) /r ) > such thatfor any < q < Θ , < s ≤ ∞ , p − − β ) < θ ≤ n and for any solution u to (1.1) with a finite measure µ ∈ L q ( θ − θ ( p − , s ( θ − θ ( p − ; q ( θ − p − (Ω) there holds k∇ u k L q,s ; q ( θ − p − (Ω) ≤ C k| µ | p − k L q ( θ − θ , s ( θ − θ ; q ( θ − p − (Ω) . (2.13) Remark 2.9 In this work, it remarks that at least for the case ≤ p ≤ n , a localbounds of (2.13) was studied by G. Mingione in [34]. This section is intended to obtain the local interior and boundary comparisonestimates that are essential to our development later.In a certain range of singular p , we always suppose that the domain Ω ⊂ R n is a bounded domain whose complement satisfies a p -capacity uniform thicknesscondition with constants c , r > 0. And for simplicity of notation, the constant C we mention in what follows always depends on some given constants n, p and α, β > A .6 .1 Interior Estimates First, we will take our attention to the interior estimates. Let us fix a point x ∈ Ω, for 0 < R ≤ r ( r was given in (1.4)) and µ ∈ M b (Ω). Assume u ∈ W ,p (Ω) being solution to (1.1) and for each ball B R = B R ( x ) ⊂⊂ Ω, weconsider the unique solution w ∈ W ,p ( B R ) + u to the following equation: (cid:26) − div ( A ( x, ∇ w )) = 0 in B R ,w = u on ∂B R . (3.1)In this section, we are going to deal with some basic estimates of renormalizedsolution u to (1.1) in comparison to the solution w to (3.1). For the convenienceof the reader, we repeat the results in relevant materials, via Lemmas 3.1, 3.2,3.3 and 3.4 herein without proofs, then making our proof of Lemma 3.5.We first recall the following version of interior Gehring’s lemma applied to thefunction w defined in equation (3.1), has been studied in [20, Theorem 6.7]. It isalso known as a kind of “reverse” H¨older inequality with increasing supports. Lemma 3.1 Let w be the solution to (3.1) . Then, there exist constants Θ =Θ( n, p, α, β ) > p and C = C ( n, p, α, β ) > such that the following estimate B ρ/ ( y ) |∇ w | Θ dx ! ≤ C B ρ ( y ) |∇ w | p − dx ! p − (3.2) holds for all B ρ ( y ) ⊂ B R ( x ) . The next lemma gives an estimate for the difference ∇ u − ∇ w . These resultswere described and proved in [27, Lemma 2.2, 2.3]. Lemma 3.2 Let w be solution to (3.1) . Then, for any − p ≤ γ < ( p − nn − ≤ ,there is a constant C = C ( n, p, α, β ) > such that: B R ( x ) |∇ ( u − w ) | γ dx ! γ ≤ C (cid:20) | µ | ( B R ( x )) R n − (cid:21) p − + C | µ | ( B R ( x )) R n − B R ( x ) |∇ u | γ dx ! − pγ . (3.3)The following lemma comes from the standard interior H¨older continuity ofsolutions, that can be found in [20, Theorem 7.7]. Lemma 3.3 Let w be solution to (3.1) . Then, there exists a constant β = β ( n, p, α, β ) ∈ (0 , / such that: B ρ ( y ) | w − w B ρ ( y ) | p dx ! p ≤ C (cid:16) ρr (cid:17) β B r ( y ) | w − w B r ( y ) | p dx ! p , or any y ∈ B R ( x ) with B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) . Moreover, there exists aconstant C = C ( n, p, α, β ) > such that we have the following estimate: B ρ ( y ) |∇ w | p dx ! p ≤ C (cid:16) ρr (cid:17) β − B r ( y ) |∇ w | p dx ! p , (3.4) holds for any y ∈ B R ( x ) such that B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) . It can be noticed that the denotation w B ρ ( y ) indicates the average integral of w over the ball B ρ ( y ). Applying Lemma 3.2, the inequality (3.4) can be furtherimproved as in the following lemma. Lemma 3.4 Let w be solution to (3.1) . Then, for any Θ ∈ (0 , p ] , there existconstants β = β ( n, p, α, β ) ∈ (0 , / , C = C ( n, p, α, β, Θ) > , there holds B ρ ( y ) |∇ w | Θ dx ! ≤ C (cid:16) ρr (cid:17) β − B r ( y ) |∇ w | Θ dx ! , for any y ∈ B R ( x ) such that B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) . Lemma 3.5 Let β ∈ (0 , / be as in Lemmas 3.3 and 3.4. Then, for any δ ∈ h − n − pp − , β (cid:17) , there exists a constant C = C ( n, p, α, β, c , β ) > such that forany B ρ ( y ) ⊂ B r ( y ) ⊂⊂ Ω : ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:16) M T θ ( | µ | )( y ) (cid:17) p − ρ nγ + δ − , (3.5) where θ = 1 + ( p − − δ ) , T = diam (Ω) , and < ρ < T . In order to prove this Lemma 3.5, it will be necessary to refer to Lemma 3.6in [29, Lemma 1.4] as follows, where its proof can be found therein. Lemma 3.6 Let φ ( t ) be a nonnegative and nondecreasing function on [0 , R ] .Suppose that φ ( ρ ) ≤ A h(cid:16) ρr (cid:17) α + ε i φ ( r ) + Br β , for any < ρ ≤ θr < R , with A, B, α, β nonnegative constants and θ ∈ (0 , and β < α . Then, for any γ ∈ ( β, α ) , there exists a constant ε = ε ( A, α, β, γ, θ ) such that if ε < ε we have for all < ρ ≤ r ≤ R : φ ( ρ ) ≤ C h(cid:16) ρr (cid:17) γ φ ( r ) + Bρ β i , where C is a positive constant depending on A, α, β, γ . In particular, we have forany < r ≤ R : φ ( r ) ≤ C (cid:20) φ ( R ) R γ r γ + Br β (cid:21) . roof of Lemma 3.5. First of all, for 0 < ρ ≤ r/ 2, let us take B r ( y ) ⊂⊂ Ω, where B ρ ( y ) ⊂ B r ( y ).By making use of Lemma 3.2 with B R = B r ( y ), one gives: B r ( y ) |∇ ( u − w ) | γ dx ! γ ≤ C (cid:20) | µ | ( B r ( y )) r n − (cid:21) p − (3.6)+ C | µ | ( B r ( y )) r n − B r ( y ) |∇ u | γ dx ! − pγ , (3.7)and applying the Lemma 3.3 with B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) and p = γ showsthat: B ρ ( y ) |∇ w | γ ! γ ≤ C (cid:16) ρr (cid:17) β − B r/ ( y ) |∇ w | γ ! γ . (3.8)Combining (3.6), (3.8) with the fact that ˆ B r/ ( y ) |∇ w | γ dx ≤ C ˆ B r ( y ) |∇ u | γ dx. we obtain B ρ ( y ) |∇ u | γ dx ! γ ≤ B ρ ( y ) |∇ w | γ dx ! γ + B ρ ( y ) |∇ u − ∇ w | γ dx ! γ ≤ C (cid:16) ρr (cid:17) β − B r ( y ) |∇ u | γ dx ! γ + C (cid:18) | µ | ( B r ( y )) r n − (cid:19) p − + C | µ | ( B r ( y )) r n − B r ( y ) |∇ u | γ dx ! − pγ , which implies ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:16) ρr (cid:17) nγ + β − ˆ B r ( y ) |∇ u | γ dx ! γ + Cρ nγ (cid:18) | µ | ( B r ( y )) r n − (cid:19) p − + Cρ n ( p − γ | µ | ( B r ( y )) r n − (cid:16) ρr (cid:17) n (2 − p ) γ ˆ B r ( y ) |∇ u | γ dx ! − pγ . ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:16) ρr (cid:17) nγ + β − ˆ B r ( y ) |∇ u | γ dx ! γ + C ε ρ nγ (cid:18) | µ | ( B r ( y )) r n − (cid:19) p − + ε (cid:16) ρr (cid:17) nγ ˆ B r ( y ) |∇ u | γ dx ! γ . (3.9)The repeated application of Lemma 3.6 enables us to set function Φ : R → R ofany t ∈ R , t > t ) = ˆ B t ( y ) |∇ u | γ dx ! γ . (3.10)Thus, (3.9) can be rewritten in term of function Φ:Φ( ρ ) ≤ C (cid:20)(cid:16) ρr (cid:17) nγ + β − + ε (cid:21) Φ( r ) + C ε ρ nγ (cid:18) | µ | ( B r ( y )) r n − (cid:19) p − . (3.11)Therefore, for any δ ∈ h − n − pp − , β (cid:17) , it satisfies thatΦ( ρ ) ≤ C (cid:20)(cid:16) ρr (cid:17) nγ + β − + ε (cid:21) Φ( r ) + C ε r nγ + δ − r − δ (cid:18) | µ | ( B r ( y )) r n − (cid:19) p − = C (cid:20)(cid:16) ρr (cid:17) nγ + β − + ε (cid:21) Φ( r ) + C ε r nγ + δ − (cid:18) | µ | ( B r ( y )) r n − − ( p − − δ ) (cid:19) p − . Choosing ε > γ = nγ + β − , β = nγ + δ − 1, with δ < β (as β < γ in Lemma 3.6) and B = (cid:16) M T θ ( | µ | )( y ) (cid:17) p − , forany 0 < ρ < r < T it is easily seen thatΦ( ρ ) ≤ C (cid:20)(cid:16) ρr (cid:17) nγ + β − Φ( r ) + Cρ nγ + δ − (cid:16) M T θ ( | µ | )( y ) (cid:17) p − (cid:21) , and we thus get ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C "(cid:18) T (cid:19) nγ + δ − (cid:18) ˆ Ω |∇ u | γ dx (cid:19) γ + (cid:16) M T θ ( | µ | )( y ) (cid:17) p − ρ nγ + δ − . (3.12)According to the Remark 2.3, it gives (cid:18) T n ˆ Ω |∇ u | γ (cid:19) /γ ≤ C γ (cid:20) | µ | (Ω) T n − (cid:21) p − , for any γ ∈ (cid:18) , ( p − nn − (cid:19) (cid:18) T (cid:19) nγ + δ − (cid:18) ˆ Ω |∇ u | γ dx (cid:19) γ ≤ C γ (cid:18) T (cid:19) δ − (cid:20) | µ | (Ω) T n − (cid:21) p − ≤ C (cid:16) M T θ ( | µ | )( y ) (cid:17) p − . (3.13)From (3.12) and (3.13) it turns to ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:16) M T θ ( | µ | )( y ) (cid:17) p − ρ nγ + δ − , and that is our desired conclusion. Next, let us give some comparison estimates on the boundary, the same conclusionas interior estimates can be drawn. First, as R n \ Ω is uniformly p -thick withconstants c , r > 0, let x ∈ ∂ Ω be a boundary point and for 0 < R < r / R = Ω R ( x ) = B R ( x ) ∩ Ω. With u ∈ W ,p (Ω) being a solutionto (1.1), we consider the unique solution w ∈ u + W ,p (Ω R ) to the followingequation: (cid:26) − div ( A ( x, ∇ w )) = 0 in Ω R ( x ) ,w = u on ∂ Ω R ( x ) . (3.14)In what follows we extend µ and u by zero to R n \ Ω and w by u to R n \ Ω R .Let us recall the following Lemma 3.7, which was stated and proved in [37]. Thisnaturally leads to Lemma 3.8 below, whose proof can be found in [40]. Lemma 3.7 Let w be the solution to (3.14) . Then, there exist constants Θ =Θ( n, p, α, β, c ) > p and C = C ( n, p, α, β, c ) > such that the following estimate B ρ/ ( y ) |∇ w | Θ dx ! ≤ C B ρ ( y ) |∇ w | p − dx ! p − (3.15) holds for all B ρ ( y ) ⊂ B R ( x ) , y ∈ B r ( x ) . Lemma 3.8 Let w be the solution to (3.14) . Then, there exist constants Θ =Θ( n, p, α, β, c ) > p and C = C ( n, p, α, β, c ) > such that we have the followingestimate B ρ/ ( y ) |∇ w | Θ dx ! ≤ C B ρ/ ( y ) |∇ w | p − dx ! p − (3.16) holds for all B ρ ( y ) ⊂ B R ( x ) , y ∈ B r ( x ) . Now, we state the following Lemma in a somewhat more general form of Lemmas3.7 and 3.8. 11 emma 3.9 Let w be the solution to (3.14) . Then, for < θ < θ < thereexist constants Θ = Θ( n, p, α, β, c ) > p and C = C ( n, p, α, β, θ , θ , c ) > suchthat we have the following estimate B θ ρ ( y ) |∇ w | Θ dx ! ≤ C B θ ρ ( y ) |∇ w | p − dx ! p − (3.17) holds for all B ρ ( y ) ⊂ B R ( x ) , y ∈ B r ( x ) . More formally, Lemmas 3.10, 3.11 and 3.12, which we state below are mainingredients for us to obtain boundary estimates. They are the boundary versionof Lemmas 3.2, 3.3 and 3.4, respectively. Lemma 3.10 Let w be the solution to (3.14) . Then, for any − p ≤ γ < ( p − nn − ≤ , there is a constant C = C ( n, p, α, β, c ) > such that: B R ( x ) |∇ ( u − w ) | γ dx ! γ ≤ C (cid:20) | µ | ( B R ( x )) R n − (cid:21) p − + C | µ | ( B R ( x )) R n − B R ( x ) |∇ u | γ dx ! − pγ . (3.18) Lemma 3.11 Let w be the solution to (3.14) . Then, there exist constants β = β ( n, p, α, β, c ) ∈ (0 , / and C = C ( n, p, α, β, c ) > such that: B ρ ( y ) |∇ w | p dx ! p ≤ C (cid:16) ρr (cid:17) β B r ( y ) |∇ w | p dx ! p , (3.19) for any y ∈ B r ( x ) such that B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) . Lemma 3.12 Let w be the solution to (3.14) . Then, for any Θ ∈ (0 , p ] , thereexist constants β = β ( n, p, α, β, c ) ∈ (0 , / and C = C ( n, p, α, β, Θ , c ) > there holds: B ρ ( y ) |∇ w | Θ dx ! ≤ C (cid:16) ρr (cid:17) β − B r ( y ) |∇ w | Θ dx ! , (3.20) for any y ∈ B r ( x ) such that B ρ ( y ) ⊂ B r ( y ) ⊂ B R ( x ) . We next state and prove the selection Lemma which establishes solution gradientestimate up to the boundary. Lemma 3.13 Let β ∈ (0 , / be as in Lemmas 3.11 and 3.12. Then, for any δ ∈ h − n − pp − , β (cid:17) , there exists a constant C = C ( n, p, α, β, c , β ) > such that forany B ρ ( y ) ∩ ∂ Ω = ∅ : ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:16) M T θ ( | µ | )( y ) (cid:17) p − ρ nγ + δ − , (3.21) where θ = 1 + ( p − − δ ) , T = diam (Ω) , < ρ < T . roof of Lemma 3.13. We begin by taking 0 < ρ ′ ≤ r such that B ρ ′ / ( y ) ∩ ∂ Ω = ∅ . Let y ∈ B ρ ′ / ( y ) ∩ ∂ Ω such that | y − y | = dist ( y, ∂ Ω) ≤ ρ ′ / 4. Note that since | y − y | ≤ ρ ′ / 4, this clearly gives B ρ ′ / ( y ) ⊂ B ρ ′ / ( y ) and B ρ ′ / ( y ) ⊂ B ρ ′ / ( y ). Otherwise,for ρ ≥ | y − y | / B ρ ( y ) ⊂ B ρ ( y ).For ρ ≤ ρ ′ / 4, let w be as in Lemmas 3.11 and 3.12 with B ρ ( y ) ⊂ B ρ ′ ( y ) ⊂ B R ( x ). Applying B R = B ρ ′ / ( y ) in Lemma 3.10 yields: B ρ ′ ( y ) |∇ ( u − w ) | γ dx ! γ ≤ C (cid:20) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:21) p − + C | µ | ( B ρ ′ ( y )) ρ ′ n − B ρ ′ ( y ) |∇ u | γ dx ! − pγ . (3.22)Applying Lemma 3.11 with B ρ ( y ) ⊂ B ρ ′ ( y ) ⊂ B R and p = γ shows that: B ρ ( y ) |∇ w | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) β B ρ ′ / y ) |∇ w | γ dx ! γ . Moreover, since: ˆ B ρ ( y ) |∇ w | γ dx ≤ C ˆ B ρ ( y |∇ u | γ dx, it follows that B ρ ( y ) |∇ w | γ dx ! γ ≤ C B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) β − B ρ ′ / ( y ) |∇ u | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) β − B ρ ′ / ( y ) |∇ u | γ dx ! γ . (3.23)According to (3.22) and (3.23), we get the estimate: B ρ ( y ) |∇ u | γ dx ! γ ≤ B ρ ( y ) |∇ w | γ dx ! γ + B ρ ( y ) |∇ u − ∇ w | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) β − B ρ ′ / y ) |∇ u | γ dx ! γ + C (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:19) p − + C | µ | ( B ρ ′ ( y )) ρ ′ n − B ρ ′ ( y ) |∇ u | γ dx ! − pγ . ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) nγ + β − ˆ B ρ ′ ( y ) |∇ u | γ dx ! γ + Cρ nγ (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:19) p − + Cρ n ( p − γ | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:18) ρρ ′ (cid:19) n (2 − p ) γ ˆ B ρ ′ ( y ) |∇ u | γ dx ! − pγ . (3.24)In the use of H¨older’s inequality for the last term, it may be concluded that: ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C (cid:18) ρρ ′ (cid:19) nγ + β − ˆ B ρ ′ ( y ) |∇ u | γ dx ! γ + C ε ρ nγ (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:19) p − + ε (cid:18) ρρ ′ (cid:19) nγ ˆ B ρ ′ ( y ) |∇ u | γ dx ! γ . (3.25)The same proof works when we take Lemma 3.6 in use. It is natural to set afunction Φ as in (3.10). From (3.25), it can be written the inequality of the formΦ( ρ ) ≤ C "(cid:18) ρρ ′ (cid:19) nγ + β − + ε Φ( ρ ′ ) + C ε ρ nγ (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:19) p − . It suffices to show that for any δ ∈ h − n − pp − , β (cid:17) , we haveΦ( ρ ) ≤ C "(cid:18) ρρ ′ (cid:19) nγ + β − + ε Φ( ρ ′ ) + C ε ρ ′ nγ + δ − ρ ′ − δ (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − (cid:19) p − = C "(cid:18) ρρ ′ (cid:19) nγ + β − + ε Φ( ρ ′ ) + C ε ρ ′ nγ + δ − (cid:18) | µ | ( B ρ ′ ( y )) ρ ′ n − − ( p − − δ ) (cid:19) p − , and here the supremum is taken for all 0 < ρ ′ < T , that yieldsΦ( ρ ) ≤ C "(cid:18) ρρ ′ (cid:19) nγ + β − + ε Φ( ρ ′ ) + C ε ρ ′ nγ + δ − (cid:16) M T θ ( | µ | )( y ) (cid:17) p − . For ε > γ = nγ + β − , β = nγ + δ − δ < β (as β < γ in Lemma 3.6) and B = (cid:16) M T θ ( | µ | )( y ) (cid:17) p − , forany 0 < ρ < ρ ′ < T one getsΦ( ρ ) ≤ C "(cid:18) ρρ ′ (cid:19) nγ + β − Φ( ρ ′ ) + Cρ nγ + δ − (cid:16) M T θ ( | µ | )( y ) (cid:17) p − . ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C "(cid:18) T (cid:19) nγ + δ − (cid:18) ˆ Ω |∇ u | γ dx (cid:19) γ + (cid:16) M T θ ( | µ | )( y ) (cid:17) p − ρ nγ + δ − . (3.26)It follows easily from Remark 2.3 that (cid:18) T n ˆ Ω |∇ u | γ (cid:19) /γ ≤ C γ (cid:20) | µ | (Ω) T n − (cid:21) p − , for any γ ∈ (cid:18) , ( p − nn − (cid:19) which implies (cid:18) T (cid:19) nγ + δ − (cid:18) ˆ Ω |∇ u | γ dx (cid:19) γ ≤ C (cid:18) T (cid:19) δ − (cid:20) | µ | (Ω) T n − (cid:21) p − ≤ C (cid:16) M T θ ( | µ | )( y ) (cid:17) p − . (3.27)From both (3.26) and (3.27) have already proved, it completes the proof.As a consequence of Lemmas 3.5 and 3.13, the supremum is taken for all0 < ρ < T and y ∈ Ω, one of our results may be summarized in the followingimportant Lemma. Lemma 3.14 Let β ∈ (0 , / be as in Lemmas 3.4 and 3.12. Then, for any δ ∈ h − n − pp − , β (cid:17) , there exists a constant C = C ( n, p, α, β, c , β ) > such that: sup y ∈ Ω ,ρ ∈ (0 ,T ) ρ − nγ − δ +1 ˆ B ρ ( y ) |∇ u | γ dx ! γ ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) , (3.28) where θ = 1 + ( p − − δ ) , T = diam (Ω) , < ρ < T . This section is devoted to separable proofs of our main results in Theorem 2.4,2.5 and 2.8. Here, our proof techniques are global up to the boundary results.We first prove the form of Theorem 2.4. The main tools are properties ofHardy-Littlewood maximal function and the following lemma, that can be viewedas a substitution for the Calder´on-Zygmund-Krylov-Safonov decomposition. Lemma 4.1 Let < ε < , R > and the ball Q := B R ( x ) for some x ∈ R n . Let E ⊂ F ⊂ Q be two measurable sets in R n +1 with L n ( E ) < ε L n ( Q ) and satisfying the following property: for all x ∈ Q and r ∈ (0 , R ] , we have B r ( x ) ∩ Q ⊂ F provided L n ( E ∩ B r ( x )) ≥ ε L n ( B r ( x )) . Then L n ( E ) ≤ Cε L n ( F ) for some C = C ( n ) . roof of Theorem 2.4. Let µ , λ + k , λ − k be as in Definition 2.1. Let u bethe renormalized solution to (1.1) and u k ∈ W ,p (Ω) be the unique solution tothe following problem: (cid:26) − div( A ( x, ∇ u k )) = µ k in Ω ,u k = 0 on ∂ Ω , where µ k = χ {| u | 0, firstly we have B ρ ( y ) χ D |∇ u | γ dx ! /γ ≤ sup (cid:8) Λ , ¯Λ (cid:9) , where Λ = sup ρ ′ 0, it can be seen clearly that:( M ( χ D |∇ u | γ )( y )) /γ ≤ max n(cid:2) M (cid:0) χ D χ B r ( x ) |∇ u | γ (cid:1) ( y ) (cid:3) γ , nγ λ o , ∀ y ∈ B r ( x ) . Therefore, for all λ > ε satisfies ε − > nγ , it shows that E λ,ε ∩ B r ( x ) = n M (cid:0) χ D χ B r ( x ) |∇ u | γ (cid:1) γ > ε − λ, ( M ( χ D µ )) p − ≤ ε p − γ λ o ∩ D ∩ B r ( x ) . (4.10)In order to prove (4.9) we separately consider for the case B r ( x ) ⊂⊂ Ω(interior) and the case B r ( x ) ∩ Ω c = ∅ (boundary). The proof falls naturally intotwo cases. Case 1: B r ( x ) ⊂⊂ Ω: Applying Lemma 3.2 for u k ∈ W ,p (Ω) and w k thesolution to: ( div( A ( x, ∇ w k )) = 0 , in B r ( x ) ,w k = u k , on ∂B r ( x ) , (4.11)with µ = µ k and B R = B r ( x ), one has a constant C = C ( n, p, α, β, c , T /r ) > B r ( x ) |∇ u k − ∇ w k | γ dx ! γ ≤ C (cid:20) | µ k | ( B r ( x )) r n − (cid:21) p − + C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ . (4.12)18therwise, Lemma 3.1 is also applied to give: B r ( x ) |∇ w k | dx ! Θ ≤ C B r ( x ) |∇ w k | p − dx ! p − ≤ C B r ( x ) |∇ u k | γ dx ! γ + C B r ( x ) |∇ u k − ∇ w k | γ dx ! γ , (4.13)where, the second inequality is obtained by using H¨older’s inequality and for γ > p − 1. On the other hand, it is easily to check that L n ( E λ,ε ∩ B r ( x )) ≤ L n (cid:16) { M (cid:0) χ D χ B r ( x ) |∇ ( u k − w k ) | γ (cid:1) γ > − γ ε − λ } ∩ B r ( x ) (cid:17) + L n (cid:16) { M (cid:0) χ D χ B r ( x ) |∇ ( u − u k ) | γ (cid:1) γ > − γ ε − λ } ∩ B r ( x ) (cid:17) + L n (cid:16) { M (cid:0) χ D χ B r ( x ) |∇ w k | γ (cid:1) γ > − γ ε − λ } ∩ B r ( x ) (cid:17) . (4.14)and by using Remark 2.6 for each term on right hand side of (4.14), one gives L n ( E λ,ε ∩ B r ( x )) ≤ C (cid:16) ε − λ (cid:17) γ " ˆ B r ( x ) χ D |∇ u k − ∇ w k | γ dx ++ ˆ B r ( x ) χ D |∇ u − ∇ u k | γ dx + C (cid:16) ε − λ (cid:17) Θ ˆ B r ( x ) χ D |∇ w k | Θ dx. (4.15)19ombining inequalities (4.12) with (4.13) to (4.15) yields L n ( E λ,ε ∩ B r ( x )) ≤ ε γ λ − γ r n " C (cid:18) | µ k | ( B r ( x )) r n − (cid:19) p − ++ C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ γ + Cε γ λ − γ ˆ B r ( x ) |∇ u − ∇ u k | γ dx + Cελ − Θ r n C ˆ B r ( x ) |∇ u k | γ dx ! γ + C (cid:18) | µ k | ( B r ( x )) r n − (cid:19) p − ++ C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ Θ . Letting k → ∞ and thanks to (4.7) and (4.8) one obtains: L n ( E λ,ε ∩ B r ( x )) ≤ ε γ λ − γ r n C | µ | ( B r ( x )) r n − ! p − ++ C | µ | ( B r ( x )) r n − B r ( x ) |∇ u | γ dx ! − pγ γ ++ Cελ − Θ r n C ˆ B r ( x ) |∇ u | γ dx ! γ + C | µ | ( B r ( x )) r n − ! p − ++ C | µ | ( B r ( x )) r n − B r ( x ) |∇ u | γ dx ! − pγ Θ . As | x − x | < r , B r ( x ) ⊂ B r ( x ). This gives: B r ( x ) |∇ u | γ dx ≤ | B (0) || B (0) | B r ( x ) |∇ u | γ dx ≤ C sup ρ> B ρ ( x ) |∇ u | γ dx = C M ( |∇ u | γ ) ( x ) . (4.16)Similarly, as | x − x | < r , we obtain B r ( x ) ⊂ B r ( x ) ⊂ D and for all ρ > 0, itfinds: | µ | ( B r ( x ))) r n − ≤ | µ | ( B ρ ( x )) ρ n − ≤ n − M ( χ D µ )( x ) . (4.17)20pplying (4.16) and (4.17) together with (4.7), (4.8) yields that: L n ( E λ,ε ∩ B r ( x )) ≤ ε γ + γ p − γ r n (cid:16) C + Cε p − γ ( p − (cid:17) γ + Cεr n (cid:16) ε p − γ + ε p − γ ( p − (cid:17) Θ ≤ C h ε γ + γ ( p − p − γ + ε i r n ≤ Cεr n . which establishes our desired (4.9). Case 2: B r ( x ) ∩ Ω c = ∅ : Let x ∈ ∂ Ω such that | x − x | = dist( x, ∂ Ω) ≤ r . Itis not difficult to check that: B r ( x ) ⊂ B r ( x ) ⊂ D . Applying Lemma 3.10 for u k ∈ W ,p (Ω) and w k being solution to: ( div( A ( x, ∇ w k )) = 0 , in B r ( x ) ,w k = u k , on ∂B r ( x ) , (4.18)for µ = µ k and B R = B R ( x ), one has a constant C = C ( n, p, α, β, c , T /r ) > B r ( x ) |∇ u k − ∇ w k | γ dx ! γ ≤ C (cid:20) | µ k | ( B r ( x )) r n − (cid:21) p − + C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ , (4.19)and for all ρ > B ρ ( y ) ⊂ B r ( x ), following Lemma 3.8 one has B ρ/ ( y ) |∇ w k | Θ dx ! ≤ C B ρ ( y ) |∇ w k | p − dx ! p − , Θ > p. (4.20)As a version of (4.15) in the ball B r ( x ), one gives: L n ( E λ,ε ∩ B r ( x )) ≤ C (cid:16) ε − λ (cid:17) γ " ˆ B r ( x ) |∇ u k − ∇ w k | γ dx ++ ˆ B r ( x ) |∇ u − ∇ u k | γ dx + C (cid:16) ε − λ (cid:17) Θ ˆ B r ( x ) |∇ w k | Θ dx. (4.21)21ince B r ( x ) ⊂ B r ( x ), similar to (4.13), we obtain: B r ( x ) |∇ w k | Θ dx ! ≤ C B r ( x ) |∇ w k | p − dx ! p − ≤ C B r ( x ) |∇ u k | γ dx ! γ + C B r ( x ) |∇ u k − ∇ w k | γ dx ! γ , (4.22)where the second inequality is obtained by using H¨older’s inequality and for γ > p − 1. On the ball B r ( x ), applying these estimates (4.19) and (4.20) with(4.22) from above to (4.21), one obtains the following estimate: L n ( E λ,ε ∩ B r ( x )) ≤ ε γ λ − γ r n " C (cid:18) | µ k | ( B r ( x )) r n − (cid:19) p − + C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ γ + Cε γ λ − γ ˆ B r ( x ) |∇ u − ∇ u k | γ dx + Cελ − Θ r n C B r ( x ) |∇ u k | γ dx ! γ + C (cid:18) | µ k | ( B r ( x )) r n − (cid:19) p − + C | µ k | ( B r ( x )) r n − B r ( x ) |∇ u k | γ dx ! − pγ Θ . Letting k → ∞ , we can assert that: L n ( E λ,ε ∩ B r ( x )) ≤ Cε γ λ − γ r n | µ | ( B r ( x )) r n − ! p − + | µ | ( B r ( x )) r n − B r ( x ) |∇ u | γ dx ! − pγ γ + Cελ − Θ r n B r ( x ) |∇ u | γ dx ! γ + | µ | ( B r ( x )) r n − ! p − + | µ | ( B r ( x )) r n − B r ( x ) |∇ u | γ dx ! − pγ Θ . x , x in the previous case and the definition of x , since dist( x, Ω) ≤ r ,we can easily check that these following bounds: B r ( x ) ⊂ B r ( x ) ⊂ B r ( x ) ⊂ D ,B r ( x ) ⊂ B r ( x ) ⊂ B r ( x ) ⊂ D , and the following estimates | µ | ( B r ( x )) r n − ≤ | µ | ( B r ( x )) r n − ≤ n − M ( χ D µ )( x )hold. On the other hand, as | x − x | = dist( x, ∂ Ω), one obtains B r ( x ) |∇ u | γ dx ! γ ≤ | B (0) || B (0) | B r ( x ) |∇ u | γ dx ! γ ≤ C sup ρ> B ρ ( x ) χ D |∇ u | γ dx ! γ = C ( M ( χ D |∇ u | γ ) ( x )) γ . (4.23)Combining these above estimates together, one finally concludes that L n ( E λ,ε ∩ B r ( x )) ≤ Cr n . According to Lemma 4.1 for E = E λ,ε , F = F λ , the proof of Theorem 2.4 iscomplete and we will refer to this result in the sequel.Now, let us give a brief proof of Theorem 2.5. Proof of Theorem 2.5. Let 0 < ρ < T and x be fixed in Ω. We first applyTheorem 2.4 with R = ρ and the corresponding sets D = B ρ ( x ), D = B ρ ( x ),there exist Θ = Θ( n, p, α, β, c ) > p and a constant C = C ( n, p, α, β, c , T /r ) > L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ, ( M ( χ B ρ ( x ) µ )) p − ≤ ε p − γ λ o ∩ B ρ ( x ) (cid:17) ≤ Cε L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > λ o ∩ B ρ ( x ) (cid:17) , (4.24)for any λ > λ ( ε ) , ε ∈ (0 , γ ∈ (cid:16) − p , ( p − nn − (cid:17) , where λ = ε − p − γ k∇ u k L γ ( B ρ ( x )) ρ − nγ . Thus, for all λ > λ ( ε ) it gives L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ o ∩ B ρ ( x ) (cid:17) ≤ Cε L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > λ o ∩ B ρ ( x ) (cid:17) + L n (cid:16)n ( M ( χ B ρ ( x ) µ )) p − > ε p − γ λ o ∩ B ρ ( x ) (cid:17) . (4.25)23t is necessary to estimate ( M ( χ B ρ ( x ) |∇ u | γ )) /γ in L q,s ( B ρ ( x )) for 0 < q < Θand 0 < s < ∞ . Firstly, let us rewrite the Lorentz norm as k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k sL q,s ( B ρ ( x )) = q ˆ ∞ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > λ o ∩ B ρ ( x ) (cid:17) s/q dλ = ε − s Θ q ˆ ∞ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ o ∩ B ρ ( x ) (cid:17) s/q dλ. (4.26)Since the estimate (4.25) only holds for all λ > λ ( ε ), one splits the integral intothe sum of integrals on (0 , λ ) and ( λ , ∞ ) to get ˆ ∞ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ o ∩ B ρ ( x ) (cid:17) s/q dλ = ˆ λ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ o ∩ B ρ ( x ) (cid:17) s/q dλ + ˆ ∞ λ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > ε − λ o ∩ B ρ ( x ) (cid:17) s/q dλ. (4.27)According to (4.26) and (4.27), it finds: k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k sL q,s ( B ρ ( x )) ≤ Cε − s Θ λ s L n ( B ρ ( x )) s/q + Cε − s Θ + sq ˆ ∞ λ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) |∇ u | γ )) /γ > λ o ∩ B ρ ( x ) (cid:17) s/q dλ + Cε − s Θ ˆ ∞ λ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) µ )) p − > ε p − γ λ o ∩ B ρ ( x ) (cid:17) s/q dλ. And thus, we check at once that k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k sL q,s ( B ρ ( x )) ≤ Cε − s Θ λ s ρ ns/q + Cε − s Θ + sq k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k sL q,s ( B ρ ( x )) + Cε − s Θ − s ( p − γ ˆ ∞ λ s − L n (cid:16)n ( M ( χ B ρ ( x ) µ )) p − > λ o ∩ B ρ ( x ) (cid:17) s/q dλ = Cε − s Θ λ s ρ ns/q + Cε − s Θ + sq k M ( χ B ρ ( x ) |∇ u | γ )) /γ k sL q,s ( B ρ ( x )) + Cε − s Θ − s ( p − γ k ( M ( χ B ρ ( x ) µ )) p − k sL q,s ( B ρ ( x )) . Then, it gives us the estimate: k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ Cε − λ ρ n/q + Cε − + q k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) + Cε − − p − γ k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) . (4.28)24f q < Θ, let us choose ε = ε > Cε − + q < / k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ C k∇ u k L γ ( B ρ ( x )) ρ − n (cid:16) γ − q (cid:17) + C k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) . The application of Lemma 3.14 enables us to obtain k∇ u k L γ ( B ρ ( x )) ≤ Cρ nγ + − θp − k M T θ ( | µ | ) k p − L ∞ (Ω) . In consequence, we get k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ Cρ − n (cid:16) γ − q (cid:17) ρ nγ + − θp − k M T θ ( | µ | ) k p − L ∞ (Ω) + C k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) = Cρ − nγ + nq + nγ + − θp − k M T θ ( | µ | ) k p − L ∞ (Ω) + C k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) that yields ρ − nq + θ − p − k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) + Cρ − nq + θ − p − k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) . Finally, by taking the supremum for all ρ ∈ (0 , T ) and x ∈ Ω, we conclude thedesired result:sup ρ ∈ (0 ,T ) ,x ∈ Ω ρ − nq + θ − p − k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ( B ρ ( x )) ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) + sup ρ ∈ (0 ,T ) ,x ∈ Ω Cρ − nq + θ − p − k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) . Proof of Theorem 2.8. Let 0 < ρ < T and x be fixed in Ω. From what has already been proved inTheorem 2.5 and the definition of Lorentz-Morrey norm (2.3) it gets k ( M ( χ B ρ ( x ) |∇ u | γ )) /γ k L q,s ; q ( θ − p − ( B ρ ( x )) ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) + sup ρ ∈ (0 ,T ) ,x ∈ Ω Cρ − nq + θ − p − k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) , that yields k∇ u k L q,s ; q ( θ − p − (Ω) ≤ C k M T θ ( | µ | ) k p − L ∞ (Ω) + C sup ρ ∈ (0 ,T ) ,x ∈ Ω ρ − nq + θ − p − k ( M ( χ B ρ ( x ) µ )) p − k L q,s ( B ρ ( x )) . (4.29)25n order to get the gradient estimate of solution in Lorentz-Morrey spaces, it issufficient to show that each term on the right side of (4.29) is bounded by theLorentz-Morrey norm of measure data µ . In particular, it is sufficient to showthat k M T θ ( | µ | ) k L ∞ (Ω) ≤ C k| µ | p − k p − L q ( θ − θ , s ( θ − θ ; q ( θ − p − (Ω) (4.30)and sup ρ ∈ (0 ,T ) ,x ∈ Ω ρ − n ( p − q + θ − k ( M ( χ B ρ ( x ) µ )) k L qp − , sp − ( B ρ ( x )) ≤ C k| µ | p − k p − L q ( θ − θ , s ( θ − θ ; q ( θ − p − (Ω) . (4.31)hold. First, we can proceed to the proof of (4.30). For 0 < ρ < T and x ∈ Ωwe have k| µ | p − k p − L q ( θ − θ , s ( θ − θ ; q ( θ − p − (Ω) = k µ k L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (Ω) ≥ k µ k L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − ( B ρ ( x )) ≥ k µ k L q ( θ − θ ( p − , ∞ ; q ( θ − p − ( B ρ ( x )) ≥ Cρ q ( θ − p − − nq ( θ − θ ( p − [ L n ( B ρ ( x ))] − q ( θ − θ ( p − | µ | ( B ρ ( x ))= Cρ θ − nθ ( p − q ( θ − ρ − n + nq ( θ − θ ( p − | µ | ( B ρ ( x ))= C | µ | ( B ρ ( x )) ρ n − θ , that leads to our desired result in (4.30) by taking the supremum both side forall 0 < ρ < T and x ∈ Ω, where it follows the definition of M T θ ( | µ | ) in (2.10).On the other hand, one refers to [38, Theorem 1.1] to get that for any x ∈ B = B ρ ( x ) one obtains: M ( χ B | µ | ) ( x ) ≤ C [ M ( χ B | µ | ) ( x )] − θ k µ k θ L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (10 B ) , (4.32)and this can be used to give the estimate in (4.31). Indeed, k M ( χ B | µ | ) k L qp − , sp − (10 B ) ≤ C k [ M ( χ B | µ | )] θ − θ k L qp − , sp − (10 B ) k µ k θ L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (10 B ) ≤ C k M ( χ B | µ | ) k θ − θ L q ( θ − θ ( p − , s ( θ − θ ( p − (10 B ) k µ k θ L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (10 B ) . M in Remark 2.7,it finds: k M ( χ B | µ | ) k L qp − , sp − (10 B ) ≤ C k µ k θ − θ L q ( θ − θ ( p − , s ( θ − θ ( p − (10 B ) k µ k θ L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (10 B ) , and from the definition of Lorentz-Morrey norm (2.3), it follows that: k M ( χ B | µ | ) k L qp − , sp − (10 B ) ≤ Cρ n − q ( θ − p − q ( θ − θ ( p − . θ − θ k µ k L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (10 B ) ≤ Cρ n ( p − q − θ +1 k µ k L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (Ω) . (4.33)Multiplying both sides of (4.33) by ρ − n ( p − q + θ − , we turn to ρ − n ( p − q + θ − k M ( χ B | µ | ) k L qp − , sp − (10 B ) ≤ C k µ k L q ( θ − θ ( p − , s ( θ − θ ( p − 1) ; q ( θ − p − (Ω) = C k| µ | p − k p − L q ( θ − θ , s ( θ − θ ; q ( θ − p − (Ω) , and the proof of (4.31) is complete. Acknowledgments The author T.N. Nguyen was supported by Ho Chi Minh City University ofEducation under grant No. B2017-SPS-12. References [1] D. R. Adams and L. I. Hedberg, Function spaces and potential theory ,Springer-Verlag, Berlin, 1996.[2] K. Adimurthi and N. C. Phuc, Quasilinear equations with natural growth inthe gradients in spaces of Sobolev multipliers . Calc. Var. Partial DifferentialEquations (2018).[3] K. Adimurthi and N. C. Phuc, Global Lorentz and Lorentz-Morrey estimatesbelow the natural exponent for quasilinear equations . Calc. Var. Partial Dif-ferential Equations (2015), 3107-3139.[4] P. Benilan, L. Boccardo, T. Gallouet, R. Gariepy, M. Pierre , and J. L.Vazquez, An L theory of existence and uniqueness of solutions of nonlinearelliptic equations, Ann. Scuola Norm. Sup. Pisa (IV) (1995), 241–273.275] A. Bensoussan, L. Boccardo, and F. Murat, On a nonlinear partial differ-ential equation having natural growth terms and unbounded solution , Ann.Inst. H. Poincar´e Anal. Non Lin´eaire. (1988), 347–364.[6] M. F. Betta, A. Mercaldo, F. Murat, and M. M. Porzio, Uniqueness ofrenormalized solutions to nonlinear elliptic equations with a lower order termand right-hand side in L (Ω), ESAIM: Control, Optimisation and Calculusof Variations (2002), 239–272.[7] M. F. Betta, A. Mercaldo, F. Murat, and M. M. Porzio, Existence of renor-malized solutions to nonlinear elliptic equations with a lower-order term andright-hand side a measure , J. Math. Pures Appl. (2003), 90–124.[8] M. F. Bidaut-Veron, M. Garcia-Huidobro, and L. Veron, Remarks on somequasilinear equations with gradient terms and measure data . Recent trendsin nonlinear partial differential equations. II. Stationary problems, 31–53,Contemp. Math., 595, Amer. Math. Soc., Providence, RI, 2013.[9] L. Boccardo, T. Gallou¨et, and L. Orsina, Existence and uniqueness of entropysolutions for nonlinear elliptic equations with measure data , Ann. Inst. H.Poincar´e Anal. Non Lin´eaire (1996), 539–551.[10] L. Boccardo, F. Murat, and J.-P. Puel, Existence of bounded solutions fornonlinear elliptic unilateral problems , Ann. Mat. Pura Appl. (1988),183–196.[11] L. Boccardo, F. Murat, and J.-P. Puel, L ∞ estimate for some nonlinearelliptic partial differential equations and application to an existence result ,SIAM J. Math. Anal. (1992), 326–333.[12] S.-S. Byun and D.K. Palagachev, Morrey regularity of solutions to quasi-linear elliptic equations over Reifenberg flat domains , Calc. Var. (2014),37–76.[13] S.-S. Byun and S. Ryu, Global weighted estimates for the gradient of solutionsto nonlinear elliptic equations , Ann. I. H. Poincar´e AN (2013), 291–313.[14] S.-S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifen-berg domains , Comm. Pure Appl. Math. (2004), 1283–1310.[15] S.-S. Byun and L. Wang, Elliptic equations with BMO nonlinearity in Reifen-berg domains , Adv. Math. (6) (2008), 1937–1971.[16] G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, Renormalized solutionsof elliptic equations with general measure data , Ann. Scuola Norm. Super.Pisa (IV) (1999), 741–808.[17] F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials ,Amer. J. Math. (2011), 1093–1149.2818] F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinearpotentials , J. Funt. Anal. (2010), 2961–2998.[19] Ciprian G. Gal and Mahamadi Warma, Existence of bounded solutions for aclass of quasilineaer elliptic systems on manifolds with boundary , Journal ofDifferential Equations (2013), 151–192.[20] E. Giusti, Direct methods in the calculus of variations , World Scientic Pub-lishing Co., Inc., River Edge, NJ, 2003.[21] L. Grafakos, Classical and Modern Fourier Analysis , Pearson/Prentice Hall,2004.[22] T. Kilpelainen, J. Maly, The Wiener test and potential estimates for quasi-linear elliptic equations , Acta Math. (1994), 137–161.[23] T. Kuusi, G. Mingione, Guide to nonlinear potential estimates , Bull. Math.Sci. (2014), 1–82.[24] T. Kuusi, G. Mingione, Vectorial nonlinear potential theory , J. Europ. Math.Soc. (2018), 929–1004.[25] Quoc-Hung Nguyen, Potential estimates and quasilinear parabolic equationswith measure data , arXiv:1405.2587.[26] Quoc-Hung Nguyen, Global estimates for quasilinear parabolic equations onReifenberg flat domains and its applications to Riccati type parabolic equa-tions with distributional data , Calc. Var. Partial Differential Equations (2015), 3927–3948.[27] Quoc-Hung Nguyen, N.C. Phuc, Good- λ and Muckenhoupt-Wheeden typebounds, with applications to quasilinear elliptic equations with gradient powersource terms and measure data , Math. Ann. (2018), 1–32.[28] Quoc-Hung Nguyen, Gradient estimates for singular quasilinear ellipticequations with measure data , arXiv: 1705.07440v2 (submitted for publica-tion).[29] Qing Han and Fanghua Lin, Elliptic Partial Differential Equations: SecondEdition , American Mathematical Soc., 2011.[30] V.G. Maz’ya, Conductor and capacity inequalities for functions on topologicalspaces and their applications to Sobolev-type imbedding , J. Func. Anal. (2005), 408–430.[31] T. Mengesha, N.C. Phuc, Weighted and regularity estimates for nonlin-ear equations on Reifenberg flat domains , J. Differential Equations (5)(2011), 2485–2507.[32] T. Mengesha, N.C. Phuc, Global estimates for quasilinear elliptic equationson Reifenberg flat domains , Arch. Ration. Mech. Anal. (1) (2012), 189–216. 2933] G. Mingione, The Calder´on-Zygmund theory for elliptic problems with mea-sure data , Ann. Scu. Norm. Sup. Pisa Cl. Sci. (5) (2007), 195–261.[34] G. Mingione, Gradient estimates below the duality exponent , Math. Ann. (2010), 571–627.[35] G. Mingione, Gradient potential estimates , Journal of the European Mathe-matical Society (2011), 459-486.[36] N. C. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flatdomains, with applications to quasilinear Riccati type equations, Adv. Math. (2014), 387–419.[37] N. C. Phuc, Global integral gradient bounds for quasilinear equations belowor near the natural exponent , Ark. Mat. (2014), 329–354.[38] N. C. Phuc, Morrey global bounds and quasilinear Riccati type equationsbelow the natural exponent , Journal de Mathematiques Pures et Appliqu´ees. (2014), 99–123.[39] E. Reifenberg, Solutions of the plateau problem for m -dimensional surfacesof varying topological type , Acta Math. (1960), 1–92.[40] Minh-Phuong Tran, Good- λ type bounds of quasilinear elliptic equations forthe singular case , Nonlinear Analysis178