Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces
aa r X i v : . [ m a t h . A P ] J un Weighted distribution approach to gradient estimates forquasilinear elliptic double-obstacle problems in Orlicz spaces
Thanh-Nhan Nguyen ∗ , Minh-Phuong Tran †‡ June 5, 2020
Abstract
We construct an efficient approach to deal with the global regularity estimates for aclass of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivationof this paper comes from the study on an abstract result in the viewpoint of thefractional maximal distributions and this work also extends some regularity resultsproved in [52] by using the weighted fractional maximal distributions (WFMDs). Wefurther investigate a pointwise estimates of the gradient of weak solutions via fractionalmaximal operators and Riesz potential of data. Moreover, in the setting of the paper,we are led to the study of problems with nonlinearity is supposed to be partiallyweak BMO condition (is measurable in one fixed variable and only satisfies locallysmall-BMO seminorms in the remaining variables).Keywords: double-obstacle problem; quasilinear elliptic; gradient estimate; weighteddistribution; Orlicz spaces.
Contents ∗ Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam; [email protected] † Corresponding author. ‡ Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University,Ho Chi Minh City, Vietnam; [email protected] Proofs of main results 21
The problem statement.
The aim of this article is to study the global regularityestimates for weak solutions to quasilinear elliptic double-obstacle problem associatedwith the operator L u = − div A ( x, ∇ u ) in Ω , in the setting of both weighted Lorentz and Orlicz-Lorentz spaces, where Ω is an openbounded domain of R n ( n ≥
2) and A : Ω × R n → R n is a Carath´eodory function (thatis continuous with respect to ξ ∈ R n for almost every x in Ω and measurable in x ∈ Ωfor every ξ in R n ). Given ψ , ψ are two fixed functions in the Sobolev space W ,p (Ω)such that ψ ≤ ψ almost everywhere in Ω and ψ ≤ ≤ ψ on ∂ Ω. More precisely, weare interested in the double-obstacle problem for operator L consists of finding unknownfunction u ∈ W ,p (Ω) satisfying ψ ≤ u ≤ ψ a.e. in Ω such that L u ≤ − div B ( x, F ) + g, (1.1)where F ∈ L p (Ω; R n ) and g ∈ L pp − (Ω) for 1 < p < ∞ . This problem naturally comes tothe variational inequality ˆ Ω hA ( x, ∇ u ) , ∇ ( u − ϕ ) i dx ≤ ˆ Ω hB ( x, F ) , ∇ ( u − ϕ ) i dx + ˆ Ω g ( u − ϕ ) dx, ( P )for all ϕ ∈ W ,p (Ω) and ψ ≤ ϕ ≤ ψ a.e. in Ω. Such function u in problem ( P ) iscalled a weak solution to the double-obstacle problem (1.1). Here, we assume further that A ( x, · ) is differentiable for almost every x in Ω, and satisfies the growth conditions: thereis 0 < L < ∞ such that |A ( x, ζ ) | + |h ∂ ζ A ( x, ζ ) , ζ i| ≤ L | ζ | p − , (1.2) hA ( x, ζ ) − A ( x, ζ ) , ζ − ζ i ≥ L − ( | ζ | + | ζ | ) p − | ζ − ζ | , (1.3)for almost every x in Ω and every ζ , ζ , ζ ∈ R n \ { } . As usual, we notice here that h· , ·i is understood as the standard inner product in R n , and ∂ ζ denotes the partial derivativewith respect to ζ . Further, the operator B is also the Carath´edory vector valued mappingsatisfying |B ( x, ζ ) | ≤ L | ζ | p − , ( x, ζ ) ∈ Ω × R n . (1.4)In the view of calculus of variations, a solution u to this problem ( P ) is also closelyrelated to the minimizer of an energy functional satisfying ψ ≤ u ≤ ψ . And the ap-pearance of such double obstacle problems is indispensable for describing many physicalphenomena, such as elasticity (to find the equilibrium position of an elastic membrane2ith additional constraints, see [56]), the Stefan’s problem (to describe the temperaturedistribution in a homogeneous medium, see [28, 29]), financial mathematics (models forpricing American options, the exercise region or price changes for market fluctuations,see [40]), Tug-of-War games (to obtain an approximation to p -Laplacian, see [23]), etc.We also refer to [33, 37, 55, 63] for physical motivation and mathematical methods forobstacle problems and their applications.The main features of this paper are the assumptions on boundary of domain Ω andthe nonlinearity of coefficients A . More specifically, in order to obtain the global regu-larity results, Ω here is assumed to be a Reifenberg flat dommain . As far as we know,in the geometrical sense, the boundaries of Reifenberg flat domains are locally well-approximated by planes or hyperplanes at every scale. The concept of Reifenberg flatdomain is a “minimal regularity hypothesis” assumed on the boundary ∂ Ω to guaranteethe main results of the geometric analysis continue to be valid in Ω. Global Calder´on-Zygmund/regularity/gradient estimates for nonlinear elliptic and parabolic equations insuch flat domains were first investigated by Byun and Wang in [13, 15] and later by othersin an extensive list of references (see Section 2 for detailed definition and description ofReifenberg flat domain). On the other-hand, instead of the assumption of locally smallBMO semi-norm in x for the nonlinearity A , in this paper we confine such small BMOcondition in ( n −
1) spatial variables of vector x ∈ R n , meanwhile no assumption on theremaining one. Particularly, the coefficients A ( x, · ) is allowed to be measurable in onesingle variable, say x , and only satisfies locally small BMO semi-norm in the remainingvariables, we say x ∗ = ( x , x , ..., x n ) ∈ R n − (note that these spatial variables may berearranged which shows x = ( x , x ∗ ) in vector space R n ). It can be seen that this addi-tional hypothesis on A (that is merely measurable in one spatial variable, but regular inthe others), called partially weak BMO condition , is weaker than the small BMO conditionin the whole space R n considered in previous studies [10, 12, 13, 15, 44, 60, 62] and so on.The study of Calder´on-Zygmund theory for linear elliptic equations with partially BMOcoefficients was introduced in [26] and [14], independently. Later, results can be extendedto higher order elliptic and parabolic systems by Dong and D. Kim in [27] and for non-linear elliptic equations of p -Laplacian type by Y. Kim in [36]. It is worth noticing thatY. Kim in his paper showed that this condition is the minimal regularity requirement on A for Calder´on-Zygmund type estimates. In general, to establish the Calder´on-Zygmundestimates for nonlinear elliptic/parabolic equations, under partially weak BMO condition ,the number of spatial variables in which the nonlinearity A assumed to be measurablecannot be larger than one. The definition of partially BMO coefficients will be describedin detail in Section 2 below.1.2. Relation to prior works.
Before stating the main results in this article, let usbriefly review some existing contributions related to regularity estimates developed inrecent years. Associated with nonlinear elliptic equations, going back to the fundamentalresult due to Iwaniec in [34], the very first nonlinear Calder´on-Zygmund type estimatesrelated to the elliptic p -Laplace equation were presented. Then, classical results of Iwaniecwere extended to the case of elliptic systems of p -Laplacian type by DiBenedetto andManfredi. There have been further interior and global regularity results established byseveral authors with suitable form of data (divergence or non-divergence form, measuredata) in some certain spaces, such as [9,22,24,47,50,58,60,61], et cetera. Equation for theconstrained problem yields the variational inequality ( P ) is quasilinear elliptic equation,3n which data mixed between divergence and non-divergence forms − div A ( x, ∇ u ) = − div B ( x, F ) + g in Ω . The global Cader´on-Zygmund estimate has recently proved by Lee and Ok in [41]motivated by preceding works by V. B¨ogelein et al. in [5] for the case of parabolic systemsof p -Laplacian type. Also, one of our recent advances is the extension of results in [41] tothe framework of Lorentz spaces in [51], which was also devoted to nonlinear problems withmixed data. Over the last years, a number of intensive studies have been developed throughthe works of many authors, that stitched together to form a panorama of regularity theoryfor nonlinear elliptic/parabolic equations. For one sided obstacle problems, regularityestimates have been extensively studied over the recent decades by many authors: Choeand Lewis in [20] proved C ,α and C ,α regularity for elliptic problems, Eleuteri in [31,32] considered H¨older continuity for solutions of minimization problems under standardand non-standard growth, as far as Calder´on-Zygmund estimates for elliptic/parabolicproblems (see for instance [5,7,21]) and a large number of works conducted, such as [3,6,57]as well as many references therein.Reaching far beyond the literature only deals with one-sided obstacle, we send thereader to some recent advances concerning the double-obstacle problems. Let us referto one of the very first studies, [25], in which the authors studied pointwise regularityproperties of solutions to ( P ) in linear case when p = 2 and right-hand side zero. Later,a great deal of progress has been made to extend to nonlinear operators, for instance,[35, 43, 45] with degenerate elliptic operators, C ,α and C ,α in [4, 19]. Recently, Calder´on-Zygmund and regularity results for a broader class of nonlinear elliptic double-obstacleproblem in certain spaces presented in [8, 11, 54] and our earlier works [52, 53].1.3. Technical tools.
Let us summarize here some important techniques regardingCalder´on-Zygmund type and regularity estimates for nonlinear elliptic and parabolic par-tial differential equations, which have been proposed and considered by many authorsduring the last years. In 1983, Iwaniec in his famous work [34] first proved the localregularity results by the use of beautiful interplay between tools from Harmonic Analysisand Nonlinear PDEs. Later Caffarelli and Peral found a different approach to the W ,p estimates based on Hardy-Littlewood maximal operators together with a new and refinedversion of Calder´on-Zygmund lemma, presented in [17]. This effective method has beenwidely used and developed through a vast array of contributions since then. We pay par-ticular attention to a very successful method by Acerbi and Mingione in [1], that allows toachieve a Calder´on-Zygmund estimates in which maximal operators and harmonic anal-ysis play no role in their proofs. And later in recent decades, the idea of this techniquebecomes enormously popular and has subsequently been developed in a rich literature andreferences at the same topic, such as [13, 30, 38, 39, 44, 47, 49, 50, 58, 59, 61] an so on. Onecan also find an extensive list of references in the recent survey paper [48].Motivated by such effective approach, in our previous work [52], we study a new point ofview and a new approach that is more interesting for pursuing regularity theory due to theso-called fractional maximal distribution functions (FMD). Our approach is inspired on theone hand from the essence behind the proofs of Calder´on-Zygmund-type estimates in [1,46,47] and on the other hand by the advantages of regularity estimates in terms of fractionalmaximal operators, proposed in preceding papers [51, 61, 62]. By introducing FMD andsome interesting properties on its own, we also prove the applicability of such abstract4esults to gradient estimates of weak solutions for both quasilinear elliptic equations and(double) obstacle problems in the same paper.Continuing and extending the theoretical ideas in [52], our goal in this paper is topresent a weighted approach in dealing with regularity issues for elliptic double obstacleproblems. By deeply using some technical tools such as the boundedness property of frac-tional maximal functions, reverse H¨older’s inequality and basic result referred to Vitali’scovering lemma (a version of Calder´on-Zygmund decomposition), we are able to prove thelevel-set inequalities by specifying via weighted fractional maximal distribution functions (WFMDs). The understanding of technical ingredients will lead us to establish a moregeneral form of weighted regularity estimates in Lorentz and generalized Orlicz spaces,respectively. Making good use of the WFMDs, we believe that our theoretical results inthis paper can provide a more complete picture in regularity for nonlinear double obstacleproblems, in which some appropriate applications (that appear in many different contexts)could be explored.1.4. Main results.
Before stating the main results in the present paper, let us introducesome important terminologies and conventions. Under some suitable assumptions on thedomain Ω, the leading nonlinearity is in the class of BMO functions satisfying partiallyweak BMO condition (see Section 2 for detailed definition and explanation), we considerthe weak solution u ∈ W ,p (Ω) of the variational inequality (double obstacle problem) ( P )satisfying double constraints ψ ≤ u ≤ ψ with ψ , ψ ∈ W ,p (Ω). Here, we note that thegiven data F ∈ L p (Ω; R n ) and g ∈ L pp − (Ω) for 1 < p < ∞ . For the sake of simplicity, inthe sequel we will often denote F = (cid:16) |∇ ψ | p + |∇ ψ | p + | F | p + | g | pp − (cid:17) p . (1.5)Assuming that the nonlinear operators A , B satisfy conditions (1.2)-(1.3) and (1.4). Thetwo-obstacle problem ( P ) will be investigated in the setting of weighted spaces associatedto a Muckenhoupt weight ω ∈ A ∞ with notation [ ω ] A ∞ = ( ν, c ). Furthermore, for brevity,we shall denote data ≡ data ( n, p, L, [ ω ] A ∞ , diam(Ω) /r ) , for the dependence on a set of parameters. It is worthwhile to note here that as our maintheorems below will show, the universal constant C may depend on data , though it isnot specified explicitly in the statements. On the other hand, throughout this paper, fora suitable regularity parameter δ > r >
0, we will simply write( H ) r,δ to say that Ω is ( r, δ )-Reifenberg flat domain and the operator A satisfies the weak( r, δ ) BMO condition, [ A ] ,r ≤ δ at the same time (see Definitions 2.1 and 2.2 in Section2 below).We are now in the position to state our main results. Firstly in Theorem 1.1, wehighlight a novelty of level-set inequality regarding to the weighted fractional maximaldistribution functions in this study. Based on the WFMD inequality in Theorem 1.1,it enable us to conclude the global regularity results in the classical Lorentz spaces viaTheorem 1.2 and in Orlicz-Lorentz spaces via Theorem 1.4, respectively. Here, it is worthemphasizing that in our main results, global gradient estimates are preserved under frac-tional maximal operators M α (where the ‘fractional derivatives’ ∂u of weak solutions canbe controlled by the norm of the data, see [39]). Once our main Theorem 1.2 is stated,5n obvious corollary now follows (see Corollary 1.3). In addition, this paper also containsthe pointwise estimate of weak solutions to ( P ) in terms of the classical Riesz potential I β , will be also indicated in Theorem 1.2 as following. Theorem 1.1 (Level-set inequality on WFMDs)
For every α ∈ [0 , n ) and < a < ν (cid:0) − αn (cid:1) , one can find ε = ε ( α, a ) > , δ = δ ( α, a, ε ) > and σ = σ ( α, a, ε ) > such that if ( A , Ω) satisfying assumption ( H ) r ,δ , then the following weighted distributioninequality D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ Cε D ω M α ( |∇ u | p ) ( λ ) + D ω M α ( | F | p ) ( σλ ) , (1.6) holds for every < ε < ε and λ > . Here the weighted distribution function is definedby D ωf ( λ ) := ˆ {| f | >λ } ω ( x ) dx, for λ ≥ . (1.7) Theorem 1.2 (Global Lorentz estimates and pointwise regularity)
Assume that givendata M α ( | F | p ) ∈ L q,sω (Ω) for some < q < ∞ , < s ≤ ∞ and α ∈ [0 , n ) . Thenone can find δ = δ ( α, q, s ) > such that if ( A , Ω) satisfies assumption ( H ) r ,δ then M α ( |∇ u | p ) ∈ L q,sω (Ω) with the following inequality k M α ( |∇ u | p ) k L q,sω (Ω) ≤ C k M α ( | F | p ) k L q,sω (Ω) . (1.8) Moreover, for any β ∈ (0 , n ) and < t < ∞ , the following point-wise estimate I β (cid:0) χ Ω | M α ( |∇ u | p ) | t (cid:1) ( x ) ≤ C I β (cid:0) χ Ω | M α ( | F | p ) | t (cid:1) ( x ) , (1.9) holds for almost everywhere x ∈ R n . We then apply Theorem 1.2 to the associated α = 0 and use the boundedness propertyof M to infer the following corollary. This may be more familiar with most of the readersin the same topic. Corollary 1.3
If given data F defined as in (1.5) belongs to the weighted Lorentz space L q,sω (Ω) for some < q < ∞ and < s ≤ ∞ then one can find δ = δ ( q, s ) > such thatif ( A , Ω) satisfies assumption ( H ) r ,δ then ∇ u ∈ L q,sω (Ω) . More precisely, there holds k∇ u k L q,sω (Ω) ≤ C k F k L q,sω (Ω) . (1.10) In addition, for every β ∈ (0 , n ) the following point-wise estimate I β ( χ Ω |∇ u | q )( x ) ≤ C I β ( χ Ω | F | q )( x ) , (1.11) holds for almost everywhere x ∈ R n . Theorem 1.4 (Global Orlicz-Lorentz estimates)
Let Φ be a Young function suchthat Φ ∈ ∆ . Assume that M α ( | F | p ) belongs to the weighted Orlicz-Lorentz space L Φ; q,sω (Ω) for some < q < ∞ , < s ≤ ∞ and α ∈ [0 , n ) . Then one can find δ = δ ( α, q, s ) > such that if ( A , Ω) satisfies assumption ( H ) r ,δ then M α ( |∇ u | p ) ∈ L Φ; q,sω (Ω) according tothe inequality k M α ( |∇ u | p ) k L Φ; q,sω (Ω) ≤ C k M α ( | F | p ) k L Φ; q,sω (Ω) . (1.12)6.5. Outline of the paper.
The remainder of this article will be organized as follows.In Section 2 we introduce much of general notation, basic definitions and a few prelim-inary results that will be needed throughout the paper. The next section 3 focuses onsome crucial ingredients of regularity theory that will be discussed in the context of ourapproach. Section 4 is devoted to proving some comparison results for double obstacleproblems. For most of the research in regularity, the main difficulty is to establish compar-ison estimates (actually the difference between gradients of our solutions and solutions ofstandard homogeneous equations). An important observation is that the step of provingsuch comparison results is one of the key ingredients of our work. Then, in Section 5, westate and prove some preparatory results for the proofs of main results in Section 6 byestablishing level-set inequalities that concerning the WFMDs.
This preparatory section is devoted to providing some notations, conventions and basicdefinitions that will be essential for our main proofs later on. Moreover, we also introducebasic assumptions on problem, state and prove some preliminary results in this section.2.1.
Notation and conventions.
In the sequel, the letter C will be employed torepresent a generic constant, whose value is larger or equal than one, may change fromline to line during chains of estimates. The dependencies of C on special parameters willbe suitably emphasized between parentheses. In what follows, according to the standardnotation, the Lebesgue measure of a measurable set K ⊂ R n is denoted by | K | and wewill use the denotation K hdx = 1 | K | ˆ K hdx as the integral average of a measurable map h ∈ L ( K ). In the paper, Ω will denote an open bounded domain in R n , n ≥ R n of center ξ and radius ρ > { z ∈ R n : | z − ξ | < ρ } ,is simply abbreviated as B ρ ( ξ ). Further, we also set Ω ρ ( ξ ) := B ρ ( ξ ) ∩ Ω, and when thecenter ξ ∈ ∂ Ω, it can be seen as the “surface ball” in R n . Throughout the paper, by anabuse of notation whenever confusion does not arise, the set { x ∈ Ω : | h ( x ) | > τ } is alsowritten as {| h | > τ } for short.2.2. Assumptions on domain and coefficients.Definition 2.1 ( ( r , δ ) -Reifenberg) For < δ < and r > , Ω is called a ( r , δ ) -Reifenberg flat domain or Ω is ( r , δ ) -Reifenberg for brevity if for each ξ ∈ ∂ Ω and each ̺ ∈ (0 , r ] , it is possible to find a coordinate system { y , y , ..., y n } with origin at ξ suchthat B ̺ ( ξ ) ∩ { y n > δ̺ } ⊂ B ̺ ( ξ ) ∩ Ω ⊂ B ̺ ( ξ ) ∩ { y n > − δ̺ } . Definition 2.2 (Partially weak ( r , δ ) − BMO condition)
The operator A is called thatsatisfying a partially weak ( r , δ ) − BMO condition with respect to δ > and r > if [ A ] ,r := sup y ∈ R n , ̺ ∈ (0 ,r ] B ̺ ( y ) θ ( A , B ̺ ( y )) ( x ) dx ≤ δ. (2.1)7 ere the function θ defined by θ ( A , B ̺ ( y )) ( x ) = sup ξ ∈ R n \{ } |A ( x, ξ ) − A B ∗ ̺ ( y ∗ ) ( x , µ ) || ξ | p − , (2.2) where x = ( x , x ∗ ) ∈ R n with x ∗ = ( x , x , ..., x n ) , and A B ∗ ̺ ( y ∗ ) denotes the integral averageof A in B ∗ ̺ ( y ∗ ) , i.e. A B ∗ ̺ ( y ∗ ) = B ∗ ̺ ( y ∗ ) A ( x , x ∗ , µ ) dx ∗ . Remark 2.3
As aforementioned in the introductory section, this type of condition isweaker than the small ( r, δ ) -BMO condition on operator A (assumed on the whole space R n ) and therefore, leading to this new assumption, results will cover a larger class of prob-lems with coefficient operators A considered in [12,13]. It means that there is no regularityrequirement in one variable x i , ≤ i ≤ n , with a little abuse of notation, we say x . Itcan be highly oscillatory (or be a big jump moving) along the x -direction and the smallBMO semi-norm only assumed in x ∗ = ( x , x , ..., x n ) . In [11], authors used a differentterminology of this condition, named ( r, δ ) -vanishing of co-dimension one. We recommendthe readers to [14, 26, 27, 36] for detailed explanations of such requirement. This kind ofassumption has its own significance, for instance, to discuss mathematical representationsof models of elastic laminates or composite materials, see [18, 42] and references relatingdirectly the topic. Definition 2.4 (Muckenhoupt classes)
A non-negative measurable function ω ∈ L p loc ( R n ) is called belonging to A p with p ∈ [1 , ∞ ) , if [ ω ] A p < ∞ , where [ ω ] A p := sup B ̺ ( ξ ) ⊂ R n B ̺ ( ξ ) ω ( z ) dz ! B ̺ ( ξ ) ω ( z ) − p − dz ! p − , if p ∈ (1 , ∞ ) and [ ω ] A := sup B ̺ ( ξ ) ⊂ R n B ̺ ( ξ ) ω ( z ) dz ! sup z ∈ B ̺ ( ξ ) [ ω ( z )] − . In particular, when p = ∞ we say that ω ∈ A ∞ if there exist constants c , ν > satisfying ω ( K ) ≤ c (cid:18) | K || B | (cid:19) ν ω ( B ) , for any measurable subset K of arbitrary ball B in R n , where ω ( K ) := ´ K ω ( z ) dz . In thiscase, we write [ ω ] A ∞ = ( c , ν ) . Such ω satisfies Definition 2.4 is called a Muckenhoupt weight. We also remark heretwo standard properties of the Muckenhoupt classes: A ⊂ A p ⊂ A ∞ for all 1 < p < ∞ and A ∞ = [ p< ∞ A p . Other definitions and Remarks.
In this section, we also give some furtherdefinitions concerning the main results of this paper.8 efinition 2.5 (Weighted Lorentz spaces)
Let < q < ∞ , < s ≤ ∞ and a Muck-enhoupt weight ω ∈ A ∞ . The weighted Lorentz space L q,sω (Ω) is the set which contains allof f ∈ L (Ω) satisfying k f k L q,sω (Ω) is finite, where k f k L q,sω (Ω) := (cid:20) q ˆ ∞ λ s − ω ( { ξ ∈ Ω : | f ( ξ ) | > λ } ) sq dλ (cid:21) s , if s < ∞ and k f k L q, ∞ ω (Ω) := sup λ> λω ( { ξ ∈ Ω : | f ( ξ ) | > λ } ) q . It can be seen that when ω ≡
1, the weighted Lorentz space L q,sω (Ω) becomes the Lorentzspace L q,s (Ω). Moreover, in a special case, the weighted Lorentz space L q,qω (Ω) coincides tothe well-known weighted Lebesgue space L qω (Ω) which contains all of measurable function f satisfying k f k L qω (Ω) := (cid:18) ˆ Ω | f ( z ) | q ω ( z ) dz (cid:19) q < ∞ . Definition 2.6 (Weighted Orlicz-Lorentz spaces)
Let q ∈ (0 , ∞ ) , < s ≤ ∞ and Φ ∈ ∆ be a Young function. A measurable functions f is called belonging to the weightedOrlicz-Lorentz class O Φ; q,sω (Ω) when k Φ( | f | ) k L q,sω (Ω) < ∞ .The weighted Orlicz-Lorentz space L Φ; q,sω (Ω) is known as the smallest linear subspacethat contains O Φ; q,sω (Ω) , equipped to the Luxemburg norm k f k L Φ; q,sω (Ω) = inf n t : t > satisfying (cid:13)(cid:13) Φ (cid:0) t − | f | (cid:1)(cid:13)(cid:13) L q,sω (Ω) ≤ o . Definition 2.7 (Maximal operators)
Let ≤ α ≤ n , we denote by M α the fractionalmaximal operator of f ∈ L ( R n ) , which is given by M α f ( y ) = sup ̺> ̺ α B ̺ ( y ) | f ( z ) | dz, y ∈ R n . Remark that M is exactly the Hardy-Littlewood operator M which is studied in manyliterature. Definition 2.8 (Riesz potential)
Let β ∈ (0 , n ) and f ∈ L ( R n ; R + ) , the fractionalintegral operator (or Riesz potential) of f , denoted by I β , is given as I β f ( x ) = ˆ R n f ( ξ ) dξ | x − ξ | n − β , x ∈ R n . (2.3) In this section, we discuss the main ingredients in our strategy to prove regularity resultsfor double obstacle problems ( P ). Proofs are based on the following three ingredients:some level-set inequalities on weighted fractional maximal distributions (WFDMs), a typeof Vitali’s covering lemma, the construction of reference homogeneous problem - to a9everse H¨older’s inequality. We shall describe each of these key ingredients of this approachbriefly below.3.1. Level-set inequalities on WFDMs.
One of the main ingredient used in this paperis the level-set inequality performed on the so-called (weighted) FDMs. More precisely, inthis paper, we depart from the approach discussed in [52] and explore more on a weightedversion.
Definition 3.1
Let ω ∈ A ∞ and a given ball B ⊂ R n . For λ ≥ the weighted distributionfunction of a measurable mapping f associated to ω in B is defined by D ωf ( λ ; B ) := ω ( { x ∈ Ω ∩ B : | f ( x ) | > λ } ) . (3.1) In particular, we will write D ωf ( λ ) instead of D ωf ( λ ; B ) when the open ball B contains Ω . Here, for two given measurable functions F and G , the important point is that we tryto construct/prove a level-set decay estimates of the type D ω G ( ε − a λ ; B ) ≤ Cε D ω G ( λ ; B ) + D ω F ( σ ε λ ; B ) , (3.2)holds for any 0 < ε ≪ a ∈ (0 , σ ε > ε , a , to concludethe gradient estimates of weak solutions, especially in terms of M α (as we shall see later,the level-set inequality (3.2) involving fractional maximal operators in F and G ). Forthe sake of clarity, in section 5, we shall exclusively concentrate our attention on the useof weighted distribution functions to prove level-set inequalities on WFMDs. This worknaturally extends the recent paper [52] to the double obstacle problems and weightedestimates.3.2. Covering Lemma.
In this study, a version of Calder´on-Zygmund (or Vitali type)covering lemma is in used: the substitution of Calder´on-Zygmund-Krylov-Safonov decom-position, that is more convenient for us to use balls instead of cubes. This lemma is astandard argument of measure theory.
Lemma 3.2 (covering lemma)
Assume that Ω is ( r , δ ) -Reifenberg and ω ∈ A ∞ . Sup-pose that two measurable subsets S ⊂ R of Ω satisfying two following hypotheses:i) ω ( S ) ≤ εω ( B r ) for given ε ∈ (0 , ;ii) for any < ̺ ≤ r and ξ ∈ Ω , if ω ( S ∩ B ̺ ( ξ )) > εω ( B ̺ ( ξ )) then B ̺ ( ξ ) ∩ Ω ⊂ R .Then one can find C > such that ω ( S ) ≤ Cεω ( R ) . To our knowledge, such well-known lemma of Calder´on-Zygmund has been widelyused in many works and developed through the years with several modified versions. Thecurrent version, Lemma 3.2 plays an important role in our main proofs in this paper. Werefer to [16, Lemma 4.2] or [17, 37] for further reading on this lemma and its proof.3.3.
The construction of reference homogeneous problem.
This crucial key step yieldsa reverse Holder’s inequality that allows us to obtain local comparison estimates betweenweak solutions in the interior and on the boundary of domain (stated and proved in Section4). The main idea is that, due to a Gehring type lemma, it enables us to confirm the higherintegrability for the gradient of weak solution V to homogeneous equations of the type − div A ( x , x ∗ , ∇ V ) = 0 in B, (3.3)10here B is any ball whose center belonging to ¯Ω. A very interesting result proved in [36,Theorem 2.1], stated that for any γ ≥
1, there exists a small δ depending on n, p, q andthe structure of A such that if V is a unique solution to the reference problem (3.3) and A satisfies the partially weak ( ρ, δ )-BMO condition, then B ρ |∇ V | γp dx ! γp ≤ C B ρ |∇ V | p dx ! p , for any B ρ ⊂ Ω and the constant C depends only on n, p and the structure of A . Thereader is referred to [14, 36] for proofs and references.3.4. Properties of fractional maximal functions.
Properties of Hardy-Littlewood max-imal function and its fractional operators play a crucial role for gradient estimates of theweak solution to our problem. The maximal function has been successfully used in study-ing regularity theory of partial differential equations. In [30], F. Duzaar and G. Mingionefirst presented the gradient estimates employing fractional maximal functions and non-linear potentials. The study of regularity estimates via fractional maximal operators hasalready been established in our previous paper [61] by using the so-called cutoff fractionalmaximal operators and later in other works [51–53, 61, 62]. An advantage of dealing with M α is that one can conclude both size and oscillations of our solutions, their derivativesincluding fractional derivatives ∂ α u controlled by given data F , see [39]. Therefore, one ofthe main ingredients in our proofs is the boundedness property of the fractional maximalfunction M α . We will use the following lemma, whose detailed proof can be found in [62]. Lemma 3.3
For any α ∈ [0 , n ) and s ≥ , if f ∈ L ( R n ) and αs < n then there holds | { z ∈ R n : M α f ( z ) > λ } | ≤ (cid:18) Cλ s ˆ R n | f ( z ) | s dz (cid:19) nn − αs . This section is intended to establish some comparison estimates, that make them necessaryto derive the estimates for solutions to our problem ( P ) later. Let us start by proving thenext lemma which gives a local comparison gradient estimate between a weak solution u toproblem ( P ) with the unique solution v solved the corresponding quasi-linear homogeneousequations. Lemma 4.1
Let us consider u ∈ W ,p (Ω) as a solution to problem ( P ) and an open ball B ⊂ R n satisfying Ω B := B ∩ Ω = ∅ . Assume that v ∈ u + W ,p (Ω B ) solves the followingequations L ( v ) = 0 in Ω B , and v = u on ∂ Ω B . (4.1) Then for any ε ∈ (0 , one may find a positive number C which still depends on ε suchthat Ω B |∇ v − ∇ u | p dx ≤ ε Ω B |∇ u | p dx + C Ω B | F | p dx. (4.2)11 roof. The idea is to build the comparison inequalities between gradients of u and severalfunctions solved the one obstacle problem and the homogeneous equations , respectively.Our proof here will be divided into three steps. The first step: comparison with the one obstacle problem.
Let us consider u ∈ u + W ,p (Ω B ) and u ≥ ψ a.e. in Ω B as the unique solution to one-sided obstacle problem asfollows L ( u ) ≤ L ( ψ ) , in Ω B . The corresponding variational inequality of this problem is written by ˆ Ω B hA ( x, ∇ u ) , ∇ ( u − ϕ ) i dx ≤ ˆ Ω B hA ( x, ∇ ψ ) , ∇ ( u − ϕ ) i dx, (4.3)for all ϕ ∈ u + W ,p (Ω B ) and ϕ ≥ ψ a.e in Ω B . Note that one may take ϕ = u − ( u − ψ ) + in (4.3) to point out that ˆ Ω B (cid:10) A ( x, ∇ u ) − A ( x, ∇ ψ ) , ∇ (cid:0) ( u − ψ ) + (cid:1)(cid:11) dx ≤ , and make use of (1.3), it yields ˆ D ( |∇ u | + |∇ ψ | ) p − |∇ u − ∇ ψ | dx ≤ , (4.4)where D = { x ∈ Ω B : u ≥ ψ } . Now, due to the following fundamental inequality | γ − γ | p ≤ ε | γ | p + C ( p, ε )( | γ | + | γ | ) p − | γ − γ | , (4.5)for every γ , γ ∈ R n and ε >
0, we deduce that ˆ B |∇ (( u − ψ ) + ) | p dx = ˆ D |∇ ( u − ψ ) | p dx ≤ ε ˆ D ( |∇ u | p + |∇ ψ | p ) dx + C ˆ D ( |∇ u | + |∇ ψ | ) p − |∇ u − ∇ ψ | dx ≤ ε ˆ D ( |∇ u | p + |∇ ψ | p ) dx. (4.6)It is noticeable here that the last estimate comes from (4.4). Letting ε ց u ≤ ψ a.e. in Ω B . It allows us to extend u to Ω \ Ω B by u such that ψ ≤ u ≤ ψ a.e. and u − u = 0 in Ω \ Ω B . Taking ϕ = u in ( P ) and plugging to (4.3) with ϕ = u , itleads to ˆ Ω B hA ( x, ∇ u ) − A ( x, ∇ u ) , ∇ ( u − u ) i dx ≤ ˆ Ω B hB ( x, F ) , ∇ ( u − u ) i dx − ˆ Ω B hA ( x, ∇ ψ ) , ∇ ( u − u ) i dx + ˆ Ω B g ( u − u ) dx. ˆ Ω B ( |∇ u | + |∇ u | ) p − |∇ u − ∇ u | dx ≤ C ( L ) (cid:18) ˆ Ω B |∇ ( u − u ) || F | p − dx + ˆ Ω B |∇ ( u − u ) ||∇ ψ | p − dx + ˆ Ω B | g || u − u | dx (cid:19) . (4.7)Since u − u ∈ W ,p (Ω B ), we are able to apply Sobolev’s inequality to find out that ˆ Ω B | u − u | p dx ≤ C ˆ Ω B |∇ u − ∇ u | p dx, and together with H¨older and Young’s inequalities, one guarantees that ˆ Ω B | g || u − u | dx ≤ ε ˆ Ω B |∇ u − ∇ u | p dx + C ( p, ε ) ˆ Ω B | g | pp − dx, (4.8)for every ε >
0. We apply again H¨older and Young’s inequalities for two remain termsto discover from (4.7) and (4.8) that ˆ Ω B ( |∇ u | + |∇ u | ) p − |∇ u − ∇ u | dx ≤ ε ˆ Ω B |∇ u − ∇ u | p dx + C ( L, p, ε ) ˆ Ω B | F | p dx. (4.9)For every ε ∈ (0 ,
1) let us apply (4.5) to have ˆ Ω B |∇ u − ∇ u | p dx ≤ ε ˆ Ω B |∇ u | p dx + C ( p, ε ) ˆ Ω B ( |∇ u | + |∇ u | ) p − |∇ u − ∇ u | dx ≤ ε ˆ Ω B |∇ u | p dx + ε C ( p, ε ) ˆ Ω B |∇ u − ∇ u | p dx + C ( L, p, ε , ε ) ˆ Ω B | F | p dx, (4.10)in which the last estimate comes from (4.9). It is very easy to take a suitable value of ε depending ε in (4.10) to arrive ˆ Ω B |∇ u − ∇ u | p dx ≤ ε ˆ Ω B |∇ u | p dx + C ( L, p, ε ) ˆ Ω B | F | p dx. (4.11) The second step: connection to the below constraint.
Next, let us consider u as thesolution to the equations L ( u ) = L ( ψ ) in Ω B , and u = u on ∂ Ω B . (4.12)Since u = u ≥ ψ a.e. on ∂ Ω B so it deduces that u ≥ ψ a.e. in Ω B by proceeding thesame method at the beginning of the proof. So we can take ϕ = u in (4.3) to get that ˆ Ω B hA ( x, ∇ u ) , ∇ u − ∇ u i dx ≤ ˆ Ω B hA ( x, ∇ ψ ) , ∇ u − ∇ u i dx.
13e then combine with testing the variational formula of (4.12) by u − u to point out ˆ Ω B hA ( x, ∇ u ) − A ( x, ∇ u ) , ∇ ( u − u ) i dx = ˆ Ω B hA ( x, ∇ ψ ) − A ( x, ∇ ψ ) , ∇ ( u − u ) i dx. (4.13)The similar technique as the proof of (4.11) will be used again to obtain from (4.13) that ˆ Ω B |∇ u − ∇ u | p dx ≤ ε ˆ Ω B |∇ u | p dx + C ( L, p, ε ) ˆ Ω B ( |∇ ψ | p + |∇ ψ | p ) dx. (4.14) The third step: comparison with the homogeneous equation.
Let us now define by v the solution to the homogeneous problem L ( v ) = 0 in Ω B , and v = u on ∂ Ω B . (4.15)Since u = u = u on ∂ Ω B so the problem (4.15) is exactly (4.1). We can obtain that ˆ Ω B |∇ u − ∇ v | p dx ≤ ε ˆ Ω B |∇ v | p dx + C ( L, p, ε ) ˆ Ω B |∇ ψ | p dx. (4.16)Finally, let us combine all estimates in (4.11), (4.14) and (4.16) to conclude (4.2), withthe fact that both terms |∇ u | p and |∇ v | p can be controlled by |∇ u | p and the data .We next consider another homogeneous equation regarding to the average of A overthe ball B ∗ ̺ in R n − , whenever B ̺ ⊂ B . The interesting character of the solution to thishomogeneous problem is that its gradient still satisfies a type of reverse H¨older inequality.Moreover, we can establish the local interior difference between gradients of v and thesolution V of this equation via the following lemma. Lemma 4.2
Consider v as a solution to (4.1) with the ball B ⊂ Ω and consider a ball B ̺ ⊂ B for ̺ > . Assume that V ∈ v + W ,p ( B ̺ ) solves the following problem − div( A B ∗ ̺ ( x , ∇ V )) = 0 , in B ̺ , and V = v, on ∂B ̺ . (4.17) Then for every γ ≥ there holds B ̺ |∇ V | γp dx ! γ ≤ C B ̺ |∇ V | p dx. (4.18) Moreover if [ A ] , ̺ ≤ δ then one has B ̺ |∇ v − ∇ V | p dx ≤ Cδ B ̺ |∇ v | p dx. (4.19) Proof.
Let us first refer to [36, Theorem 2.1] for the proof of (4.18). In order toprove (4.19), we will test the variational formulas of (4.1) and (4.17) by v − V ∈ W ,p ( B ̺ ).One obtains that B ̺ hA B ∗ ̺ ( x , ∇ v ) − A B ∗ ̺ ( x , ∇ V ) , ∇ v − ∇ V i dx = B ̺ hA B ∗ ̺ ( x , ∇ V ) − A ( x , ∇ V ) , ∇ v − ∇ V i dx, θ in (2.2) that B ̺ ( |∇ V | + |∇ v | ) p − |∇ V − ∇ v | dx ≤ L B ̺ | θ ( A , B ̺ ) ||∇ V | p − |∇ v − ∇ V | dx. (4.20)Thanks to H¨older and Young’s inequalities, we deduce from (4.20) that B ̺ ( |∇ V | + |∇ v | ) p − |∇ V − ∇ v | dx ≤ ε B ̺ |∇ V − ∇ v | p dx + C ( p, L, ε ) B ̺ | θ ( A , B ̺ ) | pp − |∇ V | p dx, (4.21)for any ε >
0. Moreover, we note that condition (1.2) ensures that | θ ( A , B ̺ ) | ≤ L .Therefore, thanks to H¨older’s inequality and (4.18) with assumption [ A ] , ̺ ≤ δ , for every ǫ > B ̺ | θ ( A , B ̺ ) | pp − |∇ V | p dx ≤ B ̺ | θ ( A , B ̺ ) | (1+ ǫ ) pp − dx ! ǫ B ̺ |∇ V | (1+ ǫ ) pǫ dx ! ǫ ǫ ≤ L pǫ +1( p − ǫ ) B ̺ | θ ( A , B ̺ ) | dx ! ǫ B ̺ |∇ V | p dx ! ≤ L pǫ +1( p − ǫ ) δ ǫ B ̺ |∇ V | p dx. (4.22)Passing ǫ to 0 in (4.22), we have B ̺ | θ ( A , B ̺ ) | pp − |∇ V | p dx ≤ L p − δ B ̺ |∇ V | p dx ≤ C ( p, L ) δ B ̺ |∇ v | p dx. (4.23)Substituting (4.23) into (4.21), we conclude that B ̺ ( |∇ V | + |∇ v | ) p − |∇ V − ∇ v | dx ≤ ε B ̺ |∇ V − ∇ v | p dx + C ( p, L, ε ) δ B ̺ |∇ v | p dx. (4.24)Moreover, the fundamental inequality (4.5) gives us |∇ V − ∇ v | p ≤ ε ( |∇ V | + |∇ v | ) p + C ( p, ε )( |∇ V | + |∇ v | ) p − |∇ V − ∇ v | , (4.25)for every ε >
0. Combining between (4.24) and (4.25), it arrives to B ̺ |∇ V − ∇ v | p dx ≤ ε B ̺ ( |∇ V | + |∇ v | ) p dx + ε C ( p, ε ) B ̺ |∇ V − ∇ v | p dx + C ( p, L, ε , ε ) δ B ̺ |∇ v | p dx ≤ [4 p ε + C ( p, ε ) ε ] B ̺ |∇ V − ∇ v | p dx + [4 p ε + C ( p, L, ε , ε ) δ ] B ̺ |∇ v | p dx. (4.26)15inally, by taking ε ∈ (0 , − p − ) satisfying ε = 4 p ε [ C ( p, ε )] − and ε = 4 − p C ( p, L, ε , ε ) δ, we may conclude (4.19) from (4.26).In order to obtain the comparison estimates near the boundary, we need an additionalassumption on ∂ Ω related to Reifenberg flatness condition (this hypothesis exhibits a verylow level of regularity). The next lemma also plays a useful tool to verify the boundaryversion of comparison estimates. Similar to the above argument as in the previous Lemma4.2, with Ω is ( r , δ )-Reifenberg flatness, we also conclude the comparison result on theboundary. The analogous proof technique can be found in several articles such as [13, 44,60]. Lemma 4.3
Let v be a solution to (4.1) and consider Ω ̺ := B ̺ ∩ Ω ⊂ Ω B for some ̺ > . Assume that V ∈ v + W ,p (Ω ̺ ) solves the following problem − div( A Ω ∗ ̺ ( x , ∇ V )) = 0 , in Ω ̺ , and V = v, on ∂ Ω ̺ . (4.27) If ( A , Ω) satisfies assumption ( H ) r ,δ then there holds Ω ̺ |∇ V | γp dx ! γ ≤ C Ω ̺ |∇ V | p dx, (4.28) for every γ ≥ and Ω ̺ |∇ v − ∇ V | p dx ≤ Cδ Ω ̺ |∇ v | p dx. (4.29) The idea of our approach in this paper is to take advantages of weighted fractional maximaldistributions to establish the “good- λ ” level-set inequalities. Therefore, the purpose of thissection is to give some inequalities associated with the WFMDs. It is worth emphasizingthat the construction of these inequalities is the key technique to prove global regularityestimates in the spirit of WFMDs.Given ω ∈ A ∞ , ξ ∈ Ω and ̺ >
0. In what follows, for f ∈ L (Ω) we will define themeasurable set W f ( λ ; B ̺ ( ξ )) as follows W f ( λ ; B ̺ ( ξ )) := { x ∈ Ω : | f ( x ) | > λ } ∩ B ̺ ( ξ ) . (5.1)For simplicity of notation, when the open ball B ̺ ( ξ ) contains Ω we will write W f ( λ ) insteadof W f ( λ ; B ̺ ( ξ )). Moreover, we remind that the distribution function D ωf mentioned inthis section is defined as in (3.1). Lemma 5.1
For every ε > and a > , one can find σ = σ ( ε, a ) > such that if thereexists ξ ∈ Ω satisfying M α ( | F | p )( ξ ) ≤ σλ for some λ > then there holds D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ εω ( B r ) . (5.2)16 roof. Thanks to Lemma 3.3 and Lemma 4.1 with B ⊃ Ω and v ≡
0, from the definitionof the set W M α in (5.1) it gives us (cid:12)(cid:12) W M α ( |∇ u | p ) ( ε − a λ ) (cid:12)(cid:12) ≤ (cid:18) Cε − a λ ˆ Ω |∇ u | p dx (cid:19) nn − α ≤ (cid:18) Cε − a λ ˆ Ω | F | p dx (cid:19) nn − α . (5.3)Recall that ξ ∈ Ω satisfying M α ( | F | p )( ξ ) ≤ σλ (the value of σ will be clarified later),it enables us to cover Ω by an open ball centered at ξ ∈ Ω and radius r = 2diam(Ω), itleads to ˆ Ω | F | p dx ≤ Cr n − α r α B r ( ξ ) | F | p dx ! ≤ Cr n − α M α ( | F | p )( ξ ) ≤ Cr n − α σλ. (5.4)Substituting (5.4) into (5.3), there holds (cid:12)(cid:12) W M α ( |∇ u | p ) ( ε − a λ ) (cid:12)(cid:12) ≤ C ( σε a ) nn − α r n ≤ C (diam(Ω) /r ) n ( σε a ) nn − α | B r | , which implies from the definitions of Muckenhoupt weight ω and function D ω M α ( |∇ u | p ) that D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ c (cid:12)(cid:12) W M α ( |∇ u | p ) ( ε − a λ ) (cid:12)(cid:12) | B r | ! ν ω ( B r ) ≤ C (diam(Ω) /r ) nν ( σε a ) nνn − α ω ( B r ) . (5.5)Let us take σ depending on ε , a and data in (5.5) such that0 < C (diam(Ω) /r ) nν ( σε a ) nνn − α < ε, (5.6)to conclude (5.2) and finish the proof. Lemma 5.2
Let a > and ξ ∈ B ̺ ( ξ ) satisfying M α ( |∇ u | p )( ξ ) ≤ λ . Then one can find ξ ∈ Ω and k ∈ N such that the following inequality D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ D ω M α ( χ Bk̺ ( ξ |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) , (5.7) holds for every ε ∈ (cid:16) , − na (cid:17) . Proof.
For every ζ ∈ B ̺ ( ξ ), it is easy to check that B r ( ζ ) ⊂ B r ( ξ ) for all r ≥ ̺ , whichallows us take into account assumption M α ( |∇ u | p )( ξ ) ≤ λ to findsup r ≥ ̺ r α B r ( ζ ) |∇ u | p dx ≤ n sup r ≥ ̺ r α B r ( ξ ) |∇ u | p dx ≤ n − α M α ( |∇ u | p )( ξ ) ≤ n λ. Therefore we may conclude that M α ( |∇ u | p )( ζ ) ≤ max ( sup 17f we choose ε = 3 − na then for every ε ∈ (0 , ε ), one has W M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) = ( sup For every < a < ν (cid:0) − αn (cid:1) , one can find a constant ε = ε ( data ) ∈ (cid:16) , − na (cid:17) and numbers σ = σ ( a, ε ) > , δ = δ ( a, ε ) such that if ( A , Ω) satisfies assumption ( H ) r ,δ and there are ξ , ξ ∈ B ̺ ( ξ ) satisfying M α ( |∇ u | p )( ξ ) ≤ λ and M α ( | F | p )( ξ ) ≤ σλ, (5.9) for some λ > then for every ε ∈ (0 , ε ) there holds D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) < εω ( B ̺ ( ξ )) . (5.10) Proof. Lemma 5.2 under condition (5.9) gives us the existence of ξ ∈ Ω and k ≤ ε ∈ (cid:16) , − na (cid:17) . We define B i = B i − k̺ ( ξ ) for i = 1 , , v ∈ u + W ,p ( B ) solves the problem L ( v ) = 0 in B ∩ Ω , and v = u on ∂ ( B ∩ Ω) . Lemma 4.1 gives us the comparison estimate between ∇ v and ∇ u as below B |∇ u − ∇ v | p dx ≤ ε B |∇ u | p dx + C ( ε ) B | F | p dx, (5.11)for any ε > 0. By setting of B , it is possible to claim that B ̺ ( ξ ) ⊂ B = B k̺ ( ξ ) ⊂ B ⊂ B ̺ ( ξ ) ⊂ B ̺ ( ξ ) ⊂ B ̺ ( ξ ) ∩ B ̺ ( ξ ) , and of course | B | ∼ ̺ n . Combining with (5.9), one has B |∇ u | p dx ≤ | B ̺ ( ξ ) || B | B ̺ ( ξ ) |∇ u | p dx ≤ C̺ − α M α ( |∇ u | p )( ξ ) ≤ Cλ̺ − α , (5.12)and similarly B | F | p dx ≤ | B ̺ ( ξ ) || B | B ̺ ( ξ ) | F | p dx ≤ C̺ − α M α ( | F | p )( ξ ) ≤ Cσλ̺ − α . (5.13)18ubstituting (5.12) and (5.13) into (5.11), we find B |∇ u − ∇ v | p dx ≤ C [ ε + C ( ε ) σ ] λ̺ − α . (5.14)Let us now consider V ∈ v + W ,p ( B ) solving the next problem ( − div( A B ∗ ( x , ∇ V )) = 0 , in B ∩ Ω ,V = v, on ∂ ( B ∩ Ω) . Lemma 4.2 and 4.3 state that if ( A , Ω) satisfies assumption ( H ) r ,δ then ∇ V satisfies thefollowing reverse H¨older’s inequality (cid:18) B |∇ V | γp dx (cid:19) γ ≤ C B |∇ V | p dx, for all γ ≥ , (5.15)and the comparison estimate with ∇ v as below B |∇ v − ∇ V | p dx ≤ Cδ B |∇ v | p dx. (5.16)On the other hand, from inequality (5.7) in Lemma 5.2 and the definition of Muckenhouptweight ω ∈ A ∞ , there holds D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ D ω M α ( χ B |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ c (cid:12)(cid:12)(cid:12) W M α ( χ B |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) (cid:12)(cid:12)(cid:12) | B ̺ ( ξ ) | ν ω ( B ̺ ( ξ )) , (5.17)where ( c , ν ) = [ ω ] A ∞ . By using an elementary inequality one deduces from (5.17) that D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ C (I + II + III) ν ̺ − nν ω ( B ̺ ( ξ )) , (5.18)where I, II and III are given byI := (cid:12)(cid:12)(cid:8) M α ( χ B |∇ u − ∇ v | p ) > − p ε − a λ (cid:9)(cid:12)(cid:12) , II := (cid:12)(cid:12)(cid:8) M α ( χ B |∇ v − ∇ V | p ) > − p ε − a λ (cid:9)(cid:12)(cid:12) , III := (cid:12)(cid:12)(cid:8) M α ( χ B |∇ V | p ) > − p ε − a λ (cid:9)(cid:12)(cid:12) . Thanks to Lemma 3.3 with s = 1, there holdsI ≤ (cid:18) C − p ε − a λ ˆ B |∇ u − ∇ v | p dx (cid:19) nn − α ≤ (cid:18) C | B | ε − a λ B |∇ u − ∇ v | p dx (cid:19) nn − α , which with (5.14) implies toI ≤ C (cid:2) ε a ( ε + C ( ε ) σ ) ̺ n − α (cid:3) nn − α ≤ C̺ n [ ε a ε + C ( ε ) ε a σ ] nn − α . (5.19)19e now apply this argument again to estimate II, by combining Lemma 3.3 with s = 1and (5.16) to arriveII ≤ (cid:18) C − p ε − a λ ˆ B |∇ v − ∇ V | p dx (cid:19) nn − α ≤ (cid:18) Cδ | B | ε − a λ B |∇ v | p dx (cid:19) nn − α . Taking into account (5.9) and (5.14), one has B |∇ v | p dx ≤ C (cid:18) B |∇ u | p dx + B |∇ u − ∇ v | p dx (cid:19) ≤ C [1 + ε + C ( ε ) σ ] λ̺ − α . Both previous inequalities give usII ≤ C [ δε a (1 + ε + C ( ε ) σ )] nn − α ̺ n . (5.20)For any θ > 1, the last term III can be bounded by using Lemma 3.3 with s = θ > γ = θ to haveIII ≤ C | B | (3 − p ε − a λ ) θ B |∇ V | pθ dx ! nn − αθ ≤ " C | B | ( ε − a λ ) θ (cid:18) B |∇ V | p dx (cid:19) θ nn − αθ . (5.21)From (5.16) with δ ∈ (0 , B |∇ V | p dx ≤ C (cid:18) B |∇ v | p dx + B |∇ v − ∇ V | p dx (cid:19) ≤ C B |∇ v | p dx, which allows us to arrive the following conclusion by collecting the previous computation B |∇ V | p dx ≤ C [1 + ε + C ( ε ) σ ] λ̺ − α . Substituting this estimate into (5.21), one can findIII ≤ C (cid:20) ̺ n ( ε − a λ ) θ (1 + ε + C ( ε ) σ ) θ λ θ ̺ − αθ (cid:21) nn − αθ ≤ C [ ε a (1 + ε + C ( ε ) σ )] nθn − αθ ̺ n . (5.22)Plugging estimations of I, II and III from (5.19), (5.20) and (5.22) respectively, one getsfrom (5.18) that D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ C n [ ε a ( ε + C ( ε ) σ )] nn − α ̺ n + [ δε a (1 + ε + C ( ε ) σ )] nn − α ̺ n + [ ε a (1 + ε + C ( ε ) σ )] nθn − αθ ̺ n o ν ̺ − nν ω ( B ̺ ( ξ )) . (5.23)In the inequality (5.23), it is possible to choose σ satisfying (5.6) and ε , δ such that( δε a ) nνn − α < ε , ε = δ ∈ (0 , , < σ < ε [ C ( ε )] − , D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ C (cid:16) ε ν + ε anθn − αθ (cid:17) ν ω ( B ̺ ( ξ )) . (5.24)The most interesting point here is that assumption 0 < a < ν (cid:0) − αn (cid:1) allows us to take θ = aν + αn > anθn − αθ = ν . With this choice of θ , inequality (5.24)becomes to D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) ≤ Cε ω ( B ̺ ( ξ )) , and it follows to (5.10) for ε small enough. The proof is then complete. Our strategy now becomes clear and with aid of preliminary lemmas and estimates provedin previous sections, we are ready to prove the main results. Proof of Theorem 1.1. One can see that the inequality (1.6) is a sequence of thefollowing inequality ω (cid:0) W M α ( |∇ u | p ) ( ε − a λ ) ∩ (cid:0) W M α ( | F | p ) ( σλ ) (cid:1) c (cid:1) ≤ Cε (cid:0) W M α ( |∇ u | p ) ( λ ) (cid:1) . (6.1)Therefore we have just determined ε = ε ( data ) ∈ (0 , 1) such that: for any λ > ε ∈ (0 , ε ) and a ∈ (cid:0) , ν (cid:0) − αn (cid:1)(cid:1) we can find δ = δ ( a, ε ) > σ = σ ( a, ε ) > r , δ )-Reifenberg and [ A ] ,r ≤ δ for some r > ω ( S λε ) ≤ Cεω ( R λ ), where S λε and R λ present thesets appeared on the left and right hand side respectively. The proof is straightforwardfrom the covering Lemma 3.2 for two sets S λε and R λ . Hence we proceed to show that twohypotheses of Lemma 3.2 are satisfied.Obviously, we can prove (6.1) with assumption S λε = ∅ which allows us to have ξ ∈ Ωsuch that M ( | F | p )( ξ ) ≤ σλ . Given r > 0, Lemma 5.1 gives us a suitable value of σ = σ ( ε )that is valid the following inequality ω ( S λε ) ≤ D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ εω ( B r ) . On the other hand, Lemma 5.3 shows that if there exist ξ ∈ B ̺ ( ξ ) ∩ Ω ∩ ( R λ ) c and ξ ∈ S λε ∩ B ̺ ( ξ ) which deduce to M ( |∇ u | p )( ξ ) ≤ λ and M ( | F | p )( ξ ) ≤ ε b λ, then one can find ε ∈ (0 , σ = σ ( a, ε ) > δ = δ ( a, ε ) such that ω ( S λε ∩ B ̺ ( ξ )) ≤ D ω M α ( |∇ u | p ) ( ε − a λ ; B ̺ ( ξ )) < εω ( B ̺ ( ξ )) , for all ε ∈ (0 , ε ) provided Ω is ( r , δ )-Reifenberg and [ A ] ,r ≤ δ . For this reason, thesecond hypothesis of Lemma 3.2 can be directly obtained by contradiction.Therefore, the inequality (6.1) holds for 0 < ε < ε , and the conclusion of weighteddistribution inequality (1.6) also follows. 21 .2 Proof of Theorem 1.2 and Corollary 1.3 Proof of Theorem 1.2. Here, our attention has been focused on the case of L q,sω (Ω) for0 < q < ∞ and 0 < s < ∞ . The proof for the case s = ∞ is also obtained with a slightchanging of calculation.Firstly, for any α ∈ [0 , n ) and ( ν, c ) = [ ω ] A ∞ let us take0 < a < min (cid:26) ν (cid:16) − αn (cid:17) ; 1 q (cid:27) . Theorem 1.1 ensures the existence of ε ∈ (0 , δ > σ > H ) r ,δ of ( A , Ω) is satisfied, then the inequality (1.6) holds for every 0 < ε < ε and λ > D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ Cε D ω M α ( |∇ u | p ) ( λ ) + D ω M α ( | F | p ) ( σλ ) . (6.2)By applying (6.2) and performing several times of changing variables, one gets that k M α ( |∇ u | p ) k sL q,sω (Ω) = ε − as q ˆ ∞ λ s − h D ω M α ( |∇ u | p ) ( ε − a λ ) i sq dλ ≤ Cε − as q ˆ ∞ λ s − h ε D ω M α ( |∇ u | p ) ( λ ) i sq dλ + Cε − as q ˆ ∞ λ s − h D ω M α ( | F | p ) ( σλ ) i sq dλ ≤ Cε sq − as k M α ( |∇ u | p ) k sL q,sω (Ω) + Cσ s ε − as k M α ( | F | p ) k sL q,sω (Ω) , which leads to the following estimate with an elementary inequality k M α ( |∇ u | p ) k L q,sω (Ω) ≤ Cε q − a k M α ( |∇ u | p ) k L q,sω (Ω) + Cσε − a k M α ( | F | p ) k L q,sω (Ω) . (6.3)Since q − a > ε in (6.3) small enough to conclude (1.8).To prove the point-wise estimate (1.9) related to the Riesz potential defined as in (2.3),we refer to [60, Lemma 4.2] for the statement: if the following inequality ˆ R n ϕ ( x ) dω ( x ) ≤ C ˆ R n ψ ( x ) dω ( x ) , (6.4)holds for any ω ∈ A and β ∈ (0 , n ), then I β ϕ ( x ) ≤ C I β ψ ( x ) , a.e. in R n . (6.5)A nice feature of the weighted Lorentz space L q,sω (Ω) is that it becomes the weightedLebesgue space L qω (Ω) in the special case q = s . Hence, for 0 < t < ∞ let us apply (1.8)for q = s = t , one obtains that ˆ R n χ Ω | M α ( |∇ u | p ) | t dω ( z ) ≤ C ˆ R n χ Ω | M α ( | F | p ) | t dω ( z ) , which is valid (6.4) with ϕ = χ Ω | M α ( |∇ u | p ) | t and ψ = χ Ω | M α ( | F | p ) | t . Therefore, onededuces to (1.9) from (6.5) directly and the proof is finished.22 .3 Proof of Theorem 1.4 Proof. Since Φ ∈ ∆ , it is well-know that one can find a constant p > tλ ) ≤ Ct p Φ( λ ) , for any t ≥ λ > . (6.6)For every 0 < q < ∞ and 0 < s < ∞ , let us choose0 < a < min (cid:26) ν (cid:16) − αn (cid:17) ; 1 p q (cid:27) . (6.7)Thanks to Theorem 1.1, for any λ > ε small enough there exist δ = δ ( a, α, ε ) > σ = σ ( a, α, ε ) > H ) r ,δ of ( A , Ω) is satisfied then thereholds D ω M α ( |∇ u | p ) ( ε − a λ ) ≤ Cε D ω M α ( |∇ u | p ) ( λ ) + D ω M α ( | F | p ) ( σλ ) . (6.8)Let us replace λ in (6.8) by ε a Φ − ( λ ), it becomes to D ω M α ( |∇ u | p ) (Φ − ( λ )) ≤ Cε D ω M α ( |∇ u | p ) ( ε a Φ − ( λ )) + D ω M α ( | F | p ) ( σε a Φ − ( λ )) , which is equivalent to D ω Φ( M α ( |∇ u | p )) ( λ ) ≤ Cε D ω Φ( ε − a M α ( |∇ u | p )) ( λ ) + D ω Φ( σ − ε − a M α ( | F | p )) ( λ ) . We may (6.6) on this inequality to arrive D ω Φ( M α ( |∇ u | p )) ( λ ) ≤ Cε D ω Φ( M α ( |∇ u | p )) ( Cε ap λ ) + D ω Φ( M α ( | F | p )) ( Cσ p ε ap λ ) . (6.9)Let us now use (6.9) into the norm expression of the weighted Lorentz space to arrive k Φ( M α ( |∇ u | p )) k sL q,sω (Ω) = q ˆ ∞ λ s − h D ω Φ( M α ( |∇ u | p )) ( λ ) i sq dλ ≤ Cε sq q ˆ ∞ λ s − h D ω Φ( M α ( |∇ u | p )) ( Cε ap λ ) i sq dλ + Cq ˆ ∞ λ s − h D ω Φ( M α ( | F | p )) ( Cσ p ε ap λ ) i sq dλ. By changing of variables, we may write k Φ( M α ( |∇ u | p )) k sL q,sω (Ω) ≤ Cε sq − sap q ˆ ∞ λ s − h D ω Φ( M α ( |∇ u | p )) ( λ ) i sq dλ + Cσ − sp ε − sap q ˆ ∞ λ s − h D ω Φ( M α ( | F | p )) ( λ ) i sq dλ ≤ Cε sq − sap k Φ( M α ( |∇ u | p )) k sL q,sω (Ω) + Cσ − sp ε − sap k Φ( M α ( | F | p )) k sL q,sω (Ω) , which implies to k Φ( M α ( |∇ u | p )) k L q,sω (Ω) ≤ Cε q − ap k Φ( M α ( |∇ u | p )) k L q,sω (Ω) + Cσ − p ε − ap k Φ( M α ( | F | p )) k L q,sω (Ω) . (6.10)23ith the value of a chosen as in (6.7), one can fix ε in (6.10) small enough to observe that k Φ( M α ( |∇ u | p )) k L q,sω (Ω) ≤ C ∗ k Φ( M α ( | F | p )) k L q,sω (Ω) . (6.11)By scaling λ − |∇ u | p , λ − | F | p on weighted distribution inequality (6.8) and using the con-vexity of Φ, we obtain a similar estimate as in (6.11) for any λ > 0. More precisely, onegets that (cid:13)(cid:13) Φ (cid:0) ( C ∗ λ ) − M α ( |∇ u | p ) (cid:1)(cid:13)(cid:13) L q,sω (Ω) ≤ C ∗− (cid:13)(cid:13) Φ (cid:0) λ − M α ( |∇ u | p ) (cid:1)(cid:13)(cid:13) L q,sω (Ω) ≤ (cid:13)(cid:13) Φ (cid:0) λ − M α ( | F | p ) (cid:1)(cid:13)(cid:13) L q,sω (Ω) , ∀ λ > . This estimate yields that H ( ∇ u ) ⊂ C ∗ H ( F ) which implies to (1.12), where H ( f ) = n λ > (cid:13)(cid:13) Φ (cid:0) λ − M α ( | f | p ) (cid:1)(cid:13)(cid:13) L q,sω (Ω) ≤ o , with f = ∇ u or f = F . A slight changing of computation allows us to prove (6.11)and (1.12) even in the case s = ∞ . References [1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems , Duke Math. J. (2007), 285–320.[2] K. Adimurthi, N.C. Phuc, Global Lorentz and Lorentz-Morrey estimates below the natural exponentfor quasilinear equations , Calc. Var. Partial Differential Equations (3) (2015), 3107–3139.[3] P. Baroni, Lorentz estimates for obstacle parabolic problems , Nonlinear Anal. (2014), 167–188.[4] M. Bildhauer, M. Fuchs, G. Mingione, A priori gradient bounds and local C ,α estimates for (double)obstacle problems under non-standard growth conditions , Z. Anal. Anwendungen (4) (2001), 959–985.[5] V. B¨ogelein, F. Duzzar, G. Mingione, Degenerate problems with irregular obstacles , J. Reine Angew.Math. (2011), 107-160.[6] V. B¨ogelein, C. Scheven, Higher integrability in parabolic obstacle problems , Forum Math. (5)(2012), 931-972.[7] S.-S. Byun, Y. Cho, L. Wang, Calder´on-Zygmund theory for nonlinear elliptic problems with irregularobstacles , J. Funct. Anal. (10) (2012), 3117-3143.[8] S.-S. Byun, S. Liang, J. Ok, Irregular Double Obstacle Problems with Orlicz Growth , J. Geom. Anal. (2020), 1965-1984.[9] S.-S. Byun, D. K. Palagachev, P. Shin, Global Sobolev regularity for general elliptic equations of p -Laplacian type , Calc. Var. Partial Differential Equations (2018), pp 135.[10] S.-S. Byun, S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear ellipticequations , Ann. Inst. H. Poincar´e AN (2013), 291–313.[11] S.-S. Byun, S. Ryu, Gradient estimates for nonlinear elliptic double obstacle problems , NonlinearAnal. (2020), 111333.[12] S.-S. Byun, L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains , Comm. PureAppl. Math. (2004), 1283–1310. 13] S.-S. Byun, L. Wang, Elliptic equations with BMO nonlinearity in Reifenberg domains , Adv. Math. (6) (2008), 1937-1971.[14] S.-S. Byun, L. Wang, Elliptic equations with measurable coefficients in Reifenberg domains , Adv.Math. (5) (2010), 2648–2673.[15] S.-S. Byun, L. Wang, S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenbergdomains , J. Funct. Anal. (1) (2007), 167-196.[16] L. A. Caffarelli, X. Cabr´e, Fully nonlinear elliptic equations , American Mathematical Society Collo-quium Publications, American Mathematical Society, Providence (1) (1995), 1-21.[17] L. A. Caffarelli, I. Peral, On W ,p estimates for elliptic equations in divergence form , Commun. PureAppl. Math. (1) (1998), 1-21.[18] M. Chipot, D. Kinderlehrer, G. Vergara-Caffarelli, Smoothness of linear laminates , Arch. Ration.Mech. Anal. (1) (1986), 81-96.[19] H. J. Choe, Regularity for certain degenerate elliptic double obstacle problems , J. Math. Anal. Appl. (1) (1992), 111-126.[20] H. J. Choe, J. L. Lewis, On the obstacle problem for quasilinear elliptic equations of p -Laplacian type ,SIAM J. Math. Anal. (3) (1991), 623-638.[21] H. J. Choe, P. Souksomvang, Elliptic gradient constraint problem , Comm. in Partial DifferentialEquations, (12) (2016), 1918-1933.[22] A. Cianchi, V.G. Mazya, Global boundedness of the gradient for a class of nonlinear elliptic systems ,Arch. Ration. Mech. Anal. (1) (2014), 129–177.[23] L. Codenotti, M. Lewicka, J. J. Manfredi, Discrete approximations to the double-obstacle problem,and optimal stopping of tug-of-war games, Trans. Amer. Math. Soc. (2017), 7387-7403.[24] M. Colombo, G. Mingione, Calder´on-Zygmund estimates ans non-uniformly elliptic operators , J.Funct. Anal. (4) (2016), 1416-1478.[25] G. Dal Maso, U. Mosco, M. A. Vivaldi, A pointwise regularity theory for the two-obstacle problem ,Acta Math. (1-2) (1989), 57-107.[26] H. Dong, D. Kim, Elliptic equations in divergence form with partially BMO coefficients , Arch. Ration.Mech. Anal. (1) (2010), 25-70.[27] H. Dong, D. Kim, On the L p solvability of higher order parabolic and elliptic systems with BMOcoefficients , Arch. Rational Mech. Anal. (3) (2011), 880-941.[28] G. Duvaut, R´esolution d’un probl´eme de Stefan (Fusion dun bloc de glace a zero degr´ees), C. R.Acad. Sci. Paris (1973), 1461-1463.[29] G. Duvaut, J. Lions, Inequalities in Mechanics and Physics , vol. 219 of Grundlehren der Matematis-chen Wissenschaften. Springer-Verlag, Berlin-New York, 1976.[30] F. Duzaar, G. Mingione, Gradient estimates via linear and nonlinear potentials , J. Funct. Anal. (2010), 2961-2998.[31] M. Eleuteri, Regularity results for a class of obstacle problems , Appl. Math. , 137-170 (2007).[32] M. Eleuteri, P. Harjulehto, T. Lukkari, Global regularity and stability of solutions to obstacle problemswith nonstandard growth , Rev. Mat. Complut. (2013), 147-181.[33] A. Friedman, Variational Principles and Free-Boundary Problems , in: WileyInterscience Publication,Pure Appl. Math., John Wiley and Sons, Inc., New York, 1982. 34] T. Iwaniec, Projections onto gradient fields and L p -estimates for degenerated elliptic operators , Stud.Math. (3) (1983), 293-312.[35] T. Kilpel¨ainen, W.P. Ziemer, Pointwise regularity of solutions to nonlinear double obstacle problems ,Ark. Mat. (1) (1991), 83-106.[36] Y. Kim, Gradient estimates for elliptic equations with measurable nonlinearities , J. Math. Pures Appl.(9) (2018), 118-145.[37] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications ,Pure Appl. Math., vol. 88, Academic Press, New York, London, 1980.[38] T. Kuusi, G. Mingione, Universal potential estimates . J. Funct. Anal. (10) (2012), 4205-4269.[39] T. Kuusi, G. Mingione, Guide to nonlinear potential estimates , Bull. Math. Sci. (1) (2014), 1-82.[40] P. Laurence, S. Salsa, Regularity of the free boundary of an American option on several assets . Comm.Pure Appl. Math. (2009), 969-994.[41] M. Lee, J. Ok, Nonlinear Caldern-Zygmund theory involving dual data , Rev. Mat. Iberoamericana (4) (2019), 10530-1078.[42] Y.-Y. Li, L. Nirenberg, Estimates for elliptic systems from composite material , Commun. Pure Appl.Math. (7) (2003), 892-925.[43] G. M. Lieberman, Regularity of solutions to some degenerate double obstacle problems , Indiana Univ.Math. J. (3) (1991), 1009-1028.[44] T. Mengesha, N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat do-mains , Arch. Ration. Mech. Anal. (1) (2012), 189-216.[45] J. H. Michael, W. P. Ziemer, Existence of solutions to obstacle problems , Nonlinear Anal. (1)(1991), 45-71.[46] G. Mingione, The Calder´on-Zygmund theory for elliptic problems with measure data , Ann. Scuola.Norm. Super. Pisa Cl. Sci. (V) (2007), 195–261.[47] G. Mingione, Gradient estimates below the duality exponent , Math. Ann. (2010), 571-627.[48] G. Mingione, G. Palatucci, Developments and perspectives in Nonlinear Potential Theory , NonlinearAnal. (2020), 111452.[49] Q.-H. Nguyen, Potential estimates and quasilinear parabolic equations with measure data ,arXiv:1405.2587.[50] Q.-H. Nguyen, N. C. Phuc, Good- λ and Muckenhoupt-Wheeden type bounds, with applications toquasilinear elliptic equations with gradient power source terms and measure data , Math. Ann. (1-2) (2019), 67-98.[51] T.-N. Nguyen, M.-P. Tran, Lorentz improving estimates for the p -Laplace equations with mixed data ,Nonlinear Anal. (2020), 111960.[52] T.-N. Nguyen, M.-P. Tran, Level-set inequalities on fractional maximal distribution functions andapplications to regularity theory , arXiv:2004.06394.[53] T.-N. Nguyen, M.-P. Tran, Lorentz estimates for quasi-linear elliptic double obstacle problems involv-ing a Schr¨odinger term , arXiv:2005.03281.[54] J. F. Rodrigues, R. Teymurazyan, On the two obstacles problem in Orlicz-Sobolev spaces and appli-cations , Complex Var. Elliptic Equ. (7-9) (2011), 769-787. 55] J. F. Rodrigues, Obstacle Problems in Mathematical Physics , North Holland, Amsterdam (1987).[56] X. Ros-Oton, Obstacle problems and free boundaries: an overview , SeMa Journal (2017), 1-21.[57] C. Scheven, Gradient potential estimates in non-linear elliptic obstacle problems with measure data ,J. Funct. Anal. (6) (2012), 2777-2832.[58] M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case , Nonlinear Anal. (2019), 266-281.[59] M.-P. Tran, T.-N. Nguyen, Lorentz-Morrey global bounds for singular quasilinear elliptic equationswith measure data , Commun. Contemp. Math. (2019). https://doi.org/10.1142/S0219199719500330.[60] M.-P. Tran, T.-N. Nguyen, Weighted Lorentz gradient and point-wise estimates for solutions to quasi-linear divergence form elliptic equations with an application , arXiv:1907.01434.[61] M.-P. Tran, T.-N. Nguyen, New gradient estimates for solutions to quasilinear divergence form ellipticequations with general Dirichlet boundary data , J. Differ. Equ. (4) (2020), 1427-1462.[62] M.-P. Tran, T.-N. Nguyen, Global Lorentz estimates for non-uniformly nonlinear el-liptic equations via fractional maximal operators , J. Math. Anal. Appl. (2020).https://doi.org/10.1016/j.jmaa.2020.124084.[63] G. M. Troianiello, Elliptic Differential Equations and Obstacle Problems , The University Series inMathematics. Plenum Press, New York, xiv+353 pp. ISBN: 0-306-42448-7, (1987)., The University Series inMathematics. Plenum Press, New York, xiv+353 pp. ISBN: 0-306-42448-7, (1987).