Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 62, pp. 1–28. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu HIGHER DIFFERENTIABILITY FOR SOLUTIONS TO NONHOMOGENEOUS OBSTACLE PROBLEMS WITH 1 < p < 2 ZHENQIANG WANG Abstract. In this article, we establish integer and fractional higher-order differentiability of weak solutions to non-homogeneous obstacle problems that satisfy the variational inequality∫ Ω 〈A(x,Du), D(ϕ− u)〉 dx ≥ ∫ Ω 〈|F |p−2F,D(ϕ− u)〉 dx, where 1 < p < 2, ϕ ∈ Kψ(Ω) = {v ∈ u0 + W 1,p 0 (Ω,R) : v ≥ ψ a.e. in Ω}, u0 ∈ W 1,p(Ω) is a fixed boundary datum. We show that the higher differ- entiability of integer or fractional order of the gradient of the obstacle ψ and the nonhomogeneous term F can transfer to the gradient of the weak solu- tion, provided the partial map x 7→ A(x, ξ) belongs to a suitable Sobolev or Besov-Lipschitz space. 1. Introduction This article is devoted to studying the higher differentiability properties of the gradient of the solutions u ∈W 1,p(Ω) to the variational inequality∫ Ω 〈A(x,Du), D(ϕ− u)〉 dx ≥ ∫ Ω 〈|F |p−2F,D(ϕ− u)〉 dx, (1.1) where Ω ⊂ Rn (n > 2) is a bounded domain, the function ψ : Ω 7→ [−∞,+∞), called obstacle, belongs to the Sobolev space W 1,p loc (Ω), and Kψ(Ω) := {v ∈ u0 +W 1,p 0 (Ω,R) : v ≥ ψ a.e. in Ω} is the class of the admissible functions, with u0 ∈ W 1,p(Ω) is a fixed boundary datum. Moreover, ϕ ∈ Kψ(Ω), F ∈ Lp(Ω,Rn) is a given exterior force and the vector field A(x, ξ) : Ω × Rn → Rn is a C1-Carathéodory function, namely, A is measurable in x for all ξ ∈ Rn and differentiable in ξ for almost all x ∈ Ω. Meanwhile, we assume that A is a p-harmonic operator, that is it satisfies the following p-ellipticity and p-growth conditions with respect to the ξ-variable. There exist positive constants ν, γ, Γ and an exponent p ∈ (1, 2) and a parameter µ ∈ [0, 1], such that 〈A(x, ξ)−A(x, η), ξ − η〉 ≥ ν|ξ − η|2(µ2 + |ξ|2 + |η|2) p−2 2 , (1.2) |A(x, ξ)−A(x, η)| ≤ Γ|ξ − η|(µ2 + |ξ|2 + |η|2) p−2 2 , (1.3) 2020 Mathematics Subject Classification. 35J87, 49J40, 47J20. Key words and phrases. Nonhomogeneous elliptic obstacle problems; higher differentiability; Sobolev coefficients; Besov-Lipschitz coefficients. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 23, 2022. Published August 22, 2022. 1 2 Z. WANG EJDE-2022/62 |A(x, ξ)| ≤ γ(µ2 + |ξ|2) p−1 2 (1.4) for all ξ, η ∈ Rn and for almost all x ∈ Ω. In the previous decades, the study of the regularity theory for obstacle problems has been developing rapidly as a popular topic in calculus of variations and par- tial differential equations. The obstacle problems can be dated back to the work of Stampacchia and Lions [9, 24]. Stampacchia discussed the special case ψ = χE firstly, and then Lions and Stampacchia proposed the theory of variational inequali- ties in order to solve the regularity of obstacle problems. In an earlier work, Fichera [10] solved the elastostatic problems with unilateral constraints, namely, the Sig- norini problem with ambiguous boundary conditions. These problems can be solved by applying methods of functional analysis, and then the regularity of the solutions were connected to the obstacle problems. It is often observed that the regularity of the solutions to the obstacle problems is affected by the obstacle itself. For linear obstacle problems, obstacle and solutions have the same regularity [3, 5, 16], but nonlinear is not like this. Therefore, people pay more attention to the nonlinear case in recent years [19, 20]. A first result was about the Hölder continuity of the weak solutions to the obstacle problems by Michael and Ziemer [23], it was related to the Hölder continuity of obstacle itself. Choe [6] established the Hölder continuity of the gradient of the weak solutions when the gradient of the obstacle is Hölder continuous. Many recent works focus on the regularity of solutions to variational problem. Most of papers give that the regularity of the solution depend on the regularity of the obstacle itself and the nonhomogeneous term F , provided an advisable assumption is given on the map x 7→ A(x, ξ). It is worth noting that there is not higher differentiability for the solution of obstacle problems (1.1) even though the obstacle and nonhomogeneous term are smooth. The aim of this article is to extend some higher differentiability results in [11] to non-homogeneous elliptic obstacle problems under the suitable conditions on the x-dependence of A. We make a suitable estimate of the nonhomogeneous term F by using the crucial Lemma 2.1 and other assumptions on A(x, ξ), thus obtaining new conclusions in Sobolev or Besov-Lipschitz space. First, we show that a higher differentiability property of integer order, provided the partial map x 7→ A(x, ξ) belongs to a suitable Sobolev class. More specifically, we assume that the partial map x 7→ A(x, ξ) belongs to W 1,n loc (Ω) for any ξ ∈ Rn. Equivalently, there exits a function ι ∈ Lnloc(Ω) such that |DxA(x, ξ)| ≤ ι(x)(µ2 + |ξ|2) p−1 2 , (1.5) see [16]. Since A(x, ξ) is a C1 function with respect to ξ, (1.3) implies |DξA(x, ξ)| ≤ c(µ2 + |ξ|2) p−2 2 , for all ξ ∈ Rn \ {0} and for almost every x ∈ Ω. Now we are in position to state our main results of this paper. For convenience, we introduce a special function Vp : Rn → Rn, defined as Vp(ξ) := (µ2 + |ξ|2) p−2 4 ξ for ξ ∈ Rn. The first result we prove reads as follows. Theorem 1.1. Let u ∈ W 1,p loc (Ω) be a solution to the obstacle problem (1.1) un- der assumptions (1.2)–(1.5) for 1 < p < 2. Let Vp(Dψ) ∈ W 1,2 loc (Ω), Du ∈ W 1,max( 4n n−2 ,n) loc (Ω), Dψ ∈ W 1,max( 2n n−2 ,n) loc (Ω), F ∈ W 1,2 loc (Ω), |τhDψ| < θ|τhDu| for EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 3 any θ > 0 small enough, |τh(u − ψ)| > 1, |τhF | > 1, then Vp(Du) ∈ W 1,2 loc (Ω). Moreover, for any BR b Ω, we have the following estimate ‖DVp(Du)‖L2(BR/2) ≤ C [ 1 + ‖Du‖ W 1,max( 4n n−2 ,n) (BR) + ‖DVp(Dψ)‖L2(BR) + ‖Dψ‖ W 1,max( 2n n−2 ,n) (BR) + ‖DF‖L2(BR) + ‖ι‖Ln(BR) ]σ , where C, and σ are positive constants depending on n, p, R, θ, ν, γ and Γ. Next we plan to prove that an analogous conclusion holds true in case the obstacle belongs to a Besov-Lipschitz space, provided the operator A is related to x-variable. Specifically, given α ∈ (0, 1) and q ∈ [1,∞), we assume that there is a sequence of non-negative measurable functions ιk ∈ L n α (Ω) such that ∞∑ k=1 ‖ιk‖qLn/α(Ω) <∞. Simultaneously, we have |A(x, ξ)−A(y, ξ)| ≤ (ιk(x) + ιk(y))|x− y|α(µ2 + |ξ|2) p−1 2 , (1.6) for each ξ ∈ Rn and almost every x, y ∈ Ω such that 2−kdiam(Ω) ≤ |x − y| < 2−k+1diam(Ω). For ease of statement, we will shortly write then that (ιk)k ∈ `q(Ln/α(Ω)). Now we state the Besov regularity of the obstacle. Theorem 1.2. Let u ∈W 1,p loc (Ω) be a solution to the obstacle problem (1.1) under assumptions (1.2)–(1.4) and (1.6) for 1 < p < 2. Let Vp(Dψ) ∈ Bα2,q,loc(Ω), D2ψ ∈ L np n−2α loc (Ω), F ∈ W 1, np n−2α loc (Ω). Then Vp(Du) ∈ Bαβ2,q,loc(Ω), for any q ≤ 2n n−2α and β ∈ (0, 1). Moreover, for any B4R b Ω, we have the estimate ‖τhVp(Du) |h|αβ ‖Lq( dh |h|n ;L2(BR/2)) ≤ C [ 1 + ‖Du‖Lp(B4R) + ‖Vp(Dψ)‖Bα2,q(B4R) + ‖D2ψ‖ L np n−2α (B4R) + ‖F‖ W 1, np n−2α (B4R) + ‖(ιk)k‖`q(Ln/α(B2R)) ]σ , where C and σ are positive constants depending on n, p, q, R, α, ν, γ and Γ. In the Besov-Lipschitz space we discussed above, if q = ∞, we still have a fractional differentiability property of the obstacle. More specifically, we prove the following result. Theorem 1.3. Let u ∈W 1,p loc (Ω) be a solution to the obstacle problem (1.1) under assumptions (1.2)–(1.4) for 1 < p < 2. If for any ξ ∈ Rn and almost every x, y ∈ Ω, there exists α ∈ (0, 1) and a function ι ∈ L n α loc(Ω) such that |A(x, ξ)−A(y, ξ)| ≤ (ι(x) + ι(y))|x− y|α(µ2 + |ξ|2) p−1 2 , (1.7) then, provided 0 < α < δ < 1, Vp(Dψ) ∈ Bδ2,∞,loc(Ω), D2ψ ∈ L np n−2α loc (Ω), F ∈ W 1, np n−2α loc (Ω), we have Vp(Du) ∈ Bαβ2,∞,loc(Ω), for any β ∈ (0, 1). Moreover, for any B4R b Ω, we have the estimate [Vp(Du)]Ḃαβ2,∞(BR 2 ) ≤ C [ 1 + ‖Du‖Lp(B4R) + ‖Vp(Dψ)‖Bδ2,∞(B4R) 4 Z. WANG EJDE-2022/62 + ‖D2ψ‖ L np n−2α (B4R) + ‖F‖ W 1, np n−2α (B4R) + ‖ι‖Ln/α(B2R) ]σ , where C and σ are positive constants depending on n, p, q, R, α, β, δ, ν, γ and Γ. It is worth mentioning that in integer and fractional order cases, the fundamental tools are the difference quotient method and Calderón-Zygmund type estimates proved in [4]. This article is organized as follows. In Section 2, we give notation and preliminary results. Section 3 is devoted to the proof of Theorem 1.1, while Section 4 is devoted to the proofs of Theorems 1.2 and 1.3. 2. Notation and preliminary results In this section we will list some definitions and recall a few of fundamental tools for the proof of our main results in the following content. We shall use C or c to denote a general constant that may depend on different parameters, even within the same line of estimates. Relevant dependencies will be appropriately emphasized using parentheses. In the following, B(x, r) = Br(x) = {y ∈ Rn : |y − x| < r} b Ω will denote the open ball centered at x of radius r > 0. For a function u ∈ L1(Br(x0)), the symbol − ∫ Br(x0) u(x) dx := 1 |Br(x0)| ∫ Br(x0) u(x) dx will denote the integral mean of the function u(x) over the open ball Br(x0). Next we recall a crucial result for the function Vp, see [1, 13]. Lemma 2.1. Let p ∈ (1,∞), µ ∈ [0, 1]. There is a constant c = c(n, p) > 0 such that c−1(µ2 + |ξ|2 + |η|2) p−2 2 ≤ |Vp(ξ)− Vp(η)|2 |ξ − η|2 ≤ c(µ2 + |ξ|2 + |η|2) p−2 2 , for any ξ, η ∈ Rn. Particularly, there is a constant c = c(n, p) > 0 such that∣∣|ξ|p−2ξ − |η|p−2η ∣∣ ≤ c|ξ − η|p−1. In addition, for any Φ ∈ C2(Rn), there exists a constant C = C(p) > 0 such that C−1|D2Φ|2(µ2 + |DΦ|2) p−2 2 ≤ |DVp(DΦ)|2 ≤ C|D2Φ|2(µ2 + |DΦ|2) p−2 2 . 2.1. Difference quotients. To obtain the higher differentiability of the gradient of the weak solution, let us recall some results of the finite difference operator. Given h ∈ Rn, for every function v : Rn → R, the finite difference operator is defined by τhv(x) := v(x+ h)− v(x). We start with the description of some basic properties that can be founded in [14]. Proposition 2.2. Let F and G be two functions such that F,G ∈ W 1,p(Ω), with p ∈ [1,∞), and we consider the set Ω|h| := {x ∈ Ω : dist(x, ∂Ω) > |h|}. Then (a) τhF ∈W 1,p(Ω|h|) and Di(τhF (x)) = τh(DiF (x)). EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 5 (b) If at least one of the functions F or G has support contained in Ω|h|, then∫ Ω F (x)τhG(x) dx = ∫ Ω G(x)τ−hF (x) dx. (c) We have τh(FG)(x) = F (x+ h)τhG(x) +G(x)τhF (x). The next result about finite difference operator is a kind of integral version of the Lagrange Theorem. Lemma 2.3. If 0 < ρ < R, |h| < R−ρ 2 , p ∈ (1,∞), and F, DF ∈ Lp(BR), then∫ Bρ |τhF (x)|pdx ≤ c(n, p)|h|p ∫ BR |DF (x)|pdx. Moreover ∫ Bρ |F (x+ h)|pdx ≤ ∫ BR |F (x)|pdx. For each function v : Rn → RN and h ∈ R, we denote τs,hv(x) := v(x+ hes)− v(x), where es is the unit vector in the xs direction for any s ∈ {1, . . . , n}. Now we recall the essential Sobolev embedding property that is proved in [14]. Lemma 2.4. Let F : Rn → RN , F ∈ Lp(BR) with p ∈ (1,∞). Suppose that there exists ρ ∈ (0, R) and M > 0 such that n∑ s=1 ∫ Bρ |τs,hF (x)|pdx ≤Mp|h|p, for all h with |h| < R−ρ 2 . Then F ∈W 1,p(Bρ) ∩ L np n−p (Bρ). Moreover ‖DF‖Lp(Bρ) ≤M, ‖F‖L np n−p (Bρ) ≤ c(M + ‖F‖Lp(BR)), with c = c(n,N, p,R, ρ). Now we introduce a fractional version of Lemma 2.4, whose proof can be found in [18]. Lemma 2.5. Let F ∈ L2(BR). Suppose that there exist ρ ∈ (0, R), α ∈ (0, 1) and M > 0 such that n∑ s=1 ∫ Bρ |τs,hF (x)|2dx ≤M2|h|2α, for all h with |h| < R−ρ 2 . Then F ∈ L 2n n−2β (Bρ) for all β ∈ (0, α). Moreover ‖F‖ L 2n n−2β (Bρ) ≤ c(M + ‖F‖L2(BR)), with c = c(n,N,R, ρ, α, β). Next we give a Sobolev embedding theorem, which includes the fractional version, see [8, 12]. Lemma 2.6. Assume that Ω ⊂ Rn has the extension property. 6 Z. WANG EJDE-2022/62 (a) Let p ∈ [1, n). Then there exists a constant c = c(n, p,Ω) > 0 such that ‖f‖Lq(Ω) ≤ c‖f‖W 1,p(Ω) for all f ∈W 1,p(Ω) and p ≤ q ≤ np n−p . (b) Let α ∈ (0, 1) and p ∈ [1, nα ). Then there exists a constant c = c(n, p, α,Ω) > 0 such that ‖f‖Lq(Ω) ≤ c‖f‖Wα,p(Ω) for all f ∈Wα,p(Ω) and p ≤ q ≤ np n−αp . 2.2. Besov-Lipschitz spaces. Definition 2.7. For a given α ∈ (0, 1), and p, q ∈ [1,∞), we say that v belongs to the Besov-Lipschitz space Bαp,q(Rn) if v ∈ Lp(Rn) and ‖v‖Bαp,q(Rn) = ‖v‖Lp(Rn) + [v]Ḃαp,q(Rn) <∞, (2.1) where [v]Ḃαp,q(Rn) := (∫ Rn (∫ Rn |τhv(x)|p |h|αp dx )q/p dh |h|n )1/q <∞. (2.2) It is worth noticing that Bαp,q(Rn) is a Banach space. We can say that a function v ∈ Lp(Rn) belongs to Bαp,q(Rn) if and only if τhv |h|α ∈ L q ( dh |h|n ;Lp(Rn) ) . In addition, for h ∈ Bδ(0) where δ > 0 is a fixed constant, we can simply (2.2) to obtain an equivalent norm of (2.1), that is ‖v‖Bαp,q(Rn) ' ‖v‖Lp(Rn) + (∫ {|h|≤δ} (∫ Rn |τhv(x)|p |h|αp dx )q/p dh |h|n )1/q <∞, this is so because(∫ {|h|≥δ} (∫ Rn |τhv(x)|p |h|αp dx )q/p dh |h|n )1/q ≤ c(n, p, q, α, δ)‖v‖Lp(Rn) <∞. Definition 2.8. For a given α ∈ (0, 1), and p∈ [1,∞), q = ∞, we say that v belongs to the Besov-Lipschitz space Bαp,∞(Rn) if v ∈ Lp(Rn) and [v]Ḃαp,∞(Rn) := sup h∈Rn (∫ Rn |τhv(x)|p |h|αp dx )1/p <∞. (2.3) Similarly, we can take the supremum over |h| ≤ δ in (2.3) and obtain an equiva- lent norm. According to the construction of the norm of Besov-Lipschitz space, it is easy to see that Bαp,q(Rn) ⊂ Lp(Rn). Now we give the Sobolev-type embeddings for Besov-Lipschitz spaces, see [15]. Lemma 2.9. Suppose that α ∈ (0, 1). (a) If 1 < p < n α and 1 ≤ q ≤ p∗α := np n−αp , then there is a continuous embedding Bαp,q(Rn) ↪→ Lp ∗ α(Rn). (b) If p = n α and 1 ≤ q ≤ ∞, then there is a continuous embedding Bαp,q(Rn) ↪→ BMO(Rn), where BMO denotes the space of functions with bounded mean oscillations [14]. The following lemma describes the relationship between Besov-Lipschitz spaces, see [15]. Lemma 2.10. Suppose that 0 < β < α < 1. EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 7 (a) If 1 < p < +∞ and 1 ≤ q ≤ r ≤ +∞, then Bαp,q(Rn) ⊂ Bαp,r(Rn). (b) If 1 < p < +∞ and 1 ≤ q, r ≤ +∞, then Bαp,q(Rn) ⊂ Bβp,r(Rn). (c) If 1 ≤ q ≤ ∞, then Bαn α ,q (Rn) ⊂ Bβn β ,q (Rn). The following lemma is follows from the definition of the local Besov-Lipschitz spaces, and its proof can be found in [2]. Lemma 2.11. A function v ∈ Lploc(Ω) belongs to the local Besov space Bαp,q,loc(Ω) if and only if ‖ τhv |h|α ‖Lq( dh |h|n ;Lp(B)) <∞ for any open ball B ⊂ 2B ⊂ Ω with radius rB. Here the measure dh |h|n is restricted to the ball BrB (0) on the h-space. As we know, the Besov-Lipschitz spaces of fractional order α ∈ (0, 1) can be characterized in pointwise terms. Given a measurable function v(x) : Rn → R, a fractional α-Haj lasz gradient for v is a sequence (ιk)k of measurable, non-negative functions ιk(x) : Rn → R, together with a null set N ⊂ Rn, such that |v(x)− v(y)| ≤ (ιk(x) + ιk(y))|x− y|α whenever k ∈ Z and x, y ∈ Rn\N are such that 2−k ≤ |x − y| < 2−k+1. We say that (ιk) ∈ `q(Z;Lp(Rn)) if ‖(ιk)k‖`q(Lp) = (∑ k∈Z ‖ιk‖qLp(Rn) )1/q <∞. Now we give a necessary and sufficient condition of a function v to belong to the Besov-Lipschitz space Bαp,q(Rn), which was proved in [17]. Theorem 2.12. Let 0 < α < 1, 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. Let v ∈ Lp(Rn). One has v ∈ Bαp,q(Rn) if and only if there exists a fractional α-Haj lasz gradient (ιk)k ∈ `q(Z;Lp(Rn)) for v. Moreover ‖v‖Bαp,q(Rn) ' inf ‖(ιk)k‖`q(Lp), where the infimum runs over all possible fractional α-Haj lasz gradients for v. Now we give a crucial lemma, which proof can be found in [11]. Lemma 2.13. Let Ω ⊂ Rn be a bounded open set, 1 < p < 2, 0 < α < 1, and 1 ≤ q ≤ ∞. Then Vp(Dψ) ∈ Bα2,q,loc(Ω)⇒ Dψ ∈ Bαp,q,loc(Ω). Moreover, for any all BR b Ω and 0 < ρ < R, we have [Dψ]Ḃαp,q(Bρ) ≤ C [ 1 + ‖Dψ‖Lp(BR) + ‖Vp(Dψ)‖Bα2,q(BR) ]σ , where C and σ are positive constants depending on n, p, q and α. 8 Z. WANG EJDE-2022/62 2.3. VMO coefficients. To prove our main results, we shall introduce the related content of VMO coefficients. For convenience, given a ball B ⊂ Ω, let us introduce the operator AB(ξ) = − ∫ B A(x, ξ) dx. Definition 2.14. Suppose that B ⊂ Ω is an open ball, 1 < p <∞, setting V (x,B) := sup ξ 6=0 |A(x, ξ)−AB(ξ)| (µ2 + |ξ|2) p−1 2 , we say that x 7→ A(x, ξ) is locally uniformly in VMO if for each set K b Ω we have that lim R→0 sup r 1 and q > p. Assume that (1.2), (1.3), (1.4) hold, and that x 7→ A(x, ξ) is locally uniformly in VMO. Let u ∈ Kψ(Ω) be the weak solution of the variational inequality (1.1). Then Dψ,F ∈ Lqloc(Ω) =⇒ Du ∈ Lqloc(Ω). Moreover, there exists a constant C = C(n, p, q, v, γ,Γ) such that − ∫ BR |Du|qdx ≤ C[1 +− ∫ B2R |F |qdx+− ∫ B2R |Dψ|qdx+ (− ∫ B2R |Du|pdx)q/p] for all ball BR such that B2R b Ω. 3. Higher order integer differentiability Existence of weak solutions to the variational inequality (1.1) can be easily proved through classical theories, so in the paper we will concentrate more on the proof of the regularity results. The key point is to choose an appropriate test function ϕ in (1.1) such that ϕ turns to be admissible for the obstacle class Kψ(Ω). It is worth noticing that for the higher order integer differentiability, unfortunately, we cannot get a similar result as [11, Theorem 2.2] so that we must require a higher regularity of u and ψ. Proof of Theorem 1.1. Let us fix a ball BR such that BR ⊂ B4R b Ω and arbitrary radii R2 < r < l1 < l2 < λr < R, with 1 < λ < 2. Let us consider a cut off function η ∈ C∞0 (Bl2) such that 0 ≤ η ≤ 1, η ≡ 1 on Bl1 and |Dη| ≤ c R . Because of the local nature of our results, with no loss of generality, we suppose R ≤ 1. Let us consider ϕ := u+ θv for any θ ∈ [0, 1] and a suitable v ∈W 1,p 0 (Ω) such that u− ψ + θv ≥ 0. (3.1) It is easy to see ϕ ∈ Kψ(Ω) since ϕ = u+ θv ≥ ψ. Now, for |h| < R 4 , we consider v1(x) = η2(x)τh(u− ψ)(x). EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 9 From the regularity of u and ψ, we have v1 ∈W 1,p 0 (Ω). Moreover, v1 satisfies (3.1). Indeed, for almost every x ∈ Ω and for all θ ∈ [0, 1] u(x)− ψ(x) + θv1(x) = u(x)− ψ(x) + θη2(x)τh(u− ψ)(x) = u(x)− ψ(x) + θη2(x)[(u− ψ)(x+ h)− (u− ψ)(x)] = θη2(x)(u− ψ)(x+ h) + (1− θη2(x))(u− ψ)(x) ≥ 0, since u ∈ Kψ(Ω) and η ∈ [0, 1]. Therefore, we can use ϕ = u + θv1 as an admissible test function in variational inequality (1.1), thus we have∫ Ω 〈A(x,Du(x)), D[η2(x)[(u− ψ)(x+ h)− (u− ψ)(x)]]〉dx ≥ ∫ Ω 〈|F (x)|p−2F (x), D[η2(x)[(u− ψ)(x+ h)− (u− ψ)(x)]]〉dx. (3.2) Similarly, if we define v2(x) = η2(x− h)τ−h(u− ψ)(x), we have v2 ∈W 1,p 0 (Ω), and v2 satisfies (3.1). So we can also use ϕ = u + θv2 as an admissible test function in (1.1), thus we obtain∫ Ω 〈A(x,Du(x)), D[η2(x− h)[(u− ψ)(x− h)− (u− ψ)(x)]]〉dx ≥ ∫ Ω 〈|F (x)|p−2F (x), D[η2(x− h)[(u− ψ)(x− h)− (u− ψ)(x)]]〉dx. (3.3) By means of a simple change of variable in (3.3), we have∫ Ω 〈A(x+ h,Du(x+ h)), D[η2(x)[(u− ψ)(x)− (u− ψ)(x+ h)]]〉dx ≥ ∫ Ω 〈|F (x+ h)|p−2F (x+ h), D[η2(x)[(u− ψ)(x)− (u− ψ)(x+ h)]]〉dx. (3.4) We can add (3.2) and (3.4), thus obtaining∫ Ω 〈A(x+ h,Du(x+ h))−A(x,Du(x)), D[η2(x)[(u− ψ)(x+ h)− (u− ψ)(x)]]〉dx ≤ ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), D[η2(x)[(u− ψ)(x+ h)− (u− ψ)(x)]]〉dx, which implies∫ Ω 〈A(x+ h,Du(x+ h))−A(x,Du(x)), η2(x)[Du(x+ h)−Du(x)]〉dx − ∫ Ω 〈A(x+ h,Du(x+ h))−A(x,Du(x)), η2(x)[Dψ(x+ h)−Dψ(x)]〉dx + ∫ Ω 〈A(x+ h,Du(x+ h))−A(x,Du(x)), 2η(x)Dη(x)τh(u− ψ)(x)〉dx 10 Z. WANG EJDE-2022/62 ≤ ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), η2(x)[Du(x+ h)−Du(x)]〉dx − ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), η2(x)[Dψ(x+ h)−Dψ(x)]〉dx + ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), 2η(x)Dη(x)τh(u− ψ)(x)〉dx. We can write the previous inequality in the form I := ∫ Ω 〈A(x+ h,Du(x+ h))−A(x+ h,Du(x)), η2[Du(x+ h)−Du(x)]〉dx ≤ ∫ Ω 〈A(x+ h,Du(x+ h))−A(x+ h,Du(x)), η2[Dψ(x+ h)−Dψ(x)]〉dx − ∫ Ω 〈A(x+ h,Du(x+ h))−A(x+ h,Du(x)), 2ηDητh(u− ψ)(x)〉dx − ∫ Ω 〈A(x+ h,Du(x))−A(x,Du(x)), η2[Du(x+ h)−Du(x)]〉dx + ∫ Ω 〈A(x+ h,Du(x))−A(x,Du(x)), η2[Dψ(x+ h)−Dψ(x)]〉dx − ∫ Ω 〈A(x+ h,Du(x))−A(x,Du(x)), 2ηDητh(u− ψ)(x)〉dx + ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), η2[Du(x+ h)−Du(x)]〉dx − ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), η2[Dψ(x+ h)−Dψ(x)]〉dx + ∫ Ω 〈|F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x), 2ηDητh(u− ψ)(x)〉dx =: II + III + IV + V + V I + V II + V III + IX. so we have I ≤ |II|+ |III|+ |IV |+ |V |+ |V I|+ |V II|+ |V III|+ |IX|. (3.5) Next, we estimate I, II, III, IV, V, V I, V II, V III and IX. By ellipticity assumption (1.2) we have I ≥ ν ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx. (3.6) For the term II, by assumption (1.3) and using the fact that |τhDψ| < θ|τhDu|, we have |II| ≤ Γ ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu||τhDψ|dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx. (3.7) For the term III, by Young’s Inequality with exponents (p, p p−1 ) and |τh(u−ψ)| > 1, we obtain |III| ≤ 2Γ ∫ Ω η|Dη|(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu| |τh(u− ψ)|dx EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 11 ≤ ε ∫ Ω η p p−1 (µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(ε) Rp |h|2 ∫ Bλr |D(u− ψ)|2dx. (3.8) To estimate IV , we use assumption (1.5), Young’s Inequality with exponents (2, 2) and Hölder’s Inequality with exponents (n2 , n n−2 ) to obtain |IV | ≤ ∫ Ω |h|ι(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−1 2 η2|τhDu|dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(ε)|h|2 (∫ Bl2 (µ2 + |Du(x)|2 + |Du(x+ h)|2) np 2(n−2) dx )n−2 n × (∫ Bl2 ιndx )2/n . (3.9) Assumption (1.5) also implies |V | ≤ ∫ Ω |h|ι(µ2 + |Du|2) p−1 2 η2|τhDψ|dx ≤ ∫ Bl2 |h|ι(µ2 + |Du|2) p−1 2 (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) 2−p 4 × (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) p−2 4 |τhDψ|dx ≤ c|h|2 (∫ Bl2 (µ2 + |Du|2) n(p−1) n−2 (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) n(2−p) 2(n−2) dx )n−2 n × (∫ Bl2 ιndx )2/n + c ∫ Bl2 |τhVp(Dψ)|2dx ≤ c|h|2 (∫ Bl2 ιndx )2/n[( ∫ Bl2 (µ2 + |Du|2) 2n(p−1) n−2 dx )n−2 n + (∫ Bl2 (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) n(2−p) (n−2) dx )n−2 n ] + c|h|2 ∫ Bλr |DVp(Dψ)|2dx, (3.10) where we also used Young’s Inequality with exponents (2, 2), Hölder’s Inequality with exponents (n2 , n n−2 ), Lemma 2.1 and Lemma 2.3. For the term V I, arguing as we did for the estimate of V , we obtain |V I| ≤ 2 ∫ Ω |h|ι(µ2 + |Du|2) p−1 2 η|Dη||τh(u− ψ)|dx ≤ c R ∫ Ω |h|ι(µ2 + |Du|2) p−1 2 η|τh(u− ψ)|dx ≤ c R |h|2 (∫ Bλr |D(u− ψ)|ndx )2/n(∫ Bλr (µ2 + |Du|2) n(p−1) 2(n−2) dx )n−2 n 12 Z. WANG EJDE-2022/62 + c R |h|2 (∫ Bl2 ιndx )2/n(∫ Bl2 (µ2 + |Du|2) n(p−1) 2(n−2) dx )n−2 n . (3.11) Using Lemma 2.1, Lemma 2.3, Young’s Inequality with exponents (2, 2) and |τhF | > 1 for the term V II, we obtain |V II| ≤ ∫ Ω η2||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)||τhDu|dx ≤ c(n, p)|h|2( ∫ Bλr |DF |2dx+ ∫ Bλr |D2u|2dx). (3.12) Arguing analogously, for the terms V III and IX, we have |V III| ≤ ∫ Ω η2||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)||τhDψ|dx ≤ c(n, p)|h|2 (∫ Bλr |DF |2dx+ ∫ Bλr |D2ψ|2dx ) , (3.13) and |IX| ≤ 2 ∫ Ω η||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)| |Dη||τh(u− ψ)|dx ≤ c(n, p) R |h|2 (∫ Bλr |DF |2dx+ ∫ Bλr |D(u− ψ)|2dx ) . (3.14) Now, plugging (3.6)-(3.14) into (3.5), choosing a sufficiently small value of ε, we obtain∫ BR η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx ≤ c Rp |h|2 ∫ Bλr |D(u− ψ)|2dx + c|h|2 (∫ Bl2 (µ2 + |Du(x)|2 + |Du(x+ h)|2) np 2(n−2) dx )n−2 n (∫ Bl2 ιndx )2/n + c|h|2( ∫ Bl2 ιndx)2/n · [ ( ∫ Bl2 (µ2 + |Du|2) 2n(p−1) n−2 dx) n−2 n + (∫ Bl2 (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) n(2−p) (n−2) dx )n−2 n ] + c|h|2 ∫ Bλr |DVp(Dψ)|2dx + c R |h|2 (∫ Bλr |D(u− ψ)|ndx )2/n(∫ Bλr (µ2 + |Du|2) n(p−1) 2(n−2) dx )n−2 n + c R |h|2( ∫ Bl2 ιndx)2/n (∫ Bl2 (µ2 + |Du|2) np 2(n−2) dx )n−2 n + c|h|2 (∫ Bλr |DF |2dx+ ∫ Bλr |D2u|2dx ) + c|h|2 (∫ Bλr |DF |2dx+ ∫ Bλr |D2ψ|2dx ) EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 13 + c R |h|2 (∫ Bλr |DF |2dx+ ∫ Bλr |D(u− ψ)|2dx ) . (3.15) By Lemma 2.1, the left-hand side of (3.5) can be bounded from below as follows∫ BR η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx ≥ ∫ BR/2 |τhVp(Du)|2dx. By Lemma 2.3 and Young’s Inequality with exponents (n2 , n n−2 ) to (3.15), we obtain∫ BR/2 |τhVp(Du)|2dx ≤ c|h|2 [ ∫ BR (µ2 + |Du|2) 2n n−2 dx+ ∫ BR ιn dx + ∫ BR (µ2 + |Dψ|2) n n−2 dx+ ∫ BR |DF |2dx + ∫ BR |D2u|2dx+ ∫ BR |D2ψ|2dx+ ∫ BR |Du|ndx + ∫ BR |Dψ|ndx+ ∫ BR |DVp(Dψ)|2dx ] =: M |h|2, for a suitable c = c(n, p,R, θ, ν, γ,Γ). So applying Lemma 2.4 for a suitable choice of C and σ, we obtain ‖DVp(Du)‖L2(BR/2) ≤ C [ 1 + ‖Du‖ W 1,max( 4n n−2 ,n) (BR) + ‖DVp(Dψ)‖L2(BR) + ‖Dψ‖ W 1,max( 2n n−2 ,n) (BR) + ‖DF‖L2(BR) + ‖ι‖Ln(BR) ]σ , so the conclusion is established. � 4. Higher order fractional differentiability This section is devoted to the proofs of Theorems 1.2 and 1.3. The proof of Theorem 1.2 is the same as the proof of the one presented in the previous section until the estimate (3.6). Differences come when starting estimate II-IX, in which the different assumptions on the partial map x 7→ A(x, ξ) and on the obstacle come into play. 4.1. Proof of Theorem 1.2. Our starting point is the estimate ν ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx ≤ |II|+ |III|+ |IV |+ |V |+ |V I|+ |V II|+ |V III|+ |IX|. (4.1) Let us observe that, since Vp(Dψ) ∈ Bα2,q,loc(Ω) for q ≤ 2n n−2α , then by Lemmas 2.9 and 2.13, Vp(Dψ) ∈ L 2n n−2α loc (Ω) and Dψ ∈ L np n−2α loc (Ω). Moreover, because F ∈ L np n−2α loc (Ω), by Theorem 2.16, we have Du ∈ L np n−2α loc (Ω). Now we consider the term II, by assumption (1.3), we obtain |II| ≤ Γ ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu||τhDψ|dx. (4.2) 14 Z. WANG EJDE-2022/62 Setting E1 := {x ∈ Ω : |Du(x)|2 + |Du(x + h)|2 > |Dψ(x)|2 + |Dψ(x + h)|2} and E2 := Ω \ E1, so (4.2) becomes |II| ≤ Γ ∫ E1 η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu||τhDψ|dx + Γ ∫ E2 η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu||τhDψ|dx =: II1 + II2. (4.3) Using Young’s Inequality with exponents (2,2) and Lemma 2.1, we have |II1| ≤ Γ ∫ E1 η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 4 |τhDu| × ( µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2 ) p−2 4 |τhDψ|dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p,Γ, ε) ∫ Bl2 |τhVp(Dψ)|2dx. (4.4) Similarly, for the term II2, we obtain |II2| ≤ C(n, p,Γ) ∫ Bl2 |τhVp(Dψ)|2dx+ C(n, p,Γ) ∫ Bλr (µp + |Dψ|p)dx. Noticing that Dψ ∈ L np n−2α loc (Ω) and p < np n−2α , then by Hölder’s Inequality with exponents ( n2α , n n−2α ), we obtain∫ BR |Dψ|pdx ≤ ( ωnR n )2α/n(∫ BR |Dψ| np n−2α dx )n−2α n ≤ C(n, p)R2α (∫ BR |Dψ| np n−2α dx )n−2α n , where ωn is the measure of the ball of radius 1 in Rn. Then |II2| ≤ C(n, p,Γ) ∫ Bl2 |τhVp(Dψ)|2dx + C(n, p,Γ)R2α [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n . (4.5) Plugging (4.4) and (4.5) into (4.3), we obtain |II| ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p,Γ, ε) ∫ Bl2 |τhVp(Dψ)|2dx + C(n, p,Γ)R2α [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n . (4.6) EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 15 For the term III, by assumption 1.3, Young’s Inequality with exponents ( p p−1 , p), Lemma 2.3 and D(u− ψ) ∈ L np n−2α loc (Ω), arguing like in (4.5), thus obtaining |III| ≤ 2Γ ∫ Ω η|Dη|(µ2 + |Du|2 + |Du(x+ h)|2) p−2 2 |τhDu||τh(u− ψ)|dx ≤ ε ∫ Ω η p p−1 (µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p,Γ, ε) Rp−2α |h|p (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n . (4.7) For the term IV , by assumption (1.6) and Young’s Inequality with exponents (2, 2), we have |IV | ≤ |h|α ∫ Ω η2(ιk(x) + ιk(x+ h))(µ2 + |Du|2) p−1 2 |τhDu|dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(ε)|h|2α ∫ Bl2 (ιk(x) + ιk(x+ h))2(µ2 + |Du(x)|2 + |Du(x+ h)|2)p/2dx, where 2−k R4 ≤ |h| ≤ 2−k+1R 4 for k ∈ N. Then by using Hölder’s Inequality with exponents ( n2α , n n−2α ) and Lemma 2.3 we obtain |IV | ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p, α, ε)|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × (∫ BR (µ np n−2α + |Du| np n−2α )dx )n−2α n . Now we apply Theorem 2.16 with q = np n−2α , thus obtaining − ∫ BR |Du| np n−2α dx ≤ C [ 1 +− ∫ B2R |F | np n−2α dx+− ∫ B2R |Dψ| np n−2α dx+ ( − ∫ B2R |Du|pdx ) n n−2α ] . Then for the term IV , we obtain |IV | ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p, α, ε)|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx + (∫ B2R |Du|pdx ) n n−2α ]n−2α n . (4.8) 16 Z. WANG EJDE-2022/62 To estimate the term V , we consider 2−k R4 ≤ |h| ≤ 2−k+1R 4 for k ∈ N, using (1.6), Young’s Inequality with exponents (2, 2) and Lemma 2.1, thus obtaining |V | ≤ |h|α ∫ Ω η2(ιk(x) + ιk(x+ h))(µ2 + |Du|2) p−1 2 |τhDψ|dx ≤ |h|α ∫ Bl2 (ιk(x) + ιk(x+ h))(µ2 + |Du|2) p−1 2 |τhDψ| × (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) 2−p 4 (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) p−2 4 dx ≤ c|h|2α ∫ Bl2 (ιk(x) + ιk(x+ h))2(µ2 + |Du|2)p−1 × (µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2) 2−p 2 dx + c ∫ BR |τhVp(Dψ)|2dx. By using Hölder’s Inequality with exponents ( n2α , n n−2α ), ( p 2(p−1) , p 2−p ), and Lemma 2.3, the previous estimate becomes |V | ≤ c|h|2α (∫ Bl2 (ιk(x) + ιk(x+ h))n/αdx )2α/n × (∫ Bl2 (µ2 + |Du|2) n(p−1) n−2α ( µ2 + |Dψ(x)|2 + |Dψ(x+ h)|2 ) n(2−p) 2(n−2α) dx )n−2α n + c ∫ BR |τhVp(Dψ)|2dx ≤ c|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × (∫ BR (µ np n−2α + |Du| np n−2α )dx ) 2(p−1) p n−2α n × (∫ BR (µ np n−2α + |Dψ| np n−2α )dx ) 2−p p n−2α n + c ∫ BR |τhVp(Dψ)|2dx. Using Young’s Inequality with exponents ( p 2(p−1) , p 2−p ), we obtain |V | ≤ c ∫ BR |τhVp(Dψ)|2dx+ c|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx + (∫ B2R |Du|pdx ) n n−2α ]n−2α n . (4.9) Now we consider the term V I, taking 2−k R4 ≤ |h| ≤ 2−k+1R 4 for k ∈ N, using as- sumption (1.6), Hölder’s Inequality with exponents ( n2α , n n−2α ), ( p p−1 , p) and Lemma 2.3, we obtain |V I| ≤ |h|α ∫ Ω η|Dη|(ιk(x) + ιk(x+ h))(µ2 + |Du|2) p−1 2 |τh(u− ψ)|dx ≤ c R |h|α (∫ BR (ιk(x) + ιk(x+ h)) n 2α dx )2α/n EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 17 × (∫ BR (µ2 + |Du|2) n(p−1) 2(n−2α) |τh(u− ψ)| n n−2α dx )n−2α n ≤ c R |h|α (∫ BR (ιk(x) + ιk(x+ h)) n 2α dx ) 2α n × [( ∫ BR ( µ2 + |Du|2 ) np 2(n−2α) dx ) p−1 p n−2α n (∫ BR |τh(u− ψ)| np n−2α dx )n−2α np ] ≤ C(n, p, α) R |h|α+1 (∫ BR (ιk(x) + ιk(x+ h)) n 2α dx )2α/n × [( ∫ BR ( µ np n−2α + |Du| np n−2α ) dx ) p−1 p n−2α n (∫ BλR |D(u− ψ)| np n−2α dx )n−2α np ] . Notice that {ιk}k ⊂ Ln/α(Ω) ⊂ L n 2α (Ω) with the estimate ‖ιk‖L n 2α (BR) ≤ (∫ BR 12dx )α/n(∫ BR |ιk|n/αdx )α/n ≤ cRα‖ιk‖Ln/α(BR). (4.10) By Young’s Inequality with exponents ( p p−1 , p) and (4.10), the previous estimate becomes |V I| ≤ C(n, p, α) R1−α |h|α+1 (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [( ∫ BR (µ np n−2α + |Du| np n−2α )dx )n−2α n + (∫ BλR |D(u− ψ)| np n−2α dx )n−2α n ] ≤ C(n, p, α) R1−α |h|α+1 (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx + (∫ B2R |Du|pdx ) n n−2α ]n−2α n . (4.11) It remains to estimate V II, V III, and IX. For the term V II, using Young’s Inequality with exponents ( p p−1 , p), Lemma 2.1, Lemma 2.3, and Hölder’s Inequality with exponents ( n2α , n n−2α ), we have |V II| ≤ ∫ Ω η2||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)||τhDu|dx ≤ C0(n, p)ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p, α, ε)|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n . (4.12) 18 Z. WANG EJDE-2022/62 For V III, using Hölder’s Inequality with exponents ( p p−1 , p) and ( n2α , n n−2α ), Lemma 2.1, and Lemma 2.3, we have |V III| ≤ ∫ Ω η2||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)||τhDψ|dx ≤ C(n, p)|h|p (∫ BR |DF |pdx ) p−1 p (∫ BR |D2ψ|pdx )1/p ≤ C(n, p)|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np . (4.13) Arguing as we did for the estimate of V III, for the term IX, we obtain |IX| ≤ 2 ∫ Ω η|Dη|||F (x+ h)|p−2F (x+ h)− |F (x)|p−2F (x)| |τh(u− ψ)|dx ≤ C(n, p) R |h|p (∫ BR |DF |pdx ) p−1 p ( ∫ BR (|Du|p + |Dψ|p)dx)1/p ≤ C(n, p) R |h|pR2α (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.14) Plugging (4.6), (4.7), (4.8), (4.9), (4.11), (4.12), (4.13), and (4.14) into (4.1), and then choosing ε = ν 2(3+C0) , we obtain∫ BR/2 |τhVp(Du)|2dx ≤ cR2α [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c Rp−2α |h|p (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n + c|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] n−2α n + c ∫ BR |τhVp(Dψ)|2dx+ c R1−α |h| α+1 (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n + c|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np + c|h|pR2α−1 (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 19 + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.15) Now, we use a covering argument [5] to the balls of the integrals in (4.15), whose radii are proportional to R, we have R ∝ |h|β for β ∈ (0, 1) and a sufficiently small value of |h|, then (4.15) becomes∫ BR/2 |τhVp(Du)|2dx ≤ c|h|2αβ [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c|h|p(1−β)+2αβ (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n + c|h|2α (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c ∫ BR |τhVp(Dψ)|2dx+ c|h|α−β+αβ+1 (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|p+2αβ (∫ BR |DF | np n−2α dx )n−2α n + c|h|p+2αβ (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np + c|h|p−β+2αβ (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.16) Since α, β ∈ (0, 1), by setting p1 = 2αβ ∈ (0, 2), p2 = p(1− β) + 2αβ ∈ (0, 4), p3 = 2α ∈ (0, 2), p4 = α− β + αβ + 1 = (α+ 1)(1− β) + 2αβ ∈ (0, 3), p5 = p+ 2αβ ∈ (1, 4), p6 = p− β + 2αβ ∈ (0, 3), we have min 1≤i≤6 pi = p1 = 2αβ. We divide both sides of (4.16) by |h|2αβ , and notice that |h|−2αβ ≤ |h|−2α for |h| < 1, 0 < α, β < 1. Then we have∫ BR/2 |τhVp(Du)|2 |h|2αβ dx ≤ c [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c|h|p(1−β) (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n 20 Z. WANG EJDE-2022/62 + c|h|2α(1−β) (∫ BR (ιk(x) + ιk(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ ( ∫ B2R |Du|pdx) n n−2α ]n−2α n + c ∫ BR |τhVp(Dψ)|2 |h|2α dx+ c|h|(α+1)(1−β) (∫ BR (ιk(x) + ιk(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|p( ∫ BR |DF | np n−2α dx) n−2α n + c|h|p (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np + c|h|p−β (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.17) Next we take the Lq norm with the measure dh |h|n restricted to the ball B(0, R4 ). For any k ∈ N, the integral in the third and fifth lines of (4.17) are taken for 2−k R4 ≤ |h| ≤ 2−k+1R 4 , so it is essential to notice that B(0, R 4 ) = ∪∞k=1 ( B(0, 2−k+1R 4 )\B(0, 2−k R 4 ) ) =: ∪∞k=1Ek. It is also worth noticing that the choice of the radius R = |h|β is possible for small values of |h|. This is because 2−k R4 ≤ |h| ≤ 2−k+1R 4 if and only if 2− k+2 1−β ≤ |h| ≤ 2− k+1 1−β , for any k ∈ N. Thus we have the estimate∫ BR 4 (0) (∫ BR 2 |τhVp(Du)|2 |h|2αβ dx )q/2 dh |h|n ≤ c ∫ BR 4 (0) [ ∫ BR (1 + |Dψ| np n−2α )dx ] q(n−2α) 2n dh |h|n + c ∫ BR 4 (0) |h| qp(1−β) 2 dh |h|n (∫ BR |D(u− ψ)| np n−2α dx ) q(n−2α) 2n + c ∞∑ k=1 ∫ Ek |h|qα(1−β) (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα n dh |h|n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] q(n−2α) 2n + c ∫ BR 4 (0) (∫ BR |τhVp(Dψ)|2 |h|2α dx )q/2 dh |h|n + c ∞∑ k=1 ∫ Ek |h| q(α+1)(1−β) 2 (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα 2n dh |h|n EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 21 × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] q(n−2α) 2n + c ∫ BR 4 (0) |h|qp/2 dh |h|n (∫ BR |DF | np n−2α dx ) (n−2α)q 2n + c ∫ BR 4 (0) |h|qp/2 dh |h|n (∫ BR |DF | np n−2α dx )n−2α n q(p−1) 2p (∫ BR |D2ψ| np n−2α dx ) q(n−2α) 2np + c ∫ BR 4 (0) |h| q(p−β) 2 dh |h|n (∫ BR |DF | np n−2α dx )n−2α n q(p−1) 2p × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] q(n−2α) 2np . (4.18) To simplify notation, we set S∗ = ∫ B2R (1 + |Du|p + |F |p̂ + |DF |p̂ + |Du|p̂ + |Dψ|p̂ + |D2ψ|p̂)dx. (4.19) where p̂ = np n−2α , 0 < α < 1. Then (4.18) can be written as∫ BR 4 (0) (∫ BR 2 |τhVp(Du)|2 |h|2αβ dx )q/2 dh |h|n ≤ C ∫ BR 4 (0) |h| qp(1−β) 2 dh |h|n + C ∞∑ k=1 ∫ Ek |h|qα(1−β) (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα n dh |h|n + C ∫ BR 4 (0) (∫ BR |τhVp(Dψ)|2 |h|2α dx )q/2 dh |h|n + C ∞∑ k=1 ∫ Ek |h| q(α+1)(1−β) 2 (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα 2n dh |h|n + C ∫ BR 4 (0) |h|qp/2 dh |h|n + C ∫ BR 4 (0) |h| q(p−β) 2 dh |h|n , (4.20) where the constant C = C(n, p, q, R, α, ν, γ,Γ, S∗). Now we apply Young’s Inequality with exponents (2, 2) to the second and the fourth integrals of the right-hand side of (4.20), thus obtaining∫ BR 4 (0) (∫ BR 2 |τhVp(Du)|2 |h|2αβ dx )q/2 dh |h|n ≤ C ∫ BR 4 (0) |h| qp(1−β) 2 dh |h|n + C ∫ BR 4 (0) |h|2qα(1−β) dh |h|n + C ∞∑ k=1 ∫ Ek (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) 2qα n dh |h|n 22 Z. WANG EJDE-2022/62 + C ∫ BR 4 (0) |h|q(α+1)(1−β) dh |h|n + C ∞∑ k=1 ∫ Ek (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα n dh |h|n + C ∫ BR 4 (0) |h|qp/2 dh |h|n + C ∫ BR 4 (0) |h| q(p−β) 2 dh |h|n + C ∫ BR 4 (0) (∫ BR |τhVp(Dψ)|2 |h|2α dx )q/2 dh |h|n . (4.21) Noticing that α, β ∈ (0, 1) and p ∈ (1, 2), if we set q1 = qp(1− β) 2 , q2 = 2qα(1− β), q3 = q(α+ 1)(1− β), q4 = qp 2 , q5 = q(p− β) 2 , then we have ϑ := min 1≤i≤5 qi > 0. Since |h| < 1, we can write (4.21) as follows∫ BR 4 (0) (∫ BR 2 |τhVp(Du)|2 |h|2αβ dx )q/2 dh |h|n ≤ C ∞∑ k=1 ∫ Ek (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) 2qα n dh |h|n + C ∞∑ k=1 ∫ Ek (∫ BR (ιk(x) + ιk(x+ h))n/αdx ) qα n dh |h|n + C ∫ BR 4 (0) (∫ BR |τhVp(Dψ)|2 |h|2α dx )q/2 dh |h|n + C ∫ BR 4 (0) |h|ϑ dh |h|n =: I1 + I2 + I3 + I4. (4.22) By assumption, we have I3 ≤ ‖Vp(Dψ)‖Bα2,q(BR) <∞. (4.23) For the term I4, by polar coordinates transformation, we obtain I4 = C ∫ BR 4 (0) |h|ϑ dh |h|n = C ∫ R/4 0 ρϑ−1dρ = C(n, p, q, R, α, β) <∞, (4.24) since ϑ > 0, i.e. ϑ− 1 > −1, which implies the last integral is finite. We now write the integral I1 in polar coordinates, so h ∈ Ek if and only if h = ρξ for 2−k R4 ≤ ρ < 2−k+1R 4 and some ξ in the unit sphere Sn−1 on Rn. We denote by EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 23 dσ(ξ) the surface measure on Sn−1, then we have I1 = C ∞∑ k=1 ∫ rk−1 rk ∫ Sn−1 (∫ BR (ιk(x) + ιk(x+ ρξ))n/αdx ) 2qα n dσ(ξ) dρ ρ ≤ C ∞∑ k=1 ∫ rk−1 rk ∫ Sn−1 ‖ιk(x) + ιk(x+ ρξ)‖2q Ln/α(BR) dσ(ξ) dρ ρ ≤ C ( ∞∑ k=1 ‖ιk‖2qLn/α(B2R) )∫ rk−1 rk dρ ρ = C ln 2 ∞∑ k=1 ‖ιk‖2qLn/α(B2R) = C‖(ιk)k‖2q`2q(Ln/α(B2R)) ≤ C‖(ιk)k‖2q`q(Ln/α(B2R)) <∞, (4.25) where we set rk = 2−k R4 . Moreover, for any ξ ∈ Sn−1 and rk ≤ ρ < rk−1, by Lemma 2.3, we have ‖(ιk(x) + ιk(x+ ρξ))‖Ln/α(BR) ≤ 2‖ιk‖Ln/α(B R+R 4 ) ≤ 2‖ιk‖Ln/α(B2R). Recalling the continuous embedding `q(Ln/α(B2R)) ↪→ `2q(Ln/α(B2R)), so the es- timate is easily established. Arguing as the estimate to I1, for the term I2, we obtain I2 ≤ C‖(ιk)k‖q`q(Ln/α(B2R)) <∞. (4.26) Inserting (4.23)-(4.26) in (4.22), we have ‖τhVp(Du) |h|αβ ‖Lq( dh |h|n ;L2(BR/2)) ≤ C ( 1 + ‖Vp(Dψ)‖Bα2,q(BR) + ‖(ιk)k‖2q`q(Ln/α(B2R)) ) . Noticing that the constant C depends on S∗ given by (4.19), then for a suitable exponent σ = σ(n, p, q, α) > 0, utilizing that p < np n−2α , Theorem 2.16, Lemma 2.9, and Lemma 2.11, we obtain ‖τhVp(Du) |h|αβ ‖Lq( dh |h|n ;L2(BR/2)) ≤ C [ 1 + ‖Du‖Lp(B4R) + ‖Vp(Dψ)‖Bα2,q(B4R) + ‖D2ψ‖ L np n−2α (B4R) + ‖F‖ W 1, np n−2α (B4R) + ‖(ιk)k‖`q(Ln/α(B2R)) ]σ , that is the conclusion. � 4.2. Proof of Theorem 1.3. To prove the theorem we use the arguments of the previous section. Differences come when we estimate IV, V and V I. Proof. By hypothesis, since Vp(Dψ) ∈ Bδ2,∞,loc(Ω) with 0 < α < δ < 1, we have Vp(Dψ) ∈ L 2n n−2α loc (Ω), and so Dψ ∈ L np n−2α loc (Ω). Similar to the proof of Theorem 1.2, the estimate to I, II, III, V II, V III and IX can be treated in the same way. We only need to use assumption (1.7) instead of (1.6) so as to estimate the term IV, V and V I. 24 Z. WANG EJDE-2022/62 For the term IV , using the assumption (1.7), Young’s Inequality with exponents (2, 2), Hölder’s Inequality with exponents ( n2α , n n−2α ) and Lemma 2.3, for |h| < R 4 , we have |IV | ≤ |h|α ∫ Ω η2(ι(x) + ι(x+ h))(µ2 + |Du|2) p−1 2 |τhDu|dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(ε)|h|2α ∫ Bl2 (ι(x) + ι(x+ h))2(µ2 + |Du(x)|2 + |Du(x+ h)|2)p/2dx ≤ ε ∫ Ω η2(µ2 + |Du(x)|2 + |Du(x+ h)|2) p−2 2 |τhDu|2dx + C(n, p, α, ε)|h|2α (∫ BR (ι(x) + ι(x+ h))n/αdx )2α/n[ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n . (4.27) For the term V , by the same arguments that we used in the previous section, we have |V | ≤ c|h|2α (∫ BR (ι(x) + ι(x+ h))n/αdx )2α/n[ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c ∫ BR |τhVp(Dψ)|2dx. (4.28) For the term V I, by assumption (1.7), for |h| < R 4 , we obtain |V I| ≤ C|h|α+1 R1−α (∫ BR (ι(x) + ι(x+ h))n/αdx )α/n[ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n (4.29) Now we plug (4.6), (4.7), (4.12),(4.13), (4.14), (4.27), (4.28), and (4.29) into (4.1), then choosing ε = ν 2(3+C0) , we obtain∫ BR/2 |τhVp(Du)|2dx ≤ cR2α [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c Rp−2α |h|p (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n + c|h|2α (∫ BR (ι(x) + ι(x+ h))n/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 25 + c ∫ BR |τhVp(Dψ)|2dx+ c R1−α |h| α+1 (∫ BR (ι(x) + ι(x+ h))n/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n + c|h|pR2α (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np + c|h|pR2α−1 (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.30) Now let us notice that |h| ≤ |h|β 4 if and only if |h| ≤ 2− 2 1−β for any β ∈ (0, 1). Recalling the covering argument that we used in the previous section, so we have R ∝ |h|β . Dividing both sides of (4.30) by |h|2αβ , using Lemma 2.3 and that for |h| < R 4 < R ≤ 1, since 0 < α, β < 1, |h|−2αβ < |h|−2α, we obtain∫ BR/2 |τhVp(Du)|2 |h|2αβ dx ≤ c [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c|h|p(1−β) (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n + c|h|2α(1−β) (∫ B2R ιn/αdx )2α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c ∫ BR |τhVp(Dψ)|2 |h|2α dx+ c|h|(α+1)(1−β) (∫ B2R ιn/αdx )α/n × [ 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n + c|h|p (∫ BR |DF | np n−2α dx )n−2α n + c|h|p (∫ BR |DF | np n−2α dx )n−2α n p−1 p (∫ BR |D2ψ| np n−2α dx )n−2α np + c|h|p−β (∫ BR |DF | np n−2α dx )n−2α n p−1 p [ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α np . (4.31) Using Young’s Inequality with exponents ( n2α , n n−2α ) and ( p p−1 , p), then (4.31) be- comes∫ BR/2 |τhVp(Du)|2 |h|2αβ dx 26 Z. WANG EJDE-2022/62 ≤ c [ ∫ BR (1 + |Dψ| np n−2α )dx ]n−2α n + c|h|p(1−β) (∫ B2R |D(u− ψ)| np n−2α dx )n−2α n + c|h|2α(1−β) [ ∫ B2R ιn/αdx+ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] + c ∫ BR |τhVp(Dψ)|2 |h|2α dx+ c|h|(α+1)(1−β) · [ ( ∫ B2R ιn/αdx) 1 2 + 1 + ∫ B2R |F | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ] + c|h|p (∫ BR |DF | np n−2α dx )n−2α n + c|h|p [( ∫ BR |DF | np n−2α dx )n−2α n + (∫ BR |D2ψ| np n−2α dx )n−2α n ] + c|h|p−β [ ∫ BR |DF | np n−2α dx+ 1 + ∫ B2R |F | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]n−2α n . (4.32) By Lemma 2.13, Vp(Dψ) ∈ Bδ2,∞,loc(Ω) implies that Dψ ∈ Bδp,∞,loc(Ω). Using that 0 < α < δ < 1 and Lemma 2.10, we have Vp(Dψ) ∈ Bα2,∞,loc(Ω) and Dψ ∈ Bαp,∞,loc(Ω). Taking the supremum for |h| < R 4 at both sides of (4.32), we obtain [Vp(Du)]Ḃαβ2,∞(BR/2) ≤ C[Vp(Dψ)]Ḃα2,∞(BR) + C [ 1 + ∫ B2R ιn/αdx+ ∫ B2R |F | np n−2α dx+ ∫ B2R |DF | np n−2α dx + ∫ B2R |Dψ| np n−2α dx+ ∫ B2R |D2ψ| np n−2α dx+ (∫ B2R |Du|pdx ) n n−2α ]σ , where the exponent σ = σ(n, p, α) and the constant C = C(n, p,R, α, ν,Γ). By the definition of the norm in Besov-Lipschitz spaces and using Lemma 2.10, we have [Vp(Du)]Ḃαβ2,∞(BR/2) ≤ C‖Vp(Dψ)‖Bδ2,∞(BR) + C [ 1 + ∫ B2R ιn/αdx+ ∫ B2R |F | np n−2α dx + ∫ B2R |DF | np n−2α dx+ ∫ B2R |Dψ| np n−2α dx+ ∫ B2R |D2ψ| np n−2α dx + (∫ B2R |Du|pdx ) n n−2α ]σ . Noticing for 0 < α < δ < 1, we have p < np n−2α < np n−2δ , then we obtain [Vp(Du)]Ḃαβ2,∞(BR 2 ) ≤ C [ 1 + ‖Du‖Lp(B2R) + ‖Vp(Dψ)‖Bδ2,∞(BR) + ‖Dψ‖ L np n−2δ (B2R) + ‖D2ψ‖ L np n−2α (B2R) + ‖F‖ W 1, np n−2α (B2R) EJDE-2022/62 HIGHER DIFFERENTIABILITY FOR OBSTACLE PROBLEMS 27 + ‖ι‖Ln/α(B2R) ]σ . 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Ziemer; Interior regularity for solutions to obstacle problems, Nonlinear Anal., 10 (1986), no. 12, 1427–1448. [24] G. Stampacchia; Formes bilineaires coercivitives sur les ensembles convexes, C. R. Acad. Sci. Paris., 258 (1964), 4413–4416. [25] J. Xiao; The transport equation in the scaling invariant Besov or Essén-Janson-Peng-Xiao space, J. Differ. Equ., 266 (2019), 7124–7151. Zhenqiang Wang School of Mathematical Sciences, Nankai University, Tianjin 300071, China Email address: 2120190058@mail.nankai.edu.cn 1. Introduction 2. Notation and preliminary results 2.1. Difference quotients 2.2. Besov-Lipschitz spaces 2.3. VMO coefficients 3. Higher order integer differentiability 4. Higher order fractional differentiability 4.1. Proof of Theorem ?? 4.2. Proof of Theorem ?? References