Higher differentiability for solutions to a class of obstacle problems

We establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫Ω⟨A(x,Du),D(φ-u)⟩dx≥0∀φ∈Kψ(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega } \langle \mathcal {A}(x, Du), D(\varphi - u) \rangle \, dx \ge 0 \qquad \forall \varphi \in \mathcal {K}_{\psi }(\Omega ) \end{aligned}$$\end{document}where A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document} is a p-harmonic type operator, ψ∈W1,p(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \in W^{1,p}(\Omega )$$\end{document} is a fixed function called obstacle and Kψ={w∈W1,p(Ω):w≥ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_{\psi }=\{w \in W^{1,p}(\Omega ): w \ge \psi \,\,$$\end{document} a.e. in Ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \}$$\end{document} is the class of the admissible functions. We prove that an extra differentiability assumption on the gradient of the obstacle transfers to Du with no losses in the natural exponent of integrability, provided the partial map x↦A(x,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto \mathcal {A}(x,\xi )$$\end{document} possesses a suitable differentiability property measured either in the scale of the Sobolev space W1,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,n}$$\end{document} or in that of the critical Besov–Lipschitz spaces Bnα,qα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^\alpha _{\frac{n}{\alpha }, q}$$\end{document}, for a suitable 1≤q≤+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le q\le +\infty $$\end{document}.


Introduction
The aim of this paper is the study of the higher differentiability properties of the gradient of the solutions u ∈ W 1, p ( ) to variational obstacle problems of the form min ˆ F(x, Dw) : w ∈ K ψ ( ) . (1.1) The function ψ : → [−∞, +∞), called obstacle, belongs to the Sobolev class W 1, p ( ) and the class K ψ ( ) is defined as follows K ψ ( ) := w ∈ W 1, p ( ) : w ≥ ψ a.e. in . (1.2) Note that the set K ψ is not empty since ψ ∈ K ψ . It is worth observing that u ∈ W 1, p ( ) is a solution to the obstacle problem in K ψ ( ) if and only if u ∈ K ψ solves the following variational inequalitŷ for all ϕ ∈ K ψ ( ), where we set We assume that there exist positive constants ν, L, and an exponent p ≥ 2 such that the following p-ellipticity and p-growth conditions are satisfied: |A(x, ξ) − A(x, η)| ≤ L |ξ − η|(μ 2 + |ξ | 2 + |η| 2 ) for a.e. x ∈ and every ξ, η ∈ R n . From now on, throughout the paper, we denote with μ ∈ [0, 1] a parameter that will allow us to consider in our analysis both the degenerate and the non-degenerate situation.
The study of the regularity theory for obstacle problems is a classical topic in Partial Differential Equations and Calculus of Variations. The obstacle problem appeared in the mathematical literature in the work by Stampacchia [44] in the special case ψ = χ E and related to the capacity of a subset E ⊂⊂ ; in an earlier independent work, Fichera [21] solved the first unilateral problem, the so-called Signorini problem in elastostatics.
It is well known that the solution to the obstacle problem cannot be of class C 2 independently of how regular the obstacle is; this led to the origin of the concept of weak solution and to the theory of variational inequalities, after the fundamental work of Lions and Stampacchia [35] (for more details we refer to classical monographs [1,12,22,29,43]); these problems can generally be solved by applying methods of functional analysis, so the question to give conditions to establish that weak solutions are, in many cases, classical ones is of fundamental importance (see [10]).
It is usually observed that the regularity of solutions to the obstacle problems is influenced by the one of the obstacle; for example, for linear obstacle problems, obstacle and solutions have the same regularity [7,11,29]. This does not apply in the nonlinear setting, hence along the years, there have been intense research activities for the regularity of the obstacle problem in this direction (see in particular the fundamental techniques established in [32,33]).
A first important result by Michael and Ziemer [36] establishes Hölder continuity of solutions to the obstacle problem when the obstacle itself is Hölder continuous. Choe [13] proved that if the gradient of the obstacle is Hölder continuous, the same happens for the gradient of solutions. Other results that deserved to be quoted are [14,23,34], in the case of a single obstacle problem, and [4] in the case of double obstacle problems. Since then, many regularity results have been obtained in different situations: for instance we quote [16] in the setting of Morrey and Campanato spaces, [8,17,19,37] in the setting of nonstandard growth conditions (see also [9,18,38] for Calderón-Zygmund case). Moreover we refer to [3,39] for gradient continuity for nonlinear obstacle problems, [20] for global results up to the boundary, [5] for the parabolic case, [6] for the porous medium problem.
As far as we know, such analysis has not been carried out in case the gradient of the obstacle ψ possesses some extra differentiability properties.
Here, assuming that the gradient of the obstacle belongs to a suitable Sobolev class, of integer or fractional order, we are interested in finding conditions on the partial map x → A(x, ξ) in order to obtain that the extra differentiability property of the obstacle transfers to the gradient of the solution with no loss in the order of differentiability.
Our analysis comes from the fact that the regularity of the solutions to the obstacle problem (1.1) is strictly connected to the analysis of the regularity of the solutions to partial differential equations of the form It is well known that no extra differentiability properties for the solutions of partial differential equations of the type divA(x, Du) = div G, (1.4) both in the setting of integer and fractional Sobolev spaces, can be expected even if G is smooth, unless some assumption is given on the x-dependence of A. On the other hand, recent results concerning the higher differentiability of solutions to (1.4) show that the weak differentiability of integer or fractional order of the map A, as function of the x-variable, is a sufficient condition (see [24,25,[40][41][42] for the case of Sobolev space with integer order and [2,15] for the fractional one).
Indeed, in [24,40,41], the higher differentiability of the solutions to the equation in (1.4) is obtained assuming a W 1,n type regularity on the partial map x → A(x, ξ), expressed through a pointwise condition on A(·, ξ) that relies on the characterization of the Sobolev spaces due to Hajłasz [27].
More precisely, for Carathéodory functions A satisfying assumptions (A1)-(A3), it is assumed that there exists a non negative function κ ∈ L n loc ( ) such that the following inequality holds true for a.e. x, y ∈ and for every ξ ∈ R n . It turns out that this condition is sufficient also in our context of obstacle problems to prove that the differentiability of the gradient of the obstacle transfers to the gradient of the solution with no loss in the order of differentiability. More precisely, our first result is the following: for an exponent 2 ≤ p < n and let u ∈ K ψ ( ) be the solution to the obstacle problem (1.3). Then we have The existing literature on the regularity of solutions to partial differential equations of the form (1.4) shows that the same phenomenon occurs also in case of a fractional Sobolev assumption on the data. In fact, it has been proven that, if G belongs to a suitable Besov-Lipschitz space, the gradient of the solution to the Eq. (1.4) gains an extra fractional differentiability that can be expressed through its belonging to a suitable Besov space, provided the partial map x → A(x, ξ) belongs to a Besov space too. Also such Besov regularity of the map A(·, ξ) can be expressed through the pointwise characterization of these spaces due to Koskela et al. ([30]).
More precisely, given 0 < α < 1 and 1 ≤ q ≤ ∞ we say that (A5) is satisfied if there exists a sequence of measurable non-negative functions g k ∈ L n α ( ) such that k g k q L n α ( ) < ∞, and at the same time for each ξ ∈ R n and almost every x, y ∈ such that 2 −k diam( ) ≤ |x − y| < 2 −k+1 diam( ). We will shortly write then that (g k ) k ∈ q (L n α ( )). If A(x, ξ) = γ (x)|ξ | p−2 ξ and = R n then (A5) says that the function γ (x) belongs to B α n α ,q , see [30,Theorem 1.2] and Sect. 2 for more details. Under assumption (A5), we are able to prove that the extra fractional differentiability of the obstacle transfers to the solutions to the problem (1.3), both measured in the Besov scale. Indeed, we have the following: Theorem 1.2 Assume that A(x, ξ) satisfies (A1)-(A3) and (A5) for an exponent 2 ≤ p < n α . Let u ∈ K ψ ( ) be the solution to the obstacle problem (1.3). Then the following implication holds locally, provided q ≤ p * α = np n−α p . Our proofs are achieved by means of the difference quotient method, that is quite natural when trying to establish higher differentiability results. Here the difficulties come from the set of admissible test functions that have to take into account the presence of the obstacle. In order to overcome this problem, we need to consider difference quotient involving both the solution and the obstacle, so that the test function satisfies the constraint of belonging to the admissible class K ψ . Moreover, in the case of Besov coefficients, the proof relies on the fact that the Besov spaces B α n α ,q , for every 1 ≤ q ≤ +∞, continuously embed into the V M O space of Sarason (see [28] and Sect. 2 below). Obstacle problems with V M O coefficients are known to have a nice L p theory (see [9]) and we take advantage from this result through the embedding Theorems in Besov spaces ( [28] and Sect. 2.) In case the order of fractional differentiability of the coefficients differs from that of the obstacle, we have that the gradient of the solution still inherits an extra fractional differentiability which is the minimum between the one of the coefficients and the one of the obstacle, all measured in the scale of Besov spaces. Indeed, combining the inclusion between Besov spaces (see Lemma 2.3 in Sect. 2) with Theorem 1.2, allows us to establish the following: Corollary 1.3 Assume that A(x, ξ) satisfies (A1)-(A3) and (A5). Let u ∈ K ψ ( ) be the solution to the obstacle problem (1.3). Then the following implication holds locally, provided q ≤ p * β = np n−β p . It is also worth mentioning that, in the scale of Besov spaces, a regularity of the type B α p,∞ is the weakest one to assume both on the coefficients and on the gradient of the obstacle (see Lemma 2.3 in Sect. 2 below). In this case, we still have that the differentiability assumption on the gradient of the obstacle transfers to the solutions, but with a small loss. The main difference is that an analogous of the Sobolev imbedding Theorem holds but in a weaker form. Actually, we have the following: for an exponent 2 ≤ p < n α and let u ∈ K ψ ( ) be the solution to the obstacle problem (1.3). If there exist 0 < α < 1 and for a.e. x, y ∈ and for every ξ ∈ R n , then the following implication holds locally, provided 0 < α < β < 1.
Previous regularity results concerning local minimizers of integral functionals of the Calculus of Variations, under the assumption (A6), have been obtained by Kristensen and Mingione [31]. The paper is organized as follows: Sect. 2 contains some notations and preliminary results, Sect. 3 is devoted to the proof of Theorem 1.1 while Sect. 4 is concerned with the proof of Theorem 1.2 and Corollary 1.3 and Sect. 5 with that of Theorem 1.4.

Notations and preliminary results
In this paper we shall denote by C or c a general positive constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies will be suitably emphasized using parentheses or subscripts. In what follows, B(x, r ) = B r (x) = {y ∈ R n : |y − x| < r } will denote the ball centered at x of radius r . We shall omit the dependence on the center and on the radius when no confusion arises. For a function u ∈ L 1 (B), the symbol will denote the integral mean of the function u over the set B. It is convenient to introduce an auxiliary function for any ξ , η ∈ R n .

Besov-Lipschitz spaces
Given h ∈ R n and v : R n → R, let us introduce the notation As in [28, Section 2.5.12], given 0 < α < 1 and 1 ≤ p, q < ∞, we say that v belongs to the Equivalently, we could simply say that v ∈ L p (R n ) and τ h v |h| α ∈ L q dh |h| n ; L p (R n ) . As usual, if one simply integrates for h ∈ B(0, δ) for a fixed δ > 0 then an equivalent norm is obtained, because Similarly, we say that Again, one can simply take supremum over |h| ≤ δ and obtain an equivalent norm. By construction, B α p,q (R n ) ⊂ L p (R n ). One also has the following version of Sobolev embeddings (a proof can be found at [28,Prop. 7.12]). For further needs, we recall the following inclusions ([28, Proposition 7.10 and Formula (7.35)]).

Given a domain
⊂ R n , we say that v belongs to the local Besov space B α p,q,loc if ϕ v belongs to the global Besov space B α p,q (R n ) whenever ϕ belongs to the class C ∞ c ( ) of smooth functions with compact support contained in . It is worth noticing that one can prove suitable versions of Lemma 2.2 and of Lemma 2.3, by using local Besov spaces.
The following Lemma is an easy exercise and its proof can be found in [2].
Here the measure dh |h| n is restricted to the ball B(0, r B ) on the h-space.
It is known that Besov-Lipschitz spaces of fractional order α ∈ (0, 1) can be characterized in pointwise terms. Given a measurable function v : The following result was proved in [30].
where the infimum runs over all possible fractional α-Hajłasz gradients for v.

Difference quotient
We recall some properties of the finite difference operator that will be needed in the sequel. Let us recall that, for every function F : R n → R the finite difference operator is defined by where h ∈ R, e s is the unit vector in the x s direction and s ∈ {1, . . . , n}. We start with the description of some elementary properties that can be found, for example, in [26]. Proposition 2.6 Let F and G be two functions such that F, G ∈ W 1, p ( ), with p ≥ 1, and let us consider the set (d2) If at least one of the functions F or G has support contained in |h| , then The next result about finite difference operator is a kind of integral version of Lagrange Theorem.
Now, we recall the fundamental Sobolev embedding property.
For the proof see, for example, [26,Lemma 8.2]. We conclude this subsection recalling a fractional version of previous Lemma (see [31]).

VMO coefficients
We shall use the fact that if A satisfies (A1), (A2) , (A3) and (A5) or (A6) then it is locally uniformly in V M O. More precisely, given a ball B ⊂ , let us introduce the operator One can easily check that A B (ξ ) also satisfies assumptions (A1), (A2) and (A3). Setting Next Lemma will be a key tool in the proof of our results. Its proof can be found in [15,Lemma 12] Lemma 2.10 Let A be such that (A1), (A2), (A3) and (A5) or (A6) hold. Then A is locally uniformly in V M O, that is (2.5) holds.
The following Theorem is a Calderón-Zygmund type estimate for solutions to the obstacle problem with V M O coefficients. Theorem 2.11 Let 2 ≤ p ≤ n, and q > p. Assume that (A1), (A2), (A3) hold, and that

3). Then the following implication holds
Moreover, there exists a constant C = C(n, ν, , L, p, q) such that the following inequality

Proof of Theorem 1.1
Existence of solutions to the obstacle problem (1.1) can be easily proved through classical results regarding variational inequalities, so in this paper we will mainly concentrate on the regularity results. In particular, this Section is devoted to the proof of Theorem 1.1. The main point will be the choice of suitable test functions ϕ in (1.3) that involve the difference quotient of the solution but at the same time turns to be admissible for the obstacle class K ψ ( ).

Proof Let us consider
It is easy to see that such function ϕ belongs to the obstacle class Due to the local nature of our results, there is no loss of generality in supposing R ≤ 1, that we will do from now on. Then, for |h| < R 4 , we consider From the regularity of u and ψ, it is immediate to check that v 1 ∈ W 1, p 0 ( ). Moreover v 1 fulfills (3.1). Indeed, for a.e. x ∈ and for any t ∈ [0, 1) By using in (1.3) as an admissible test function ϕ = u + tv 1 , with v 1 introduced in (3.2), we obtain On the other hand, if we define then v 2 ∈ W 1, p 0 ( ) and (3.1) still is trivially satisfied, due to the fact that By means of a simple change of variable, we obtain We can add (3.3) and (3.5), thus obtaining We can write previous inequality as follows The ellipticity assumption expressed by (A1) implies By virtue of assumption (A2) and Young's inequality, we get where we used Hölder's inequality, the properties of η and the second estimate of Lemma 2.7. Using the assumption Dψ ∈ W 1, p and first estimate of Lemma 2.7 in the second integral of the right hand side of (3.9), we obtain Arguing analogously, we get where we used the fact that |Dη| ≤ c R . Since u −ψ ∈ W 1, p ( ), we may use the first estimate of Lemma 2.7 to control last integral in the right hand side of previous estimate, obtaining that where we used also Young's inequality. In order to estimate the integral I V , we use assumption (A4), Young's and Hölder's inequalities as follows where we also used that suppη ⊂ B R . Now, the assumption on the obstacle, Theorem 2.11 and classical Sobolev embedding Theorem imply for a constant C = C(n, ν, L, , p). Observing that (3.14) Inserting (3.14) in (3.12), we get where now C ε = C(ε, n, ν, L, , p, R) and we used the fact that 0 ≤ μ ≤ 1. Assumption (A4) also entails where we used the properties of η, Hölder's inequality and, in the last line, we used estimate (3.13), since Dψ ∈ L p * and since By virtue of the assumption Dψ ∈ W 1, p loc ( ), we can use the first inequality of Lemma 2.7 to estimate the first integral in the right hand side of (3.16) and the second inequality of Lemma 2.7 to estimate the second one, thus getting with C = C(n, ν, L, , p, R). Finally, arguing as we did for the estimate of V we get where we used the fact that |Dη| ≤ C R . Using Lemma 2.7 and (3.13) in the right hand side of previous estimate, we get  .7), we infer the existence of constants C ε ≡ C ε (ε, ν, L, , n, p, R) and C ≡ C(ν, L, , n, p, R) such that where we also used Lemma 2.7. Choosing ε = ν 6 we get

Proof of Theorem 1.2 and Corollary 1.3
This section is devoted to the proof of Theorem 1.2 and Corollary 1.3. The proof of Theorem 1.2 goes along the lines of the one presented in the previous Section until the estimates of the first three terms, I , I I , I I I in (3.6). Differences come when starting estimate the last three integrals, in which the different assumption on the partial map x → A(x, ·) and on the obstacle come into the play.
Proof Our starting point is the following estimate that is obtained inserting (3.8), (3.9) and (3.11) in (3.7) of previous Section and where I V , V and V I are those defined in (3.6) in the previous Section. In order to estimate the integral I V , we use this time assumption (A5), Young's and Hölder's inequalities as follows with a constant C = C(n, ν, L, , p, α). Observing that by Hölder's inequality and (4.3), we have Inserting (4.4) in (4.2), we get with a constant C ε = C ε (ε, n, ν, L, , p, α, R).
For h such that 2 −k R 4 ≤ |h| ≤ 2 −k+1 R 4 , k ∈ N, assumption (A5) also yields that with a constant C = C(n, ν, L, , p, α, R); here we used the properties of η, Hölder's inequality and, in the last line, we used estimate (4.3), since Dψ ∈ L p * α and since Finally, arguing as we did for the estimate of V , we get since |Dη| ≤ C R . Using the first estimate of Lemma 2.7 and (4.3) in the right hand side of previous estimate, we get with a constant C = C(n, ν, L, , p, α, R). Inserting estimates (4.5), (4.6) and (4.7) in (4.1), we infer the existence of constants C ε = C ε (ε, n, ν, L, , p, α, R) and C = C(n, ν, L, , p, α, R) such that Using Lemma 2.1 in the left hand side of previous estimate, recalling that η ≡ 1 on B R 2 and dividing both sides by |h| 2α , we get In order to conclude, we need now to take the L q norm with the measure dh |h| n restricted to the ball B(0, R 4 ) on the h-space of the L 2 norm of the difference quotient of order α of the function V p (Du). Since the functions g k are defined We obtain the following estimatê inequality (4.9) can be simplified as followŝ where now the constant C = C(ν, , L, n, p, q, α, R,H ). Furthermore, the use of Young's inequality with exponent 2 in the third and fourth lines of estimate (4.10) yieldŝ (4.11) Note that I 1 is controlled by the norm in the Besov space B α p,q on B R of the gradient of the obstacle. More precisely which is finite by the assumptions. The integral I 2 can be easily calculated in polar coordinates as follows since α ∈ (0, 1). We now write the integral I 3 in polar coordinates, so h ∈ E k if and only if h = r ξ for some 2 −k+1 R 4 ≤ r < 2 −k R 4 and some ξ in the unit sphere S n−1 on R n . We denote by dσ (ξ) the surface measure on S n−1 . We bound the term I 3 by where we set r k = 2 −k R 4 . We note that for each ξ ∈ S n−1 and r k−1 ≤ r ≤ r k Hence .
Inserting the estimate of I i , i = 1, 2, 3 in (4.11) and taking the power 1 q to both sides, we get . Lemma 2.4 now yields that V p (Du) ∈ B α 2,q locally, i.e. estimate (1.8).
We are in position to give the

Proof of Theorem 1.4
This section is devoted to the proof of Theorem 1.4 that is achieved using the arguments of previous sections, with the modifications that take into account the different assumptions on the gradient of the obstacle and on the map x → A(x, ξ).
Proof The proof goes exactly as that of Theorem 1.2 until we arrive at estimate (4.1) of previous Section. We need to treat differently only the integrals I V , V and V I in which the new assumption (A6) comes into the play. Indeed, assumption (A6) together with Young's and Hölder's inequalities yields that for every h ∈ R n such that |h| < R 4 . Since p ≥ 2, the use of Hölder's inequality implieŝ where the first inequality is given by Lemma 2.1. Therefore, dividing previous inequality by |h| 2β and using (5.2), we get where C = C(n, ν, L, , p, α, β). Inserting (5.4) in (5.1), we get Again by virtue of assumption (A6), we get By virtue of (5.2), we may estimate the first integral in the right hand side of (5.6) as follows with a constant C depending also on , and where we also used the second inequality of Lemma 2.7. Finally, arguing as we did for the estimate of V we get |V I | ≤ 2|h| αˆ η|Dη|(g(x + h) + g(x))(1 + |Du| 2 ) Inserting estimates (5.5), (5.7) and (5.8) in (4.1) we infer the existence of constants C ε ≡ C(ε, ν, L, , n, p, q, R, ||Dψ|| B β p,∞ ) and C ≡ C (ν, L, , n, p, q, R,