On a Class of Nonlocal Obstacle Type Problems Related to the Distributional Riesz Fractional Derivative

In this work, we consider the nonlocal obstacle problem with a given obstacle $\psi$ in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^{d}$, such that $\mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset$, given by \[u\in\mathbb{K}_\psi^s:\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi,\] for $F\in H^{-s}(\Omega)$, the dual space of $H^s_0(\Omega)$, $0<s<1$. The nonlocal operator $\mathcal{L}_a:H^s_0(\Omega)\to H^{-s}(\Omega)$ is defined with a measurable, bounded, strictly positive singular kernel $a(x,y)$, possibly not symmetric, by \[\langle\mathcal{L}_au,v\rangle=P.V.\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}v(x)(u(x)-u(y))a(x,y)dydx=\mathcal{E}_a(u,v),\] with $\mathcal{E}_a$ being a Dirichlet form. Also, the fractional operator $\tilde{\mathcal{L}}_A=-D^s\cdot AD^s$ defined with the distributional Riesz $s$-fractional derivative and a bounded matrix $A(x)$ gives a well defined integral singular kernel. The corresponding $s$-fractional obstacle problem converges as $s\nearrow1$ to the obstacle problem in $H^1_0(\Omega)$ with the operator $-D\cdot AD$ given with the gradient $D$. We mainly consider problems involving the bilinear form $\mathcal{E}_a$ with one or two obstacles, and the N-membranes problem, deriving a weak maximum principle, comparison properties, approximation by bounded penalization, and the Lewy-Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in $L^\infty(\Omega)$, local H\"older regularity when $a$ is symmetric, and local regularity in $W^{2s,p}_{loc}(\Omega)$ and $C^1(\Omega)$ for fractional $s$-Laplacian obstacle-type problems. These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of $H^s_0(\Omega)$ and some remarks on the associated $s$-capacity for general $\mathcal{L}_a$.

for F in H −s (Ω), the dual space of the fractional Sobolev space H s 0 (Ω), 0 < s < 1. The nonlocal operator La : H s 0 (Ω) → H −s (Ω) is defined with a measurable, bounded, strictly positive singular kernel a(x, y) : R d × R d → [0, ∞), by the bilinear form which is a (not necessarily symmetric) Dirichlet form, whereũ,ṽ are the zero extensions of u and v outside Ω respectively. Furthermore, we show that the fractional operatorLA = −D s · AD s : H s 0 (Ω) → H −s (Ω) defined with the distributional Riesz fractional D s and with a measurable, bounded matrix A(x) corresponds to a nonlocal integral operator L k A with a well defined integral singular kernel a = kA. The corresponding s-fractional obstacle problem forLA is shown to converge as s 1 to the obstacle problem in H 1 0 (Ω) with the operator −D · AD given with the classical gradient D.
We mainly consider obstacle-type problems involving the bilinear form Ea with one or two obstacles, as well as the N-membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization, and also the Lewy-Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in L ∞ (Ω), local Hölder regularity of the solutions when a is symmetric, and local regularity in fractional Sobolev spaces W 2s,p loc (Ω) and in C 1 (Ω) when La = (−∆) s corresponds to fractional s-Laplacian obstacle-type problems
Recently, in a series of two interesting papers [50] and [51], Shieh and Spector have considered a new class of fractional partial differential equations based on the distributional Riesz fractional derivatives. These fractional operators satisfy basic physical invariance requirements, as observed by Silhavy [52], who developed a fractional vector calculus for such operators. Instead of using the well-known fractional Laplacian, their starting concept is the distributional Riesz fractional gradient of order s ∈ (0, 1), which will be called here the s-gradient D s , for brevity: for u ∈ L p (R d ), p ∈ (1, ∞), we set where ∂ ∂xj is taken in the distributional sense, for every v ∈ C ∞ c (R d ), with I s denoting the Riesz potential of order s, 0 < s < 1: (1.2) Conversely, by Theorem 1.12 of [50], every u ∈ C ∞ c (Ω) can be expressed as where R j is the Riesz transform, which we recall, is given by y j |y| d+1 f (x − y) dy, j = 1, . . . , d.
Thus, we can write the s-gradient (D s ) and the s-divergence (D s ·) for sufficiently regular functions u and vector φ ( [16], [50], [51], [52]) in integral form, respectively, by where χ (x, z) is the characteristic function of the set {(x, z) : |z − x| > } for > 0. As it was shown in [50], D s has nice properties for u ∈ C ∞ c (R d ), namely it coincides with the fractional Laplacian as follows: where, for 0 < s < 1, Observe that for the s-gradient (D s ) and the s-divergence (D s ·) we need to consider the Cauchy principal value (P.V.) in the first expressions, but not in the second ones. This is because for the second expressions, we can estimate the integrand, as in [16], by separating the integrals into the parts {|y − x| ≤ 1} and {|y − x| > 1}. Then the first integral can be controlled by ω d Lip(u)´1 0 r −s dr, while the second integral can be controlled by 2ω d u L ∞ (R d )´+ ∞ 1 r −(1+s) dr, where Lip(u) is the Lipschitz constant for the function u and ω d is the spherical measure ω d =´{ |x|=1} dσ. Therefore, the second expressions are well-defined for all Lipschitz functions u with compact support, in particular for u ∈ C ∞ c (R d ). Similarly, the fractional Laplacian for smooth u, by Lemma 3.2 of [23], has two representations, with the second one being Lebesgue integrable by using a second order Taylor expansion. As a matter of fact, the s-divergence, s-gradient and s-Laplacian are linear operators from C ∞ functions with compact support into C ∞ functions that are rapidly decreasing at ∞ and are in L p (R d ) for any p ∈ [1, ∞].
Observe that for u, v ∈ C ∞ c (R d ), by using the duality between s-divergence and s-gradient as in Lemma 2.5 of [16], we have where we have used the above definitions together with the Lebesgue and Fubini theorems.
In this work, we are concerned with the classical fractional Sobolev space H s 0 (Ω) in a bounded domain Ω ⊂ R d with Lipschitz boundary, for 0 < s < 1, defined as where u is extended by 0 in R d \Ω, so that this extension is also in H s (R d ). By the Sobolev-Poincaré inequality (see Theorem 1.7 of [50] and Lemma 3.2), we may consider the space H s 0 (Ω) with the following equivalent norms, owing to (1.4), On the other hand, since the Riesz kernel is an approximation to the identity as 1−s → 0, the s-derivatives approach the classical derivatives as s → 1 in appropriate spaces, as it was observed in Rodrigues-Santos [44], Comi-Stefani [15] and Belido et al [8].
We consider the closed convex set with a given obstacle ψ, such that K s ψ = ∅, and the obstacle problem for F in H −s (Ω). Here, the nonlocal operator L a : H s 0 (Ω) → H −s (Ω) is a generalization of the fractional Laplacian for a measurable, bounded, strictly positive kernel a : R d × R d → [0, ∞) satisfying (2.2), and is defined in the duality sense for u, v ∈ H s 0 (Ω), extended by zero outside Ω: (1.7) Physically, the operator L a corresponds to the class of uniformly irreducible random walks that admit a cycle decomposition with bounded range, bounded length of cycles, and bounded jump rates [22].
To better characterize the properties of the operator L a , in Section 2, we show that the bilinear form is a (not necessarily symmetric) Dirichlet form over H s 0 (Ω) × H s 0 (Ω), whereũ andṽ are the zero extensions of u and v outside Ω respectively. This provides us with many known properties of Dirichlet forms that can be applied to the bilinear form E a , including the truncation property and some regularity results [29] [37]. As a corollary, we obtain that E a is a closed, coercive, strictly T-monotone and regular Dirichlet form in H s 0 (Ω). Furthermore, we use the results of the nonlocal vector calculus developed by Du, Gunzburger, Lehoucq and coworkers in [19], [24], [25] and [31] to show that the fractional operatorL A : corresponds to a nonlocal integral operator L k A with a (not necessarily symmetric) singular kernel k A (x, y) defined by (2.9). We were also motivated by the issues raised by Shieh and Spector in [51] In Section 3, we make use of the comparison property to consider obstacle-type problems involving the bilinear form E a , thereby considering the nonlocal obstacle problem for which we derive results similar to the classical case in H 1 0 (Ω) as in [55], [35] and [42], such as the weak maximum principle and comparison properties. Making use of convergence properties of the fractional derivatives when s 1 to the classical derivatives, as already observed in [44], [8] and [15], we show that the solution of the fractional obstacle problem forL A converges to the solution of the classical case corresponding to s = 1.
By considering the approximation of the obstacle problem by semilinear problems using a bounded penalization, in Section 4, we give a direct proof of the Lewy-Stampacchia inequalities for the obstacletype problems. Here we consider the non-homogeneous data F = f ∈ L 2 # (Ω) not only for the one obstacle problem but also for the two obstacles problem and for the N-membranes problem in the nonlocal framework, extending the results of [49]. In particular, we extend the estimates in energy of the difference between the approximating solutions and the solutions of the one and the two obstacle problems, which may be useful for numerical applications such as in [10] or [41]. More important is the use of the Lewy-Stampacchia inequalities that, upon restricting a to the symmetric case, allows the application of the results of [26] to obtain locally the Hölder regularity of the solutions to those three nonlocal obstacle type problems. Such regularity results are weaker than those obtained from the fractional Laplacian (such as in [47]) or other commonly considered nonlocal kernels [27], since a is in general not a constant multiple of |x − y| −d−2s . In this special case, when L a = (−∆) s , the one obstacle problem can be written for as well as for the corresponding inequalities for the two obstacles and the N-membranes problems. We are then able to use the results of [9] together with the Lewy-Stampacchia inequalities to obtain locally regular solutions in the fractional Sobolev space W 2s,p loc (Ω) for p ≥ 2 # and also in C 1 (Ω) for s > 1/2 and p > d/(2s − 1) when D s · f ∈ L p (R d ).
In Section 5, we further consider some properties related to the fractional s-capacity extending some classical results of Stampacchia [55]. We characterize the order dual of H s 0 (Ω) as the dual space of L 2 Cs (Ω), i.e. the space of quasi-continuous functions with respect to the s-capacity which are in absolute value quasieverywhere dominated by H s 0 (Ω) functions, extending results of [6]. That dual space corresponds then to the elements of H −s (Ω) that are also bounded measures, i.e. (L 2 Cs (Ω)) = H −s (Ω) ∩ M (Ω). Therefore, using the strict T-monotonicity of L a , we state the Lewy-Stampacchia inequalities in this dual space. This section ends with some new remarks on the relations of the E a obstacle problem and the s-capacity.
2 The Anisotropic Non-symmetric Nonlocal Bilinear Form

The nonlocal bilinear form as a coercive Dirichlet form
Our first main result is to show that the nonlocal bilinear form together with its domain, (E a , H s 0 (Ω)), defined as whereũ,ṽ are the zero extensions of u, v ∈ H s 0 (Ω) to Ω c and a : is in fact a regular (not necessarily symmetric) Dirichlet form. This will also imply that the nonlocal bilinear form is also strictly T-monotone and will give us many properties, including Harnack's inequality (see for example [11]), Hölder regularity of solutions of equations involving this bilinear form (see for instance [32], [33], or [34]), and other results in stochastic processes (as given in [29]). We will begin with a remark on the symmetric case.
assuming that the integrands are summable for each fixed ε > 0, where we have set χ ε (x, y) as the characteristic function of the set {|x − y| > ε} for ε > 0.
Proof. We first show the result for u, v ∈ C ∞ c (Ω), and extend it by density to u, v ∈ H s 0 (Ω). The integral term can also be written in the form, using the symmetry of a, Then, by Fubini theorem, Taking the sum of this and the first equation above, we obtain the result 2J =ˆR dˆRd (ṽ(x) −ṽ(y))(ũ(x) −ũ(y))a(x, y)χ ε (x, y) dy dx.
with, applying Fubini's theorem, dy dx which is integrable by Cauchy-Schwarz inequality when u, v ∈ H s (R d ).
It is well-known that H s 0 (Ω) is complete with respect to its norm, as in (1.5), therefore E a is closed. Furthermore, it can be shown that (E a , H s 0 (Ω)) is regular, i.e. H s 0 (Ω) ∩ C c (Ω) is dense in H s 0 (Ω) in the H s 0 (Ω)-norm, and dense in C c (Ω) with uniform norm. The first density result follows from the density of compactly supported smooth functions in the space H s 0 (Ω). The second density result follows since C ∞ c (Ω) ⊂ H s 0 (Ω) and C ∞ c (Ω) is dense in C c (Ω) with uniform norm, by considering the mollification of any C c (Ω) function.
Recall that a coercive closed (not necessarily symmetric) bilinear form E on L 2 (Ω) is a Dirichlet form ( [37] Proposition I.4.7 and equation (4.7) pages [34][35] if and only if the following property holds: For each ε > 0, there exists a real function φ ε (t), t ∈ R, such that A classic example of φ ε is the mollification of a cut-off function (see [29] Example 1.2.1). Specifically, consider a mollifier such as where γ is a positive constant such thatˆ| x|≤1 j(x) dx = 1.
With this φ ε , recalling that a ≥ 0 (as in section II.2(d) of [37]), by Fatou's lemma, we conclude that E a is a Dirichlet form. Therefore, we obtained the following theorem: ) is a closed, regular Dirichlet form, which is also bounded and coercive.
Proof. It remains only to show that the bilinear form E a : by the Cauchy-Schwarz inequality, and coercive From this theorem, we obtain that E a possesses the property of unit contraction. Indeed, by Proposition 4.3 and Theorem 4.4 of [37], we have the following corollary. (a) the unit contraction acts on E a , i.e. v : This result in fact follows from the fact that E a is a Dirichlet form, and holds even if it is not regular.
Since E a is a regular Dirichlet form, it is possible to consider truncations, so we can introduce the positive and negative parts of and we have the Jordan decomposition of v given by It is well-known that such operations are closed in H s 0 (Ω) for 0 < s ≤ 1.

The strict T-monotonicity of E a
Since the assumptions on the kernel a imply, in particular, that it is non-negative, we can easily prove the following important property.
Theorem 2.5. E a is also strictly T-monotone in the following sense: which is strictly greater than 0 if v + = 0.

The fractional bilinear form as a nonlocal bilinear form
In this section, we consider the linear operatorL A defined with the fractional derivatives D s by the continuous fractional bilinear form for a matrix A with bounded and measurable coefficients. The integral in (1.8) is well defined and we are going to show that this bilinear form can be rewritten with a measurable kernel k A : whereũ,ṽ are the zero extensions of u, v ∈ C ∞ c (Ω) to Ω c .
Theorem 2.6. Given a matrix A : R d → R d×d with bounded and measurable coefficients, there exists a kernel k A (x, y) independent of u, v satisfyinĝ Proof. Expanding the fractional bilinear form, we have, setting for simplicity´R d as´, where χ η (x, y) is the characteristic function on the set {|x − y| > η} and similarly defined for χ ε and χ δ . The limit can be exchanged with the integral by the Fubini and Lebesgue theorems because the integrand is Lebesgue integrable. Therefore, Note that we can exchange the limit in η with the integrals because the functions are Lebesgue integrable as |x − y| → 0.

Adding and subtracting
where we make use of the fact that´( y−z)χ δ (y,z) |y−z| d+s+1 · f (z) dy = 0 for all δ > 0 for any finite function f (z). Once again, the limit in η can be interchanged with triple integrals, because the factor (y−z)χ δ (y,z) |y−z| d+s+1 is integrable for δ > 0. Also, the function is a finite function of z, because the integrand has a singularity only at x = z and we have introduced the characteristic function χ ε (x, z). Furthermore, the Lipschitz continuity of u guarantees that the singularity is removable, since we have the factor u(x) − u(z). This is also the reason why we used the first expression for the expansion of D s u, rather than the second one, which will only give us a factor of u(x) when we add and subtract u(x). Therefore, we can take the limit η → 0, so that it is just a function of z. Next, we apply Fubini's theorem, since the integrand is Lebesgue integrable for fixed ε, δ, η > 0. Therefore, Finally, regarding this limit as a double limit, in η and separately in ε and δ, which exists, we can consider the iterated limit in the following form where we may interpret the term in the parentheses as the Cauchy principal value about the singularities z = x and z = y, i.e. as a function in x, y defined for x = y, by (2.9) Remark 2.7. Note that in general, k A is neither translation nor rotation invariant, unlike the case for the fractional Laplacian. In particular, k A may not have the form j(x − y)|x − y| −d−2s . Therefore, the kernel k A may have relevance for non-homogeneous, non-isotropic and nonlocal problems. Even in the case for A being the constant coefficient matrix when k A is translation invariant, it may not be rotation invariant, unless if A is a constant multiple of the identity matrix.
Remark 2.8. Suppose that the matrix A is given by αI for a strictly positive finite constant α and the identity matrix I. Then, by (1.4) and Proposition 2.1,L αI defined by (1.8) can also be defined by a symmetric bilinear form as in (2.8) with a kernel α given by which is, up to a constant, the kernel of the fractional Laplacian and satisfies (2.2). However, we observe that this representation may not be unique, and k αI may not be equal to αc 2 d,s |x − y| −d−2s . Indeed, consider an unbounded nonzero L 2 -integrable function h(x) : R d → R which integrates to 0 over R d and has support outside Ω. Let a(x, y) be a kernel satisfying (2.2) and definẽ which is possible since the kernel is defined over all (R d × R d )\{x = y} and the integrability of h means that Lã is well defined. Since´h = 0 by the construction of h and, for any u, v ∈ C ∞ c (Ω),´ũh = 0 since they have disjoint supports, we have This example gives a class of non-uniqueness for the representation of the kernel. There may be more similar classes, and it will be interesting to know a characterization for the equivalent class of kernels. Furthermore, even if the kernel a satisfies the compatibility condition (2.2), since h may change sign and may be unbounded, our construction of the kernelã may not satisfy the compatibility condition (2.2), nor the weaker compatibility conditions for some a * , a * , M, s > 0, as given in equation (1.11) of [51].
However, (2.2) is not satisfied for the kernel k A for a general matrix A. Indeed, we can construct a numerical counterexample as follows.
Observe that the function has the shape Created in Symbolab is integrable but not absolutely integrable, and is strictly increasing and strictly negative in the interval [0.9, 1.6] with values (computed in Wolfram Alpha) Then, computing (2.9) for Compare with Open Problem 1.10 of [51].
The Theorem 2.6 and the last two remarks were inspired by the Open Problem 1.10 of [51], which asked if, given a symmetric matrix A satisfying (2.13), it is possible to find a kernel k A satisfying (2.12) such that (2.8) holds. Complementing this open problem, we propose the following conjecture (see also Remark 3.1 of [20]): Open Problem. Suppose A : R d → R d×d is a bounded, measurable and strictly elliptic matrix such that for some a * , a * > 0 for all x ∈ R d and all z, z * ∈ R d . Let k A be a corresponding kernel which is continuous outside the diagonal x = y and satisfieŝ Then, there exists an equivalent kernel k A satisfying for some a * , a * > 0 if and only if A is a bounded small perturbationα(x) of the identity matrix (up to a positive constant α), i.e. A = (α +α(x))I for some strictly positive finite constant α >> sup x |α(x)|.

The Nonlocal and the Fractional Obstacle Problem
We start by considering the linear form for F ∈ H −s (Ω) defined by d+2s when s < d 2 , and if d = 1, 2 # = q for any finite q > 1 when s = 1 2 and 2 # = 1 when s > 1 2 . By the Riesz representation theorem, we have Therefore, F ∈ H −s (Ω) may be given by F = −D s · D s φ = −D s · g for some g = (g 1 , . . . , g d ) ∈ [L 2 (R d )] d and, by the Sobolev-Poincaré inequality, it satisfies In order for to lie in the positive cone of H −s (Ω), it is enough for f # to be non-negative almost everywhere in Ω and D s · f ≤ 0 in the distributional sense in R d . We next recall the following useful inequalities.

The nonlocal obstacle problem and its properties
As a consequence of the properties of the bilinear form E a defined by (2.1) in H s 0 (Ω) for 0 < s < 1, we can derive classical properties of the fractional obstacle problems, following most of the approach of Section 4:5 in [42]. there exists a unique u ∈ K s ψ such that Moreover, suppose F,F are given as in the beginning of this section for two different obstacle problems defined in (3.1), then the solution map F → u is Lipschitz continuous, i.e.
Proof. This is just a direct application of the Stampacchia theorem, since the bilinear form E a : H s 0 (Ω) × H s 0 (Ω) → R is bounded and coercive by Theorem 2.3. For the continuous dependence on data, if u,û are the solutions corresponding to different data F and F for the obstacle problem respectively, we set v =û in the inequality for u and v = u in the inequality for u, and take the difference to obtain By the fractional Sobolev inequality and the Cauchy-Schwarz inequality, we have where C S is the Sobolev constant of Lemma 3.1.
Furthermore, we have the following properties of the solution as in the classical case, making use of the strict T-monotonicity of E a and the fractional Poincaré inequality. See for example, Chapter IV of [35] or Section 4:5-6 of [42], where the proofs can be transferred to the nonlocal case almost in the same manner. with data F and convex set K s ψ , andû be the solution with dataF and convex set K ŝ ψ . If ψ ≥ψ and F ≥F , then u ≥û a.e. in Ω.
(ii) (Weak maximum principle). In the obstacle problem (3.1), one has u ≥ 0 a.e. in Ω, if F ≥ 0; and where a supersolution is an element w ∈ H s 0 (Ω) satisfying w ≥ ψ and L a w − F ≥ 0 in the sense of order dual.
(v) The solution u of the obstacle problem (3.1) is the unique function in H s 0 (Ω), such that, (vi) (L ∞ estimates) The following estimate holds: Similarly, as in Theorem 6.1 of Chapter 4 of [42], we can prove the following additional result for the Dirichlet form E λ with λ > 0.
Moreover in this case, when f ≡ 0, the following maximum principle holds a.e. in Ω: Remark 3.6.
If Ω = R d and since the kernel a is defined in the whole R d , the domain of E λ , D(E λ ), is instead given by H s (R d ), and the Dirichlet form E λ is coercive for λ > 0.

Convergence of the fractional s-obstacle problem as s 1
We start with a continuous dependence property of the Riesz derivatives as s varies.  As it could be expected, we have a continuous transition from the fractional obstacle problem to the classical local obstacle problem as s 1 in the following sense. We now consider the obstacle problem Theorem 3.8. Suppose ψ is such that K 1 ψ := {v ∈ H 1 0 (Ω) : v ≥ ψ a.e. in Ω} = ∅. Let u s ∈ K s ψ for 0 < s < 1 be the solution to the fractional obstacle problem, i.e.
where A : R d → R d×d is a bounded, measurable and strictly elliptic matrix satisfying a * |z| 2 ≤ A(x)z · z and A(x)z · z * ≤ a * |z||z * |, (2.13) and F ∈ H −σ (Ω). Then, there exists a unique solution u s ∈ H s 0 (Ω). Furthermore, the sequence (u s ) s converges strongly to u in H σ 0 (Ω) as s 1 for any fixed 0 < σ < 1, where u ∈ K 1 ψ solves uniquely the obstacle problem for s = 1, i.e.ˆΩ Proof. The existence of a solution follows from a direct application of the Stampacchia theorem, since the bilinear form . For uniqueness, if u,û are the solutions corresponding to the same data F for the obstacle problem, we set v =û in the inequality for u and v = u in the inequality forû, and take the difference to obtain Next, we want to show the convergence of the fractional obstacle problem to the classical one. We first prove an a priori estimate for D s u s . For v 1 ∈ K 1 ψ = σ≤s<1 K s ψ , by Cauchy-Schwarz inequality and Sobolev inequality, by taking = a * 2a * 2 and = a * 2 and c 2 σ may be chosen independent of s for 0 < s ≤ 1, as a consequence of (3.4) for the dual norms · H −s (Ω) . Therefore, we have where the constant C = C(σ, a * , a * ) > 0 is independent of s ≥ σ. Also by (3.4), for σ ≤ s < 1, we have for some constant C independent of s, σ ≤ s < 1, and we may take a sequence for some ζ. By compactness, since u s is also uniformly bounded in H σ 0 (Ω), there exists a subsequence and a limit u ∈ L 2 (Ω) such that u s − −− →

But by the a priori estimate on
This means that Dũ ∈ [L 2 (R d )] d , and since Ω has a Lipschitz boundary, Together with the first inequality in (3.4) which implies that u ∈ K σ ψ for any σ < 1, we have u ∈ K 1 ψ . Furthermore, by Lemma 3.7, Finally, it remains to show that u satisfies the obstacle problem for s = 1. For any v ∈ K 1 ψ ⊂ K s ψ , since D s u s are uniformly bounded, we have, up to a subsequence and using the lower-semicontinuity of The conclusion follows by the compactness of the inclusion of H σ 0 (Ω) in H σ 0 (Ω) when σ > σ .
Remark 3.9. The case with A = I corresponds to the obstacle problem for the fractional Laplacian and was first considered by Silvestre in [54]. Indeed, from (1.4), sincê also holds for u, v ∈ H s 0 (Ω), Theorem 3.8 gives the convergence of the solution u s to the nonlocal obstacle problem (3.1), which is equivalent, up to a constant, to towards the solution u of the classical problem

Lewy-Stampacchia Inequalities and Local Regularity
In this section, we take f = f # and f = 0. We give a direct proof of the Lewy-Stampacchia inequalities. This will follow much of the approach of Section 5:3.3 in [42] or Chapter IV of [35]. The Lewy-Stampacchia inequalities will allow us to apply the results of [26][28] to obtain local Hölder regularity of the solutions when a is symmetric, and additional regularity on fractional Sobolev spaces when L a = (−∆) s using [9].

Bounded penalization of the obstacle problem in H s 0 (Ω)
Assume now that the obstacle ψ ∈ H s (R d ), so that we may define L a ψ ∈ H −s (Ω) by (1.7) for any test function v ∈ H s 0 (Ω), and ψ is such that the convex set K s ψ = ∅. Consider the approximation to the obstacle problem, where the penalization is based on any nondecreasing Lipschitz function θ : R → [0, 1] such that θ ∈ C 0,1 (R), θ ≥ 0, θ(+∞) = 1 and θ(t) = 0 for t ≤ 0; Then, for any ε > 0, consider the family of functions θ ε (t) = θ t ε , t ∈ R, which converges as ε → 0 to the multi-valued Heaviside graph. Examples of such sequences of functions include θ(t) = t/(1 + t), θ(t) = (2/π) arctan t, or from any non-decreasing Lipschitz function 0 ≤ θ ≤ 1 such that θ(t) = 1 for consider now the one parameter family of approximating semilinear problems in variational form Arguing as in the proof of Theorem 5:3.1 of [42], we have the following theorem.
(ii) One can also consider the translated penalization, given bȳ for t ∈ R and ε > 0, to approach the solution of the obstacle problem from below using monotonicity. Then we have that the unique solutionū ε of the penalized problem defines a monotone increasing sequence converging to the solution u of the obstacle problem (3.1) weakly in H s 0 (Ω).
(iii) For special choices of the function θ, we can estimate the uniform convergence for the approximations of u by penalization. Suppose that, in addition to the conditions on θ, then the approximating solution u ε of (3.1 ε ) verifies, for each ε > 0, the approximating solutionū ε yields in Ω.
From this theorem, we can derive the Lewy-Stampacchia inequality. In particular, L a u ∈ L 2 # (Ω).
Proof. Choosing ζ = (L a ψ − f ) + in (3.1 ε ), and making use of the property of θ that 0 ≤ 1 − θ ε ≤ 1, then for any ε > 0 and any v ∈ H s 0 (Ω), v ≥ 0, we havê from the variational form, and on the other hand holds a.e. in Ω. Together, these givê Letting ε → 0, this holds for u. Since v is arbitrary, we have the result.
Remark 4.4. The Lewy-Stampacchia inequalities for nonlocal obstacle problems have been first obtained in [49] for a class of symmetric integrodifferential operators L K , with even kernels K, which are also strictly T-monotone and include the fractional Laplacian, and with f and L K ψ ∈ L ∞ (Ω).

Two obstacles problem
We next consider the two obstacles problem with a Dirichlet boundary condition in a bounded domain Ω ⊂ R d with Lipschitz boundary, which consists of finding u ∈ K s ψ,ϕ such that where f ∈ L 2 # (Ω) and K s ψ,ϕ = {v ∈ H s 0 (Ω) : ψ ≤ v ≤ ϕ a.e. in Ω}. (4.4) Assume that ϕ and ψ are measurable and admissible obstacles in Ω such that K s ψ,ϕ = ∅. When ϕ, ψ ∈ H s (R d ), a sufficient condition for these two assumptions to hold is to assume ϕ ≥ ψ a.e. in Ω and ϕ ≥ 0 ≥ ψ a.e. in Ω c .
In addition, if f =f , we have the L ∞ estimate Proof. The existence and uniqueness follows as, in the previous sections, from the monotonicity, coercivity, continuity and boundedness of the operator L a and the Stampacchia theorem. The comparison property follows also as previous by the T-monotonicity of L a . The L ∞ estimate follows as well, as in the classical one obstacle problem.
Corresponding to the two obstacles problem, we also have the Lewy-Stampacchia inequality.
Theorem 4.6. The solution u of the two obstacles problem, for f, L a ϕ, L a ψ ∈ L 2 # (Ω) such that ϕ, ψ ∈ H s (R d ) are compatible and L a ϕ, L a ψ are given by (1.7), satisfies in Ω, (4.5) and therefore L a u ∈ L 2 # (Ω).
Proof. The proof is similar to that of the classical case s = 1, now for two obstacles. Consider the penalized problem given by with θ ε (t) = 1 for t ≥ ε. Then, there is a unique solution u ε ∈ H s 0 (Ω) such that ψ ≤ u ε ≤ ϕ + ε for each ε > 0. Indeed, we obtain the existence and uniqueness of the solution by the Stampacchia theorem as before. To show that ψ ≤ u ε ≤ ϕ + ε, we have Taking v = (ψ − u ε ) + ∈ H s 0 (Ω) and using the strict T-monotonicity of L a , we have because the first term is non-positive, while the factors in the second term are all non-negative. Therefore, Take v = w − u ε in the penalized problem above for arbitrary w ∈ K s ψ,ϕ , then Now, taking w = u ∈ K s ψ,ϕ , we obtain Taking the difference of these two equations, by the linearity of E a , we have Using the ellipticity of a, we have Therefore, u ε → u in H s 0 (Ω).

N-membranes problem
We consider now the N-membranes problem, which consists of: To find u = (u 1 , u 2 , . . . , and f i , . . . , f N ∈ L 2 # (Ω). As in the previous sections, the existence and uniqueness follows easily. Furthermore, the following Lewy-Stampacchia type inequality also holds.
Finally, u N solves the one obstacle problem with an upper obstacle ϕ = u N −1 , and so by the symmetric Lewy-Stampacchia estimates given in Theorem 4.3, we have The proof concludes by simple iteration (see Theorem 5.1 of [43]).

Local regularity of solutions
We make use of the Lewy-Stampacchia inequalities to show local regularity for the three types of nonlocal obstacle problems, but first it is useful to obtain the global boundedness of the solutions.
Let s ∈ (0, 1). Suppose that (a) f, L a ψ ∈ L p (Ω) for some p > d 2s for the one obstacle problem, (b) f ∧ L a ϕ and f ∨ L a ψ are in L p (Ω) for some p > d 2s for the two obstacles problem, or (c) f i ∈ L p (Ω) for i = 1, . . . , N for some p > d 2s for the N-membranes problem. Theorem 4.9. Let u denote the solutions of the one obstacle problem (3.1), or the two obstacles problem (4.3), or u = u i for i = 1, . . . , N of the N-membranes problem (4.6), respectively, under the assumptions (a), (b) or (c) above. Then g = L a u ∈ L p (Ω), with p > d 2s and there exists a constant C, depending only on a * , a * , d, Ω, u H s 0 (Ω) , g L p (Ω) and s, such that Proof. Assume that Φ : R → R is a Lipschitz convex function such that Φ(0) = 0, then if u ∈ H s 0 (Ω), we have, by repeating the proof of Proposition 4 of [36], We can then repeat the proof of Theorem 13 of [36] using the Moser technique to obtain the theorem.
Observe that in general our kernel does not satisfy the usual regularity of the kernel of the fractional Laplacian [45] [47] or other commonly considered fractional kernels [12] [27], since in general it does not satisfy the "symmetry" condition a(x, y) = a(x, −y) unless a is a constant multiple of the kernel of the fractional Laplacian. However, it will still be possible to obtain local Hölder regularity on the solution with the properties of our kernel, if we assume it is symmetric, i.e. if it satisfies a(x, y) = a(y, x).
for dµ x (dz) a measure depending on a as defined in [26] and [34], and the positive constants p 0 ∈ (0, 1) and c depend only on d, s, a * , a * . Theorem 4.11 (Hölder regularity). Let u denote the solutions of the one obstacle problem (3.1), or the two obstacles problem (4.3), or u = u i for i = 1, . . . , N of the N-membranes problem (4.6), respectively, under the assumptions (a), (b) or (c) above. Suppose B ρ ⊂⊂ Ω is a ball of radius ρ and a is symmetric. Then, there exists c ρ ≥ 0 and β ∈ (0, 1), independent of u, such that the following Hölder estimate holds for almost every x, y ∈ B ρ/2 : Proof. Since the Lewy-Stampacchia inequalities in Theorems 4.3, 4.6 and 4.7 hold a.e. in B ρ ⊂ Ω for L a u for the one obstacle, the two obstacles problem and the N-membranes problem respectively, and L a u = g in B ρ , therefore g lies in L p (Ω) for p > d 2s , and we have the result making use of Theorem 4.10 following the classical approach.
Remark 4.12. (i) The local Hölder continuity of the solutions of a two membranes problem was obtained for different operators with translation invariant kernels in [12], as well as the local C 1,γ regularity in the case of the fractional Laplacian as in the case of the regular obstacles of [53].
(iii) Since the L ∞ bound also works in the non-symmetric case, we conjecture that the weak Harnack inequality and the Hölder continuity are also true without the symmetry assumption, but this is an open problem.
In the case where a corresponds to the kernel of the fractional Laplacian, L a = (−∆) s , we can use Corollary 1.15 of [50] and apply local elliptic regularity of weak solutions in fractional Sobolev spaces W r,p associated to Dirichlet fractional Laplacian problems as in Theorem 1.4 of [9]. Theorem 4.13. Let u denote the solutions of the one obstacle problem (3.1) for f ∈ L 2 # (Ω) in the form or the corresponding two obstacles problem (4.3), or u = u i for i = 1, . . . , N of the corresponding Nmembranes problem (4.6), respectively, under the assumptions (a), (b) or (c) above, with 2 # ≤ p < ∞ and 0 < s < 1. Then, (−∆) s u ∈ L p loc (Ω) and u ∈ W 2s,p loc (Ω). In particular, u ∈ C 1 (Ω) if s > 1/2 and p > d/(2s − 1), by Theorem 7.57(c) of [4].
This theorem, which seems new, is an extension to nonlocal obstacle type problems of the well-known W 2,p loc (Ω) regularity of solutions of the classical local obstacle problem corresponding to s = 1.

s-capacity and Lewy-Stampacchia Inequalities in H −s (Ω)
In this section, we extend the results on the Lewy-Stampacchia inequalities obtained in the previous section to data in the dual space H −s (Ω). We first characterize the order dual of H s 0 (Ω), which is related to the theory of the s-capacity. This follows much of the results in the classical obstacle problem [55], [2] and [42].
In [3] and [29] the more general capacities are considered for general bilinear forms. Recently the fractional capacity for the Neumann problem was considered in [56]. In order to extend the results in Theorems 4.3, 4.6 and 4.7 to data in H −s (Ω), we may apply the general results of [38] for the one obstacle problem and [43] for two obstacles.

A characterization of the order dual
Associated with any Dirichlet form, there is a Choquet capacity. We denote by C s the capacity associated to the norm of H s 0 (Ω). For any compact set K ⊂ Ω, it is defined by For an arbitrary open set G ⊂ Ω, A function u ∈ H s 0 (Ω) is said to be quasi-continuous if for every ε > 0, there exists an open set G ⊂ Ω such that C s (G) < ε and u| Ω\G is continuous. A property is said to hold quasi-everywhere (q.e.) if it holds except for a set of capacity zero.
It is well-known (by [3] Proposition 6.1.2 page 156 or [29] Theorem 2.1.3 page 71) that for every u ∈ H s 0 (Ω), there exists a unique (up to a set of capacity 0) quasi-continuous functionū : Ω → R such thatū = u a.e. on Ω. Therefore, we have the following theorem (see also Theorem 3.7 of [56]). For K ⊂ Ω, recall that one says that u 0 on K (or u ≥ 0 on K in the sense of H s 0 (Ω)) if there exists a sequence of Lipschitz functions u k → u in H s 0 (Ω) such that u k ≥ 0 on K. Let K ⊂ Ω be any compact subset. Define the nonempty closed convex set of H s 0 (Ω) by Consider the following variational inequality This variational inequality has clearly a unique solution and consequently we can also extend to the sfractional framework the following theorem which is due to Stampacchia [55] in the case s = 1.
Theorem 5.3 (Radon measure for the bilinear form E a ). For any compact K ⊂ Ω, the unique solution u of (5.1), which is called the (s, a)-capacitary potential of K, is such that Moreover, for the non-negative Radon measure µ, one has and this number is called the (s, a)-capacity of K with respect to E a (·, ·) (or to the operator L a ).
Proof. The proof follows a similar approach to the classical case ([55] Theorem 3.9 or [42] Theorem 8.1). Taking ≤ 0 since the s-grad of a constant is zero. Hence u 1 in Ω. But u ∈ K s K , so u 1 on K. Therefore, the first result u = 1 on K follows.
We observe that when a corresponds to the kernel of the fractional Laplacian, the (s, a)-capacity corresponds to the s-capacity and the s-capacitary potential of a compact set K is the solution of the obstacle problem (5.1) when the bilinear form is the inner product in H s 0 (Ω) and we have a simple comparison of the capacities in the following proposition.
Proposition 5.4. For any compact subset E ⊂ Ω, Proof. Let u be the (s, a)-capacitary potential of E, andū be the s-capacitary potential of E. Sinceū 1 on E, we can choose v =ū ∈ K s E in (5.1) to get by Cauchy-Schwarz inequality and the coercivity of a. Similarly, we can choose v = u ∈ K s E for (5.1) with a being the identity for C s (E) to get, using the coercivity of a, Using this definition of the Radon measure, we recall that two quasi-continuous functions which are equal (or, ≤) µ-a.e. on an open subset of R d are also equal (or, ≤) q.e. on that set (see [29] Lemma 2.1.4).
Recall that a Radon measure µ is said to be of finite energy relatively to H s 0 (Ω) if its restriction to H s 0 (Ω) ∩ C c (Ω) is continuous for the topology of H s 0 (Ω), by means of Such a finite energy measure can in fact be defined for any Dirichlet form E (see [29] Section 2. , by the mapping µ → w µ . Moreover, whenever µ ∈ E + (H s 0 (Ω)), the mapping u ∈ H s 0 (Ω) →ū is continuous from H s 0 (Ω) in L 1 (µ) and whenever u ∈ H s 0 (Ω),´Ωū dµ = w µ , v . Note that in the particular case of the space H s 0 (Ω), the mapping u ∈ H s 0 (Ω) →ū ∈ L 1 (µ) is compact; this follows from the fact that´Ω |ū n | dµ = w µ , |u n | and that if u n 0 in H s 0 (Ω) then |u n | 0 in H s 0 (Ω). Extending these results to L 2 Cs (Ω), we have the following result.
Let (u n ) be a sequence in L 2 Cs (Ω) such that R Cs (u n ) → 0. Then there exists (v n ) ∈ H s 0 (Ω) such that v n ≥ |u n | q.e., and therefore µ-q.e., and v n H s 0 (Ω) → 0. As a result,´Ω |u n | dµ ≤´Ωv n dµ = w µ , v n ≤ w µ H −s (Ω) v n H s 0 (Ω) → 0. Therefore u n → 0 in L 1 (µ). Having these results, we can now identify the dual space of L 2 Cs (Ω) with the order dual of H s 0 (Ω), as given in the following theorem. Proof. According to Proposition 5.2, C c (Ω) is dense in L 2 Cs (Ω) and moreover this injection is continuous; therefore the dual of L 2 Cs (Ω) is a space of measures.
Theorem 5.7. The unique solution u to the obstacle problem (3.1) with compatible obstacle ψ ∈ H s (R d ) and F, L a ψ ∈ H −s ≺ (Ω), satisfies F ≤ L a u ≤ F ∨ L a ψ in H −s ≺ (Ω).

(5.2)
Proof. Since L a is strictly T-monotone, this is a direct consequence of the abstract Lewy-Stampacchia inequality obtained by Mosco in [38] (see also Theorem 2.1 of [42]).
We next consider the generalizations to the two obstacles problem and to the N-membranes problem. Similarly, as a direct consequence of Theorem 4.2 of [43], we may also state the Lewy-Stampacchia inequality for the two obstacles problem. Then, applying the general Lewy-Stampacchia inequalities for the one obstacle and for the two obstacles problem iteratively in the previous theorem as in the proof of Theorem 4.7, we obtain Theorem 5.9. The solution u of the N-membranes problem u ∈ K s N : with K s N given by (4.7) with data F = (F 1 , . . . , F N ) for F i ∈ H −s ≺ (Ω) satisfies in H −s ≺ (Ω). Remark 5.10. In the symmetric case, the Lewy-Stampacchia inequalities also follow from the general results of [30]. The application to Theorem 5.9 for the N-membranes problem is new.

The E a obstacle problem and the s-capacity
As a simple application of s-capacity, we consider the corresponding nonlocal obstacle problem extending some results of [55] and [2] (see also [42]). In this section we start by the following comparison property for the (s, a)-capacity which proof is exactly as in Theorem 3.10 of [55], which states that in the case when the kernel a is symmetric the (s, a)-capacity is an increasing set function. in particular µ does not charge on sets of capacity zero.
Proof. By the maximum principle Proposition 3.4(ii), taking v = u+u − , the solution is non-negative. Hence, the variational inequality (5.4) is equivalent to solving the variational inequality withK s ψ replaced byK s ψ + . Since ψ + ∈ L 2 Cs (Ω), by definition of L 2 Cs (Ω),K s ψ + = ∅ and we can apply the Stampacchia theorem to obtain a unique non-negative solution.
Remark 5.14. Further properties on the s-capacity and the regularity of the solution to the s-obstacle problem are an interesting topic to be developed. For instance, as in the classical local case s = 1 [2], it would be interesting to show that ψ is compatible, i.e.K s ψ = ∅, if and only if ∞ 0 C s ({|ψ + | > t})dt 2 < ∞.