Unilateral problems for quasilinear operators with fractional Riesz gradients

1 Introduction

In 2015, Shieh and Spector considered a new type of fractional differential operator in [64], combining results of harmonic analysis, especially pseudodifferential operators, with calculus of variations. The functionals in [64] defining those operators are given with the distributional Riesz fractional gradient, denoted by D s .

When applied to smooth functions, D s can be computed as the gradient D of the convolution of the function with the Riesz kernel. More precisely, the operator D s can be expressed as

where I 1 − s ( x ) = μ d , s d + s − 1 ∣ x ∣ 1 − s − d denotes the Riesz kernel in R d with s ∈ ( 0 , 1 ) and μ d , s is given by (2.2). These fractional derivatives have some advantages in some types of fractional differential equations. First, they can be extended to more general classes of function spaces of Sobolev type, and they approach, in a natural way, the classical derivatives due to the fact that the Riesz kernel I 1 − s is an approximation of the identity as s ↗ 1 . Second, being vector-valued like the classical gradient, in contrast with the scalar fractional Laplacian ( − Δ ) s , they are suitable for inhomogeneous and anisotropic problems, since duality methods apply to the Riesz fractional s -gradient. Moreover, it can also provide an 11th characterization of the fractional Laplacian in R d with respect to [41]. Indeed, denoting by Φ ↦ div s Φ = D s ⋅ Φ the s -divergence, we have in the distributional sense

Subsequently, Ponce made a brief introduction to the fractional gradient and divergence operators for smooth functions with compact support, [53, pp. 246–251], and Šilhavý published in [70] a fractional vector calculus developing ideas presented in a seminar [69]. In the introduction of [65], the equivalence of the definition of the distributional Riesz fractional gradient (1.1), with definition (2.6) of s -fractional gradient introduced by Šilhavý in [69,70], which corresponds to

with N − s ( x ) = − D I 1 − s ( x ) = μ d , s x ∣ x ∣ − 1 − s − d being the vector-valued kernel of Horváth [34]. Actually, this definition can be seen as a special case of a definition used by Sobolev and Nikol’skiĭ in [63] in a more general framework, with generalized homogeneous functions of degree − d − s (a class of functions containing the Horváth kernel as a special case), under the name of Liouville derivatives of order s. The proof of that equivalence was proved in [53] and in the work of Comi and Stefani [24], respectively, for smooth and Lipschitz functions with compact support.

Šilhavý’s treatment was different from the one of Shieh and Spector, as he was interested in showing that these nonlocal operators satisfy the “natural qualitative requirements on the fractional operators.” Notably, Šilhavý has shown that D s is, up to a multiplicative constant, the only operator applying scalar functions into vector-valued functions that satisfies: ( i ) rotational and translational invariance, important for the operators to be independent of the chosen basis; ( i i ) a mild continuity hypothesis, D s is sequential continuous for test functions in a suitable topology, and ( i i i ) s-homogeneity under isotropic scaling, which characterizes the fractional nature of the operator. In addition, using the weaker form of the definitions of the fractional gradient and divergence, he derived explicit formulas for these operators, in particular, showing that ( − Δ ) ( s + r ) ⁄ 2 u = − D s ⋅ D r u and computing directly the fractional gradient of distributions like the Dirac delta or the Heaviside function in R , [70, Examples 6.2 and 6.3].

With this notion of fractional gradient, for 0 < s < 1 and 1 < p < ∞ , in [64], the following Sobolev-type space was also introduced:

i.e., the completion of the space of test functions for the norm

These spaces, that we call here Lions-Calderón spaces, were not new, as they were shown, for 1 < p < ∞ , also in [64], to be equivalent to the so-called Bessel potential spaces in [1,2,67] or generalized (inhomogeneous) Sobolev spaces in [33]. With this new approach, they are well suited to the study of some fractional partial differential equations, including nonhomogeneous and anisotropic variants of the fractional Laplacian. Therefore, they have been well studied in the literature, as they have a long and rich history. Indeed, the oldest reference we could find for these spaces is due to Aronzajn and Smith, who in [7,8] defined them in the Hilbertian case p = 2 . As is well-known, this case coincides with the Sobolev-Gagliardo space W s , 2 ( R d ) = H s ( R d ) . Later, those spaces were independently generalized to the non-Hilbertian case 1 < p < ∞ , by Lions [42], using complex interpolation between L p ( R d ) and W 1 , p ( R d ) , and by Calderón [21], using Bessel potentials. The initial definition in terms of Bessel potentials G s , with their different notations H s , p ( R d ) = L s p ( R d ) = G s ( L p ( R d ) ) = Λ s , p ( R d ) , is not sufficient to justify their terminology as “Bessel potential spaces.” In fact, both the Besov and the Triebel-Lizorkin spaces can be also obtained by the image of Bessel potential, respectively, B q s , p ( R d ) = G s ( B q 0 , p ( R d ) ) and F q s , p ( R d ) = G s ( F q 0 , p ( R d ) ) , and we have B p s , p ( R d ) = W s , p ( R d ) and F 2 s , p ( R d ) = Λ s , p ( R d ) , see [71] for instance.

Conversely, as in [24] under the name of distributional fractional Sobolev spaces for 1 ≤ p ≤ ∞ , the Lions-Calderón spaces were defined directly in terms of the s -fractional gradient, i.e.

which equivalence with the definition (1.4) was proved for 1 < p < ∞ in [10] and independently in [39], while for p = 1 , it has been earlier proven in [24]. Actually, this definition was also suggested for general spaces of functions with Liouville derivatives by Sobolev and Nikol’skiĭ in the survey [63, p. 150]. The alternative names “generalized Sobolev spaces” or “distributional fractional Sobolev spaces,” although possibly more appropriate to this general class of functional spaces of the form { u ∈ L p : D u ∈ L p } , with D being some broadened notion of derivatives, collides with the Sobolev-Gagliardo spaces W s , p ( R d ) , see [1,44], which are also well-known as the fractional Sobolev spaces and do not coincide with the spaces Λ s , p ( R d ) , except when p = 2 or when s is an integer.

Another important property of the fractional gradient D s is the continuous dependence with respect to the parameter s . In fact, it was noted, in [69] and in [53] without proofs, in [56] using Kurokawa’s results [40] on the approximating the identity by Riesz kernels, and in [12] using Fourier analysis and an interpolation inequality, that the following property for sufficiently regular functions u :

These results were obtained concurrently and independently in [25] and later on generalized in [10] using direct estimates and properties of the singular integrals. In these works, the authors were able to show that

and also that

Here, R is the Riesz transform (see [32,67]), which motivates the notation D 0 = − R .

Concerning applications to partial differential equations, these spaces had also appeared in [35,44], to study the regularity theory for linear elliptic partial differential equations, and in [46], to study a variety of questions about solutions to dispersive partial differential equations. However, the new characterization of these spaces using the distributional Riesz fractional gradient allows the study of fractional partial differential equations, as proposed in [64], as well as extensions to unilateral problems, as in [5,6,47,56] and other problems of the calculus of variations, as in [65], [11,12] and [39], in particular with applications in Peridynamics [59,61] to model fractional hyperelasticity, see also [58]. Conversely, this functional framework for the Riesz fractional gradient is also well suited for certain classes of evolution problems, such as fractional Stefan-type problems [48] and hyperbolic obstacle-type problems [23].

Paraphrasing Jean Mawhin in his Foreword of [49], it is therefore a natural question to see which results “survive” when the gradient is replaced by the fractional gradient, which is also fruitful because the extension of classical results to new situations also sheds light on a better and deeper understanding of the classical results. In this work, we are interested in more general types of stationary problems, with or without constraints, in (fractional) divergence form, including the variants of the ( s , p ) -Laplacian-type:

This will be done in the general framework of monotone and pseudomonotone operators involving fractional Riesz gradients, in bounded and in unbounded domains Ω ⊂ R d , in classes of convex constrained Dirichlet-type problems. In particular, this allows us to consider the existence of solutions to problems of the type (1.7), their extensions with obstacle or s -gradient constraints and to study their dependence with respect to the parameter s , including the limit cases s = 1 and s = 0 .

In Section 2, we start with a summary of the natural functional framework of the Riesz fractional gradient D s , with the precise definition of the Lions-Calderón spaces Λ s , p ( R d ) and Λ 0 s , p ( Ω ) , with special emphasis on the differences between the properties of the latter spaces when Ω is bounded and when it is not. In particular, we recall the equivalent definitions of the fractional gradient and their important duality property in the Lions-Calderón spaces, the characterization of their dual spaces Λ − s , p ′ ( R d ) , p ′ = p ⁄ ( p − 1 ) , by showing the representation F = f 0 − D s ⋅ f , for all F ∈ Λ − s , p ′ ( R d ) , the contiguity relation between Λ s , p ( R d ) and W s , p ( R d ) and the monotone inclusions of the spaces Λ s , p ( R d ) and Λ 0 s , p ( Ω ) , with respect to the inverse monotone variation of the parameter s ∈ [ 0 , 1 ] , for 1 < p < ∞ . We also recall the fractional Gagliardo-Nirenberg inequalities and the continuity of D s with respect to s ∈ [ 0 , 1 ] in those spaces. We note that these results also hold for Λ 0 s , p ( Ω ) , since it may be considered a subspace of Λ s , p ( R d ) . Some properties of these spaces are then also complemented in the case of bounded Ω , like in the classical case, as the fractional Poincaré inequality, emphasizing the dependence on the fractional parameter, and the fractional Rellich-Kondrachov compactness embedding. In particular, we prove the new explicit estimate ‖ D t u ‖ L p ( R d ; R d ) ≤ C t 1 + 1 p ‖ D s u ‖ L p ( R d ; R d ) of the embedding Λ 0 s , p ( Ω ) ⊂ Λ 0 t , p ( Ω ) , 1 < p < ∞ , depending on the fractional parameter t but not on s , 0 < t < s ≤ 1 , and we show the compact embedding Λ 0 s , p ( Ω ) ⋐ Λ 0 t , q ( Ω ) for all 1 < p ≤ q < ∞ , 0 ≤ t < s ≤ 1 , satisfying 1 p − s d < 1 q − t d . We conclude this functional section with a relevant property on the density of the non-negative smooth functions with compact support, in a bounded open set Ω , in the non-negative cone of Λ 0 s , p ( Ω ) .

In the functional framework of the Lions-Calderón spaces, we study in Section 3 the existence of solutions u = u s ∈ K s ⊂ Λ 0 s , p ( Ω ) to fractional variational inequalities of the type

for all v ∈ K s . Here, K s denotes a non-empty closed convex subset of Λ 0 s , p ( Ω ) for s ∈ [ 0 , 1 ] and p ∈ ( 1 , ∞ ) . Some examples of these convex sets that are used throughout the text are the following:

In this new framework, we develop the theory of monotone operators, initiated in [37,50], and pseudomonotone operators, introduced by Brézis [18] and extended by Browder [19]. These methods, combined with the functional properties of the fractional Riesz gradients, allow us to obtain a variety of interesting existence results for different types of problems, with Ω bounded or unbounded, in this novel framework. In the monotone case, it is possible to let s = 0 , which corresponds to a new class of nonlinear problems involving an important Calderón-Zygmund operator, namely the Riesz transform. When Ω is bounded and s > 0 , we apply the method of [18] for pseudomonotone operators to obtain the novel existence results, while when Ω is unbounded and s > 0 we extend the method of [19] to the fractional framework. When Ω is unbounded but s ≥ 0 we restrict ourselves to monotone operators and make use of more classical arguments. We observe that, as in the classical pseudomonotone case s = 1 , when Ω is unbounded the type of functions a and b for which we can obtain existence results is more restrictive.

In particular, we study specially fractional p -Laplacian-type operators, for which the typical example is the variational ( s , p ) -Laplacian, or the s -fractional p -Laplacian

that constitutes a continuous one parameter family of nonlinear operators as s varies through Λ 0 s , p ( Ω ) , from 0 in L 0 0 ( Ω ) till 1 in W 0 1 , p ( Ω ) . In addition, we prove that the solution operator

for the variational inequality (1.8) associated with fractional p -Laplacian-type operators, for 0 ≤ s ≤ 1 , is 1 p − 1 -Hölder continuous for p ≥ 2 , locally Lipschitz continuous for 1 < p < 2 and, for Ω bounded, also compact in Λ 0 t , q ( Ω ) for 0 < t < s and 1 < p , q < ∞ satisfying 1 p − s d < 1 q − t d . These results are new for 0 ≤ s ≤ 1 with p ≠ 2 and K ⊊ Λ 0 s , p ( Ω ) , extending those of [62] for the classical Dirichlet problem corresponding to s = 1 .

The following section in this article, Section 4, contains the main results of this work and concerns the continuous dependence of the solutions to (1.8) with respect to the parameter s , which implies the variation of the convex sets K s and the respective functional spaces Λ 0 s , p ( Ω ) . Introducing a generalized Mosco convergence (1.15) for the convex sets, we show that the solutions to those problems converge, up to subsequences, to a solution of the limit problems as s → σ ∈ [ s * , 1 ] , following the four settings of Section 3, respectively, the case of pseudomonotone operators with 0 < s * < s ≤ 1 , with Ω bounded, where we prove

for pseudomonotone operators with 0 < s * < s ≤ 1 , with Ω unbounded, where the result is weaker

and similarly for monotone operators with 0 ≤ s * ≤ σ ≤ 1 and with Ω unbounded, where σ = 0 is admissible with Λ 0 0 , p ( Ω ) = L 0 p ( Ω ) and σ = 1 corresponds to Λ 0 1 , p ( Ω ) = W 0 1 , p ( Ω ) . In the case of p -Laplacian-type coercive operators we have, in addition to the convergence (1.12) for the whole sequence s → σ , as the solutions are unique, also the stronger convergences for 0 ≤ σ ≤ 1

and, in the case σ ≥ s * > 0 with bounded Ω , the first convergence in (1.11) also holds.

The proofs of these stability results are based in a refinement of the classical monotonicity and compactness methods used to study the existence problems and a generalization of the Mosco convergence [51] in a fixed Banach space V :

Although this notion of convergence is a powerful tool to study stability problems associated with variational inequalities with respect to the variation of the convex sets, as one can see, for example, in [4,16,51,52,54], it cannot be used directly in our framework, since the spaces Λ 0 s , p ( Ω ) change with s . The particular case s ↗ σ is simpler, since the convex sets K s are non-increasing and K σ ⊂ K s , and it has been considered in [22,47] for obstacle problems with σ = 1 , by using Mosco’s arguments directly in the space Λ 0 s * , p ( Ω ) where 0 ≤ s * < s < σ ≤ 1 .

The general case requires a generalization of Mosco’s notion of convergence, which in our situation consists of working with the convergences of the pair ( u , D s u ) in the space L p ( Ω ) = L p ( Ω ) × L p ( R d ; R d ) , instead of working directly with the functions u in Λ 0 s , p ( Ω ) . This allows the ambient space L p ( Ω ) to be independent of s without losing information about the regularity of u and to introduce, for converging sequences [ 0 , 1 ] ∋ s → σ ∈ [ 0 , 1 ] , the convergence of convex sets in the generalized sense of Mosco

With this approach, we are able to obtain the new results, with different combinations of weak and strong convergences of the solutions u s and their fractional gradients D s u , given in (1.11), (1.12), and (1.13). We emphasize that the convergences s → σ may be arbitrary, not requiring monotonicity even in the limit cases σ = 1 and σ = 0 .

To complement the existence and stability results obtained in Sections 3 and 4, respectively, we provide some new examples and new applications in Section 5. We start in Section 5.1 with specific examples of sets that converge in the generalized sense of Mosco that was introduced in Section 4. We show that Λ 0 s , p ( Ω ) ⟶ s − M Λ 0 σ , p ( Ω ) , which implies, for example, that, under suitable assumptions on the nonlinear coefficients, the solutions u s ∈ Λ 0 s , p ( Ω ) to (1.7), as s ↗ 1 , converge to solutions u ∈ W 0 1 , p ( Ω ) of the classical quasilinear Dirichlet problem:

In order to study the stability, as s → σ ∈ [ 0 , 1 ] , of the solutions to obstacle problems we give examples of K ≥ ψ s s ⟶ s − M K ≥ ψ σ σ , and also one example of K g s s ⟶ s − M K g σ σ , with constraint on the fractional gradient D s u instead of u , which we restrict to g s = I σ − s * g σ and s ↗ σ ∈ ( 0 , 1 ] .

In Section 5.2, we provide applications to quasi-variational problems, i.e., problems where the convex sets K = K ( u ) are allowed to depend on the solution u of the problem, a class of problems considered, for instance, in the classical framework by [9,13,14,31,36,52]. Fractional problems of this type were, as far as we know, first studied in the Hilbertian framework by Antil and Rautenberg [3] using methods that are dependent on a comparison principle. Since it is not known if the fractional operators studied in our work satisfy any type of comparison principles, we use fixed-point arguments together with variational methods, generalizing some results that were obtained in [4,47,56] to the non-Hilbertian framework. We give two examples with strictly monotone coercive operators, one of obstacle type, where the obstacle is given by ψ = Ψ ( u ) , and another with a fractional gradient constraint g = G ( u ) , for suitable compact operators Ψ and G , respectively. We conclude this subsection with two examples of continuous dependence with respect to s : (i) by showing the convergence (1.11) for s → σ ∈ [ s * , 1 ] in the case of subsequences of solutions u s ∈ K ≥ Ψ ( u s ) s to the quasi-variational implicit obstacle problem towards a solution of the limit obstacle problem u σ ∈ K ≥ Ψ ( u σ ) s , and (ii) applying a result of [5] proving the localization of solutions of fractional quasi-variational inequalities with s -gradient constraints in the Hilbertian framework p = 2 with a linear operator when s ↗ 1 .

2 Lions-Calderón spaces