Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

Abstract

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian (−∆) s . The weight is a power of the distance to the boundary that accounts for the singular boundary behavior of the solution for any 0 < s < 1. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.definitions of the operator (−∆) s , motivated by applications, here we choose the so-called integral one; that is, for a sufficiently smooth function v : R n → R we setOur choice of definition is justified by the fact that, unlike the regional or the spectral ones, the integral fractional Laplacian of order s is the infinitesimal generator of a 2s-stable Lévy process. These processes have been widely employed for modeling market fluctuations, both for risk management and option pricing purposes. It is in this context that, as mentioned above, the fractional obstacle problem arises as a pricing model for American options. More precisely, if u represents the rational price of a perpetual American option, modeling the assets prices by a Lévy process X t and denoting by χ the payoff function, then u solves (1.3). We refer the reader to [16] for an overview of the use of jump processes in financial modeling.Taking into account their applications in finance, it is not surprising that numerical schemes for integrodifferential inequalities have been proposed and analyzed in the literature; we refer the reader to [26] for a survey on these methods. These applications aim to approximate the price of a number of assets; therefore, the consideration of a logarithmic price leads to problems posed in the whole space R n . For the numerical solution, it is usual to perform computations on a sufficiently large tensor-product domain. Among the schemes based on Galerkin discretizations, reference [46] utilizes piecewise linear Lagrangian finite elements, while [29] proposes the use of wavelet bases in space. As for approximations of variational inequalities involving integral operators on arbitrary bounded domains, an a posteriori error analysis is performed in [36].Since the seminal work of Silvestre [45], the fractional obstacle problem started to draw the attention of the mathematical community. Using potential theoretic methods, reference [45] shows that if the obstacle is of class C 1,s , then the solution to the fractional obstacle problem is of class C 1,α for all α ∈ (0, s); optimal C 1,s regularity of solutions was derived assuming convexity of the contact set. The pursuit of the optimal regularity of solutions without a convexity hypothesis, in turn, motivated the celebrated extension by Caffarelli and Silvestre [11] for the fractional Laplacian in R n . Using this extension technique, Caffarelli, Salsa and Silvestre proved, in [10], the optimal regularity of solutions (cf. Proposition 3.4 below). It is important to notice, however, that this is only an interior regularity result. Nothing is said about the boundary behavior o...

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

“…Note that due to the strong boundary singularity of the solution the adaptive refinement is particularly strong near the boundary, as well as near the free boundary. This observation underlines the recent analysis of the obstacle problem in [12]. Figure 7 shows the error in the H s (Ω) norm for the different meshes in terms of the degrees of freedom.…”

“…This behaviour has been exploited in [2] who showed that the solution admits 1+s−ε derivatives in an appropriate weighted Sobolev space. For further discussion of the expected regularity of solutions of variational inequalities, see [13], as well as [11,12] for refined estimates in the case of the nonlocal obstacle problem.…”

“…This in particular applies to the thin obstacle and friction problems, and even for the standard parabolic obstacle problem regularity is only understood under strong hypotheses [20]. See [12,56] for regularity results in the elliptic case and [26] for the challenges of optimal a priori estimates for the classical Signorini problem.…”

“…Variational inequalities for time-dependent nonlocal differential equations have attracted recent interest in a wide variety of applications. Classically, parabolic obstacle problems arise in the pricing of American options with jump processes [57,60]; current advances include their regularity theory [10,56] and the a priori analysis of numerical approximations [12,38]. Mechanical problems naturally involve contact and friction at the boundary with surrounding materials.…”

Finite element discretizations of nonlocal minimal graphs: Convergence

“…We refer to the survey [9] for additional discussion, comparison of methods, and references. Moreover, the fractional obstacle problem for (−∆) s has been studied in [11,13,17], where regularity estimates and convergence rates are derived.…”