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 ψ
in a bounded Lipschitz domain Ω in R^d, such that K^s_Ψ = {ν ∈ H^s₀(Ω):ν ≥ ψ a.e. in Ω ≠ Ø, given by u ∈ K^s_Ψ : ⟨L_au, v - u⟩ ≥  ⟨F, v - u⟩ ∀ν ∈ K^s_Ψ, for F in H^-s (Ω), the dual space of the fractional sobolev space H^s₀ (Ω),0 < s < 1. The nonlocal operator L_{a :} H^s₀ (Ω) → H^-s (Ω) is defined with a measurable, bounded, strictly positive singular kernel a(x, y) : R^d x R^d → [0, ∞), by the bilinear form ⟨L_au, v⟩ = P.V. ∫_R^d ∫_R^d v(x)(ũ(x) - ũ(y)a(x, y) d y d x = E_a (u, v), which is a (not necessarily symmetric) Dirichlet form, where ũ, v are the zero extensions of u and v outside Ω respectively. Furthermore, we show that the fractional operator L_a = -D^s·AD^s : H^s₀ (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_kA with a well-defined integral singular kernel a = k_A. The corresponding s-fractional obstacle problem for L_A is shown to converge as s ↗ 1 to the obstacle problem in H¹₀(Ω) with the operator -D·AD given with the classical gradient D. We mainly consider obstacle type problems involving the bilinear form E_a 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 C¹(Ω) when L_a = (-∆)^s corresponds to fractional s-Laplacian obstacle type problems u ∈ K^s_Ψ : ∫_Rd(D_su - f)·Ds(v - u) dx ≥ 0 ∀ν ∈ K^s_Ψ, for f ∈ [L² (Rd)]d. These novel results are complemented with the extension of the Lewy–Stampacchia inequalities to the order dual of H^s₀(Ω) and some remarks on the associated s-capacity and the s-nonlocal obstacle problem for a general L_a. © 2023 Sociedade Portuguesa de Matemática