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 in a bounded Lipschitz domain center dot in Rd , such that Ks = {v E H0s.center dot/: v > a.e. in center dot} 0 , given by u E Ks : (Zau; v -u) > (F; v -u) Vv E Ks ; for Fin H -s.center dot/ , the dual space of the fractional Sobolev space H0s.center dot/ , 0 < s < 1. The nonlocal operator Za : H0s.center dot/ H -s.center dot/ is defined with a measurable, bounded, strictly positive singular kernel a.x; y/ : Rd x Rd OE 0; oo/ , by the bilinear form Z Z (Zau; v) = P:V: Rd vQ.x/.uQ.x/ -uQ.y//a.x; y/ dy dx = Ea.u; v/; Rd which is a (not necessarily symmetric) Dirichlet form, where uQ; vQ are the zero extensions of u and v outside center dot respectively. Furthermore, we show that the fractional operator -Ds center dot ADs: H0s.center dot/ H -s.center dot/ defined with the distributional Riesz fractional Dsand with a measurable, bounded matrix A.x/ corresponds to a nonlocal integral operator ZkA with a well-defined integral singular kernel a = kA. The corresponding s-fractional obstacle problem for ZzA is shown to converge as sJ' 1to the obstacle problem in H01.center dot/with the operator -D center dot 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 degrees'''.center dot/ , local Holder regularity of the solutions when a is symmetric, and local regularity in fractional Sobolev spaces W 2s;p loc .center dot/ and in C1.center dot/ when Za = .-center dot/s corresponds to fractional s-Laplacian obstacle type problems ZzA = Zu E Ks .center dot/ : Rd .Dsu -f / center dot Ds.v -u/ dx > 0 Vv E Ks for f E OE L2.Rd/center dot d: These novel results are complemented with the extension of the Lewy-Stampacchia inequalities to the order dual of H0s.center dot/and some remarks on the associated s-capacity and the s-nonlocal obstacle problem for a general Za.