The obstacle problem for a higher order fractional Laplacian

In this paper, we consider the obstacle problem for the fractional Laplace operator (- Δ) ^s in the Euclidian space Rⁿ in the case where 1 < s< 2 . As first observed in Yang (On higher order extensions for the fractional Laplacian arXiv:1302.4413 , 2013), the problem can be extended to the upper half-space R+n+1 to obtain a thin obstacle problem for the weighted b-biharmonic operator Δb2 , where Δ _bU= y^-b∇ · (y^b∇ U) . Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and Cloc1,1(Rn)∩H1+s(Rn) -regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild’s work in Schild (Ann Scuola Norm Sup Pisa Cl Sci (4) 11(1):87–122, 1984) and Schild (Ann Scuola Norm Sup Pisa Cl Sci (4) 13(4):559–616, 1986) from the case b= 0 to the general case - 1 < b< 1 .