On the Obstacle Problem in Fractional Generalised Orlicz Spaces

We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic g-Laplacian Lgˢ, with $0<s<1$. We prove the strict T-monotonicity of Lgˢ and we obtain the Lewy-Stampacchia inequalities. We consider the approximation of the solutions through semilinear problems, for which we prove a global L^∞-estimate, and we extend the local H\"older regularity to the solutions of the obstacle problems in the case of the fractional $p(x,y)$-Laplacian operator. We make further remarks on a few elementary properties of related capacities in the fractional generalised Orlicz framework, with a special reference to the Hilbertian nonlinear case in fractional Sobolev spaces.