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 ${\mathcal{L}}_g^s$, with $0<s<1$. We prove the strict T-monotonicity of ${\mathcal{L}}_g^s$ and we obtain the Lewy-Stampacchia inequalities. We consider the approximation of the solutions through semilinear problems, for which we prove a global $L^{\mathrm{\infty }}$-estimate, and we extend the local Hölder 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.