On the obstacle problem in fractional generalised Orlicz spaces

Abstract

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 $ F\leq\mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $ and $ F\wedge\mathcal{L}_g^s\varphi\leq \mathcal{L}_g^su\leq F\vee\mathcal{L}_g^s\psi $, respectively, for the one obstacle solution $ u\geq\psi $ and for the two obstacles solution $ \psi\leq u\leq\varphi $, with given data $ F $. We consider the approximation of the solutions through semilinear problems, for which we prove a global $ L^\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.

Keywords

• fractional generalised orlicz spaces
• nonlocal nonlinear anisotropic operators
• one and two obstacles problems