On the obstacle problem in fractional generalised Orlicz spaces - Catharine W. K. Lo; José Francisco Rodrigues (Mathematics in Engineering, 2024)
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≤\mathcal{L}_g^su≤ F\vee\mathcal{L}_g^sψ $ and $ F\wedge\mathcal{L}_g^sφ≤ \mathcal{L}_g^su≤ F\vee\mathcal{L}_g^sψ $, respectively, for the one obstacle solution $ u≥ψ $ and for the two obstacles solution $ ψ≤ u≤φ $, with given data $ F $. We consider the approximation of the solutions through semilinear problems, for which we prove a global $ L^∞ $-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.