On the Obstacle Problem in Fractional Generalised Orlicz Spaces

TL;DR

This paper investigates obstacle problems involving nonlocal, nonlinear, anisotropic operators in fractional Orlicz spaces, establishing key properties, inequalities, and regularity results for solutions.

Contribution

It introduces the strict T-monotonicity of the fractional g-Laplacian, proves Lewy-Stampacchia inequalities, and extends regularity results to solutions in fractional Orlicz spaces.

Findings

Proved strict T-monotonicity of the nonlocal operator

Established Lewy-Stampacchia inequalities for obstacle problems

Extended local Hölder regularity to fractional Orlicz space solutions

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. We consider the approximation of the solutions through semilinear problems, for which we prove a global $L^\infty$-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.