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On the Obstacle Problem in Fractional Generalised Orlicz Spaces (2405.17014v1)

Published 27 May 2024 in math.AP

Abstract: We consider the one and the two obstacles problems for the nonlocal nonlinear anisotropic $g$-Laplacian $\mathcal{L}_gs$, with $0<s<1$. We prove the strict T-monotonicity of $\mathcal{L}_gs$ 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.

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