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arXiv · 2405.17014

On the Obstacle Problem in Fractional Generalised Orlicz Spaces

Catharine W. K. Lo·José Francisco Rodrigues
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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.

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Catharine W. K. Lo, José Francisco Rodrigues. 2024-05-27. On the Obstacle Problem in Fractional Generalised Orlicz Spaces. https://doi.org/10.3934/mine.2024026

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