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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17689779 |
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Table of Contents:
- This paper investigates the quantitative regularity properties of solutions to partial differential equations driven by anisotropic non-local operators. Non-local operators, characterized by their integral nature, play a pivotal role in modeling diverse phenomena across physics, finance, and image processing. The inclusion of anisotropy, reflecting direction-dependent behavior, significantly complicates the analysis of solution smoothness. Our work develops a comprehensive framework to establish precise quantitative estimates for solutions in various function spaces, including Höldér and Sobolev spaces. We employ a combination of advanced harmonic analysis techniques, potential theory, and variational methods, carefully adapted to account for the intrinsic non-locality and directional preferences. The derived estimates provide explicit bounds on solution regularity, shedding light on how the anisotropy parameters influence the smoothness of solutions. These quantitative results are crucial for a deeper understanding of anisotropic anomalous diffusion processes and for guiding the development of numerical schemes with verifiable convergence rates.