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About Sobolev spaces on fractals: fractal gradians and Laplacians

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Abstract

The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for \(F^{\alpha }\)-integrable functions, introduces local fractal integrable functions and fractal distribution functionals, defines the dual space of a fractal function space, proves completeness for \(F^{\alpha }\)-differentiable function spaces, defines Fractal Sobolev spaces, and introduces fractal gradian and fractal Laplace operators on fractal Hilbert spaces. It also presents criteria for the existence of unique solutions to fractal differential equations.

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Acknowledgements

Cristina Serpa acknowledges partial support from the Portuguese National Funding from FCT-Fundação para a Ciência e a Tecnologia under the project: UIDB/04561/2020.

Author information

Authors and Affiliations

  1. Department of Physics, Urmia Branch, Islamic Azad University, Urmia, West Azerbaijan, 63896, Iran

    Alireza Khalili Golmankhaneh

  2. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Palle E. T. Jørgensen

  3. Instituto Superior de Engenharia de Lisboa (ISEL), Instituto Politécnico de Lisboa, Lisbon, Portugal

    Cristina Serpa

  4. CMAFcIO - Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Universidade de Lisboa, Lisbon, Portugal

    Cristina Serpa

  5. Faculty at California, Institute of Integral Studies, San Francisco, CA, USA

    Kerri Welch

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  1. Alireza Khalili Golmankhaneh
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  2. Palle E. T. Jørgensen
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  3. Cristina Serpa
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Contributions

A.K. Golmankhaneh: Investigation, Formal analysis, Methodology, Writing P. E. T. Jørgensen: Investigation, Writing—Review & Editing Cristina Serpa: Investigation, Writing—Review & Editing Kerri Welch: Writing—Review & Editing.

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Correspondence to Alireza Khalili Golmankhaneh.

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Dedicated to the memory of Robert S. Strichartz.

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Khalili Golmankhaneh, A., Jørgensen, P.E.T., Serpa, C. et al. About Sobolev spaces on fractals: fractal gradians and Laplacians. Aequat. Math. (2024). https://doi.org/10.1007/s00010-024-01060-6

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