About Sobolev spaces on fractals: fractal gradians and Laplacians

Khalili Golmankhaneh A., Jørgensen P. E. T., Serpa C., Welch K.

Aequationes Mathematicae, 2024 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1007/s00010-024-01060-6
  • Journal Name: Aequationes Mathematicae
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, MathSciNet, zbMATH, DIALNET, Civil Engineering Abstracts
  • Keywords: 28A80, 32K15, 35D30, 43A15, 46E20, 46E36, Fractal Laplace operator, Fractal Lipschitz continuity, Fractal Sobolev spaces, Fractal weak derivative
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for Fα-integrable functions, introduces local fractal integrable functions and fractal distribution functionals, defines the dual space of a fractal function space, proves completeness for Fα-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.