关于分形上的索波列夫空间：分形梯度和拉普拉奇

Pub Date : 2024-04-16 DOI:10.1007/s00010-024-01060-6

Alireza Khalili Golmankhaneh, Palle E. T. Jørgensen, Cristina Serpa, Kerri Welch

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引用次数: 0

摘要

论文涵盖了分形曲线上分形微积分的基础，定义了不同的函数类，建立了 $F^{\alpha }$-integrable 函数的向量空间，介绍了局部分形可积分函数和分形分布函数、定义了分形函数空间的对偶空间，证明了 $F^{alpha }$- 可微函数空间的完备性，定义了分形索波列夫空间，并介绍了分形希尔伯特空间上的分形梯度算子和分形拉普拉斯算子。它还提出了分形微分方程唯一解存在的标准。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

$About Sobolev spaces on fractals: fractal gradians and Laplacians$

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About Sobolev spaces on fractals: fractal gradians and Laplacians

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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