Fractional Sobolev Spaces and Variational Problems with Variable-Order Operators on Time Scales
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| Format: | Preprint |
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2026
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| author | Abbas, Hafida Azzouz, Abdelhalim |
| author_facet | Abbas, Hafida Azzouz, Abdelhalim |
| contents | We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W^{α(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ through a variable-order Gagliardo-type seminorm and prove completeness and compact embedding properties under standard boundedness assumptions on the order. We then extend the framework to rectangles $\mathcal R=\mathcal I_1\times \mathcal I_2\subset\mathbb T_1\times\mathbb T_2$, introducing the product spaces $W^{(α,β),p}_{\mathrm{rd}}(\mathcal R)$ and establishing completeness, reflexivity, separability, and compact embeddings. To support boundary-value problems, we propose a boundary decomposition of $\partial\mathcal R$ into four sides and a corresponding trace framework (first on $C_{\mathrm{rd}}(\mathcal R)$ and then by density). We also define variable-order Riemann--Liouville and Caputo fractional operators on time scales and derive an Euler--Lagrange equation for variational functionals depending on these operators. The resulting toolkit provides a functional-analytic basis for fractional dynamic equations on mixed time scales and for anisotropic nonlocal models on product time scales. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_00872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional Sobolev Spaces and Variational Problems with Variable-Order Operators on Time Scales Abbas, Hafida Azzouz, Abdelhalim Dynamical Systems Mathematical Physics Analysis of PDEs Primary 46E35, 34N05, Secondary 26A33, 46E30, 35R11, 49J40 We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W^{α(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ through a variable-order Gagliardo-type seminorm and prove completeness and compact embedding properties under standard boundedness assumptions on the order. We then extend the framework to rectangles $\mathcal R=\mathcal I_1\times \mathcal I_2\subset\mathbb T_1\times\mathbb T_2$, introducing the product spaces $W^{(α,β),p}_{\mathrm{rd}}(\mathcal R)$ and establishing completeness, reflexivity, separability, and compact embeddings. To support boundary-value problems, we propose a boundary decomposition of $\partial\mathcal R$ into four sides and a corresponding trace framework (first on $C_{\mathrm{rd}}(\mathcal R)$ and then by density). We also define variable-order Riemann--Liouville and Caputo fractional operators on time scales and derive an Euler--Lagrange equation for variational functionals depending on these operators. The resulting toolkit provides a functional-analytic basis for fractional dynamic equations on mixed time scales and for anisotropic nonlocal models on product time scales. |
| title | Fractional Sobolev Spaces and Variational Problems with Variable-Order Operators on Time Scales |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs Primary 46E35, 34N05, Secondary 26A33, 46E30, 35R11, 49J40 |
| url | https://arxiv.org/abs/2603.00872 |