On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913602686943232 |
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| author | Campos, Pedro Miguel |
| author_facet | Campos, Pedro Miguel |
| contents | We introduce a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces $Λ^{s,A}_0(Ω)$, where $A$ is a generalized $Φ$-function satisfying the $(\mathrm{Inc})_{p}$ and $(\mathrm{Dec})_{q}$ conditions for $1<p\leq q<\infty$, as an extension of the Lions-Calderón spaces (also known as Bessel potential spaces) $Λ^{s,p}_0(Ω)$ when $0<s<1$ to the generalized Orlicz framework. We obtain some continuous and compact embeddings for these spaces and study the continuous dependence of the Riesz fractional gradient $D^s$ with respect to $s\in[0,1]$ as $s\to σ\in[0,1]$. Finally, we apply these results to study the existence, uniqueness and continuous dependence of a family of partial differential equations depending on the Riesz fractional gradient as $s\toσ\in(0,1]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient Campos, Pedro Miguel Analysis of PDEs We introduce a new family of function spaces, the fractional generalized Sobolev-Orlicz spaces $Λ^{s,A}_0(Ω)$, where $A$ is a generalized $Φ$-function satisfying the $(\mathrm{Inc})_{p}$ and $(\mathrm{Dec})_{q}$ conditions for $1<p\leq q<\infty$, as an extension of the Lions-Calderón spaces (also known as Bessel potential spaces) $Λ^{s,p}_0(Ω)$ when $0<s<1$ to the generalized Orlicz framework. We obtain some continuous and compact embeddings for these spaces and study the continuous dependence of the Riesz fractional gradient $D^s$ with respect to $s\in[0,1]$ as $s\to σ\in[0,1]$. Finally, we apply these results to study the existence, uniqueness and continuous dependence of a family of partial differential equations depending on the Riesz fractional gradient as $s\toσ\in(0,1]$. |
| title | On generalized Sobolev-Orlicz spaces associated to the Riesz fractional gradient |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.06346 |