Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914161199415296 |
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| author | Pandey, Ankur Karak, Nijjwal |
| author_facet | Pandey, Ankur Karak, Nijjwal |
| contents | In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,Φ(\cdot,\cdot)}(Ω)$ where $Φ(x,t):=t^{p(x)}+ a(x) t^{q(x)}\log^{r(x)}(e+t)$ with $1\leq p(x)\leq q(x).$ We have established subcritical embedding in bounded John domains under some regularity assumptions on exponents $p,$ $q,$ $r$, and $a$. Conversely, we have proved that if the embedding holds in any domain $Ω$ in $\mathbb{R}^n,$ then $Ω$ must satisfy the log-measure density condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions Pandey, Ankur Karak, Nijjwal Functional Analysis In this paper, we study the necessary and sufficient conditions in the domain for Sobolev-type embedding of the space $W^{1,Φ(\cdot,\cdot)}(Ω)$ where $Φ(x,t):=t^{p(x)}+ a(x) t^{q(x)}\log^{r(x)}(e+t)$ with $1\leq p(x)\leq q(x).$ We have established subcritical embedding in bounded John domains under some regularity assumptions on exponents $p,$ $q,$ $r$, and $a$. Conversely, we have proved that if the embedding holds in any domain $Ω$ in $\mathbb{R}^n,$ then $Ω$ must satisfy the log-measure density condition. |
| title | Logarithmic double phase embeddings with variable exponents: Necessary and Sufficient Conditions |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2511.13286 |