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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.19965 |
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| _version_ | 1866915318601875456 |
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| author | Düzgün, Fatma Gamza Iannizzotto, Antonio Vespri, Vincenzo |
| author_facet | Düzgün, Fatma Gamza Iannizzotto, Antonio Vespri, Vincenzo |
| contents | We prove a general clustering result for the fractional Sobolev space $W^{s,p}$: whenever the positivity set of a function $u$ in a square has measure bounded from below by a multiple of the cube's volume, and the $W^{s,p}$-seminorm of $u$ is bounded from above by a convenient power of the cube's side, then $u$ is positive in a universally reduced cube. Our result aims at applications in regularity theory for fractional elliptic and parabolic equations. Also, by means of suitable interpolation inequalities, we show that clustering results in $W^{1,p}$ and $BV$, respectively, can be deduced as special cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19965 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A clustering theorem in fractional Sobolev spaces Düzgün, Fatma Gamza Iannizzotto, Antonio Vespri, Vincenzo Analysis of PDEs 35R11, 46E35, 35B65 We prove a general clustering result for the fractional Sobolev space $W^{s,p}$: whenever the positivity set of a function $u$ in a square has measure bounded from below by a multiple of the cube's volume, and the $W^{s,p}$-seminorm of $u$ is bounded from above by a convenient power of the cube's side, then $u$ is positive in a universally reduced cube. Our result aims at applications in regularity theory for fractional elliptic and parabolic equations. Also, by means of suitable interpolation inequalities, we show that clustering results in $W^{1,p}$ and $BV$, respectively, can be deduced as special cases. |
| title | A clustering theorem in fractional Sobolev spaces |
| topic | Analysis of PDEs 35R11, 46E35, 35B65 |
| url | https://arxiv.org/abs/2305.19965 |