A clustering theorem in fractional Sobolev spaces

Fatma Gamza Düzgün; Antonio Iannizzotto; Vincenzo Vespri

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arXiv · 2305.19965

A clustering theorem in fractional Sobolev spaces

Fatma Gamza Düzgün·Antonio Iannizzotto·Vincenzo Vespri

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Abstract

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.

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Fatma Gamza Düzgün, Antonio Iannizzotto, Vincenzo Vespri. 2023-05-31. A clustering theorem in fractional Sobolev spaces. https://arxiv.org/abs/2305.19965

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