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The extension problem for fractional Sobolev spaces with a partial vanishing trace condition
Archiv der Mathematik ( IF 0.5 ) Pub Date : 2021-03-25 , DOI: 10.1007/s00013-021-01594-0
Sebastian Bechtel

We construct whole-space extensions of functions in a fractional Sobolev space of order \(s\in (0,1)\) and integrability \(p\in (0,\infty )\) on an open set O which vanish in a suitable sense on a portion D of the boundary \({{\,\mathrm{\partial \!}\,}}O\) of O. The set O is supposed to satisfy the so-called interior thickness condition in \({{\,\mathrm{\partial \!}\,}}O {\setminus } D\), which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case \(D=\emptyset \) using a geometric construction.



中文翻译:

局部消失迹条件下分数阶Sobolev空间的扩展问题

我们在一个消失的开放集O上以分数\(s \ in(0,1)\)和可积性\(p \ in(0,\ infty)\)的分数Sobolev空间构造函数的全空间扩展。上的部分的合适的感d边界\({{\,\ mathrm {\局部\!} \,}}ø\)Ó。集合O应该满足\({{\,mathrm {\ partial \!} \,}} O {\ setminus} D \)中的所谓内部厚度条件,该条件比全局内部弱得多厚度条件。该证明通过使用几何构造将情况简化为\(D = \ emptyset \)来工作。

更新日期:2021-03-25
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