Extending Sobolev Functions with Partially Vanishing Traces from Locally (epsilon,delta)-Domains and Applications to Mixed Boundary Problems
Extending Sobolev Functions with Partially Vanishing Traces from Locally (epsilon,delta)-Domains and Applications to Mixed Boundary Problems
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用局部（epsilon，delta）域部分消失迹扩展 Sobolev 函数及其在混合边界问题中的应用
DOI：
10.1016/j.jfa.2014.02.001
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
M. Mitrea
中科院分区：
文献类型：
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作者：
Kevin Brewster;D. Mitrea;I. Mitrea;M. Mitrea
证明了对任意k∈ N，对任意开集Ω <$Rn和Ω <$的任意闭子集D，使得Ω是Ω <$D附近的局部（ε，δ）-域，存在线性有界扩张算子Ek，D映射，对任意p∈[1，∞]，空间WDk，p（Ω）到WDk，p（Rn）.这里，O表示Ω或Rn，空间WDk，p（O）被定义为来自Cc ∞（Rn）的支撑与D不相交的函数（对O的限制）在经典Sobolev空间Wk，p（O）中的完备化.反过来，这个结果被用来发展类W D k，p（Ω）的泛函分析理论（包括内在特征，边界迹和扩张结果，插值定理等），然后用于处理局部（ε，δ）-域上的混合边值问题。最后，我们证明了Ahlfors正则集上迹部分为零的（ε，δ）-域上Besov和Bessel势空间在尺度上的扩张结果，并探讨了这些扩张结果的一些意义.
We prove that given any k∈ N, for each open set Ω⊆ R n and any closed subset D of Ω¯ such that Ω is locally an (ε, δ)-domain near∂ Ω∖ D, there exists a linear and bounded extension operator E k, D mapping, for each p∈[1,∞], the space W D k, p (Ω) into W D k, p (R n). Here, with O denoting either Ω or R n, the space W D k, p (O) is defined as the completion in the classical Sobolev space W k, p (O) of (restrictions to O of) functions from C c∞(R n) whose supports are disjoint from D. In turn, this result is used to develop a functional analytic theory for the class W D k, p (Ω)(including intrinsic characterizations, boundary traces and extensions results, interpolation theorems, among other things) which is then employed in the treatment of mixed boundary value problems formulated in locally (ε, δ)-domains. Finally, we also prove extension results on the scales of Besov and Bessel potential spaces on (ε, δ)-domains with partially vanishing traces on Ahlfors regular sets and explore some of the implications of such extension results.