Pathological solutions to elliptic problems in divergence form with continuous coefficients
Pathological solutions to elliptic problems in divergence form with continuous coefficients
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连续系数散度形式椭圆问题的病态解
DOI：
10.1016/j.crma.2009.05.008
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发表时间：
2009
影响因子：
0.8
通讯作者：
Jean Van Schaftingen
中科院分区：
文献类型：
--
作者：
Tianling Jin;V. Maz'ya;Jean Van Schaftingen
我们构造一个函数 u∈ W 1, 1 loc (B (0, 1))，它是分布意义上的 div (A∇ u)= 0 的解，其中 A 是连续的并且 u/ε W 1, p loc (B (0, 1)) 对于 p> 1。我们还给出一个函数 u∈ W 1, 1 loc (B (0, 1)) 使得 u∈ W 1, p loc (B (0, 1)) 对于每个 p<∞，u 满足 div (A∇ u)= 0，A 连续，但 u/ε W 1,∞ loc (B (0, 1))。这回答了 H. Brezis 提出的问题（On a conjection of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338）。引用本文：T. Jin 等人，CR Acad。科学。巴黎，爵士。我 347（2009）。
We construct a function u∈ W 1, 1 loc (B (0, 1)) which is a solution to div (A∇ u)= 0 in the sense of distributions, where A is continuous and u/∈ W 1, p loc (B (0, 1)) for p> 1. We also give a function u∈ W 1, 1 loc (B (0, 1)) such that u∈ W 1, p loc (B (0, 1)) for every p<∞, u satisfies div (A∇ u)= 0 with A continuous but u/∈ W 1,∞ loc (B (0, 1)). This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338). To cite this article: T. Jin et al., CR Acad. Sci. Paris, Ser. I 347 (2009).