Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand
Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand
复制标题
索博列夫空间中低于被积函数增长指数的下半连续性
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Jan Kristensen
中科院分区：
文献类型：
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作者：
Jan Kristensen
设一个非负的拟凸函数F满足某个p∈]1，∞[的增长条件。对于开的有界集Ω⊂ℝm，证明了如果变分积分在W1，p函数序列上是下半连续的，p函数在W1，q中弱收敛.在证明中，我们利用一个扩张算子来确定边值.这一思想源于Meyers[26]和Maly[22]，这里的主要贡献包含在引理4.1中，其中使用了比[22](和[14])中的算子更有效的扩张算子。这种扩张算子的性质在某种意义上是最好的。
Let there be given a non-negative, quasiconvex function F satisfying the growth condition for some p ∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if then the variational integral is lower semicontinuous on sequences of W1, p functions converging weakly in W1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.