论文标题
可变指数Bochner-Lebesgue空间具有对称梯度结构
Variable exponent Bochner-Lebesgue spaces with symmetric gradient structure
论文作者
论文摘要
我们介绍功能空间，用于处理具有可变$ \ log $-Hölder连续指数的非线性抛物线方程，该方程仅包含梯度对称部分的信息。由于无法使用Korn对这些功能空间的不平等现象，因此，适当的平滑方法的构建证明了自己很困难。为此，我们证明了仅涉及对称梯度的有限Lipschitz域的边界附近的点上的庞加莱不平等。使用这种不平等，我们构建具有方便属性的平滑操作员。特别是，这种平滑操作员会导致几个密度结果，因此达到了按时间相对于时间的部分进行整合的广义集成公式。使用此公式和最大单调算子的理论，我们证明了抽象的存在结果。
We introduce function spaces for the treatment of non-linear parabolic equations with variable $\log$-Hölder continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analogue of Korn's inequality for these functions spaces is not available, the construction of an appropriate smoothing method proves itself to be difficult. To this end, we prove a point-wise Poincaré inequality near the boundary of a bounded Lipschitz domain involving only the symmetric gradient. Using this inequality, we construct a smoothing operator with convenient properties. In particular, this smoothing operator leads to several density results, and therefore to a generalized formula of integration by parts with respect to time. Using this formula and the theory of maximal monotone operators, we prove an abstract existence result.
