发表机构
Technische Universität Darmstadt（达姆施塔特工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明双侧抛物型障碍问题解映射在Lebesgue和Sobolev空间间方向可微，基于单侧结果和新不动点定理，可用于最优控制条件推导。
AI 中文摘要
我们证明了经典双侧抛物型障碍问题的解映射在适当的Lebesgue和Sobolev空间之间作为函数解释时是方向可微的。据我们所知，本文是首次针对具有双侧完全分布的点态不等式约束的抛物型演化变分不等式建立此类可微性。我们的分析依赖于单侧抛物型障碍问题的已知可微性结果，以及关于非扩张参数化不动点方程解算子方向可微性的一个新定理。后者也可能对其他应用具有参考价值。我们的结果可用于，例如，推导由双侧抛物型障碍问题控制的最优控制问题的最优性条件。
英文摘要
We prove that the solution map of the classical bilateral parabolic obstacle problem is directionally differentiable when interpreted as a function between suitable Lebesgue and Sobolev spaces. To the best of our knowledge, this paper is the first to establish this kind of differentiability for a parabolic evolution variational inequality with two-sided fully distributed pointwise inequality constraints. Our analysis relies on known differentiability results for unilateral parabolic obstacle problems and a new theorem on the directional differentiability of solution operators of non-expansive parameterized fixed-point equations. The latter may also be of interest for other applications. Our results can be used, for example, to derive optimality conditions for optimal control problems governed by bilateral parabolic obstacle problems.
