Parabolic Obstacle-Type Problems: Regularity, Existence, and Deviation

抛物线障碍类型问题：规律性、存在性和偏差

基本信息

批准号：
407265145
负责人：
Privatdozentin Dr. Darya Apushkinskaya
金额：
--
依托单位：
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/407265145?language=en
关键词：
Parabolic Obstacle Type Problems Regularity

项目摘要

The proposed project is aimed at studying several parabolic free boundary problems of obstacle-type. The research program includes issues concerning well-posedness of parabolic problems involving equations with a hysteretic discontinuity in the source term, the study of regularity properties of the free boundary and the qualitative behavior of solutions for the above-mentioned problems, and the derivation of functionals that give realistic fully computable upper bounds of the difference between the exact solutions of the parabolic Signorini problems and any approximative solution regardless of the approximation method.In recent years the proposed research directions become very popular in the world. The regularity topics belong to the mainstream of investigations. Existence and nonexistence results are also not evident, especially for problems with non-variational nature. The estimates of deviation from the exact solution (upper bounds) lie on the crossroad of analytical and numerical studies of free boundary problems.All the suggested problems are new and original. Notice also that for obstacle-type problems the most part of the classical parabolic techniques is not applicable. The reason for this is an absence of any a priori information about the regularity of the free boundary. So, we have to combine ideas from the calculus of variations, some geometrical observations, rescaling and blow-up techniques, analysis of caloric functions and various monotonicity formulas.

该项目旨在研究几个障碍物类型的抛物线自由边界问题。研究计划包括涉及源项中具有滞后不连续性的方程的抛物型问题的适定性问题、自由边界的正则性研究和上述问题的解的定性行为以及泛函的推导给出了抛物型 Signorini 问题的精确解与任何近似解之间的差异的现实的完全可计算的上限，无论近似方法如何。近年来，所提出的研究方向在世界上变得非常流行。规律性话题属于调查的主流。存在和不存在的结果也不明显，特别是对于非变分性质的问题。对精确解（上限）的偏差的估计位于自由边界问题的分析和数值研究的十字路口。所有提出的问题都是新的和原创的。还要注意，对于障碍物类型的问题，大部分经典抛物线技术都不适用。其原因是缺乏有关自由边界规则性的任何先验信息。因此，我们必须结合变分法、一些几何观察、重新缩放和爆炸技术、热量函数分析和各种单调性公式的思想。