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

本课题旨在研究几种抛物型无障碍边界问题。研究内容包括源项具有迟滞不连续的抛物型方程的适定性问题、自由边界的正则性研究以及上述问题解的定性行为。以及给出抛物型西格里尼问题的精确解与任何近似解之间的实际完全可计算的上界的泛函的推导，而不考虑近似方法。近年来，所提出的研究方向在国际上非常流行。规律性课题属于研究的主流。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。精确解（上界）偏差的估计是自由边界问题解析和数值研究的交叉点。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，大多数经典抛物线技术是不适用的。其原因是缺乏关于自由边界的规则性的任何先验信息。因此，我们必须结合变分法的思想，一些几何观察，缩放和放大技术，热函数分析和各种单调性公式。

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.