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

该项目旨在研究几个障碍型抛物型自由边界问题。研究程序包括：具有滞后间断源项的抛物型问题的适定性问题，自由边界的正则性和上述问题解的定性行为的研究，以及泛函的推导，这些泛函给出抛物型SNONORINI问题的精确解与任何近似解之间的差异的实际可计算的上界，而不考虑逼近方法。近年来，所提出的研究方向在世界上非常流行。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。与精确解(上界)的偏差估计位于自由边界问题的解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性的。另请注意，对于障碍型问题，经典抛物线技术的大部分都不适用。这样做的原因是缺乏关于自由边界正则性的任何先验信息。因此，我们必须结合变分、一些几何观察、重定标和爆破技术、热函数分析和各种单调性公式的思想。

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.