Parabolic Obstacle-Type Problems: Regularity, Existence, and Deviation
Final Report Abstract
The project is focused on parabolic free boundary problems of the obstacle-type. We concentrate our efforts on the study of deviation from exact solutions for elliptic and parabolic problems including the parabolic Signorini problem as well as on the study of regularity and existence issues for parabolic problems involving equations with a hysteretic discontinuity in thee source term. The results of this project regarding the deviation estimates are quite complete: the fully computable functional error majorants were derived for solutions of the elliptic obstacle problem for the biharmonic operator, for classical parabolic obstacle problem, and for the parabolic Signorini problem. Verification of the functional majorants is almost complete for the elliptic thin obstacle problem. For the one-space dimensional hysteresis type parabolic problem we achieved partial results. Namely, we prove the existence and uniqueness theorem under weaker assumptions. We also improve the regularity result for the free boundary in the one-space dimensional case. Although the Covid-19 pandemic has brought its own amendments and obstacles to the project, there are so many threads to follow now, in particular inspired by the recent research activities. Among them are the rest of the programme for hysteresis-type operators, verification of deviation majorants for parabolic problems, and an entirely new activity – the development of a method of quantitative analysis of the free boundary in obstacle-type problems.
Publications
- Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions. Computational Mathematics and Mathematical Physics, 16 (2020), no. 11, 1823–1838
Darya E. Apushkinskaya and Sergey I. Repin
(See online at https://doi.org/10.1134/S0965542520110032) - Functional a posteriori error estimates for parabolic obstacle problems, 2021
Darya E. Apushkinskaya and Sergey I. Repin
