2021 Fiscal Year Research-status Report
Development of Viscosity and Variational Techniques for the Analysis of Moving Interfaces
| Project/Area Number |
18K13440
|
|---|
| Research Institution | Kanazawa University |
|---|
Principal Investigator |
POZAR Norbert 金沢大学, 数物科学系, 准教授 (00646523)
|
|---|
| Project Period (FY) |
2018-04-01 – 2023-03-31
|
| Keywords | mean curvature / viscosity solutions / crystalline curvature |
|---|
| Outline of Annual Research Achievements |
In this and a previous Kakenhi grant we focused on the theory of viscosity solutions for the crystalline mean curvature flow. This is an important mathematical model of the evolution of crystals, with their facets and sharp edges and vertices. Over the years we have a published a number of papers developing this theory, which is now somewhat complete. There has also been a separate approach to the theory of the solutions of this problem pursued by another research group. We have therefore wrote a review paper that summarizes the obtained results on this topic in the recent years, unifying the notation and streamlining the definitions and proofs and giving a number of new examples with the hope that this will make the results more accessible to other researchers. I also gave an intensive course on this topic at Kyoto University.
A paper on the application of viscosity solutions for the crystalline mean curvature to the problem with a volume constraint, modelling the important situation when an evolving crystal keeps the same volume or its volume is controlled, has been published. Since this is a very challenging problem even in the non-crystalline case, our idea is to focus on evolving crystals satisfying a certain reflection property that resembles starshapedness. Under this condition, we have been able to show the existence of solutions.
|
| Current Status of Research Progress |
Current Status of Research Progress
4: Progress in research has been delayed.
Reason
The COVID-19 pandemic travel restrictions continued throughout the year which prevented me from visiting my collaborators and conferences and delayed the start of new researched topics. We have been using online communication, however, it has showed its limitations in achieving planned research goals.
|
| Strategy for Future Research Activity |
As planned in the final year of this project, we will pursue the notion of viscosity solutions to other problems in the original proposal, with focus on non-purely crystalline anisotropies and the inverse mean curvature flow.
We have started the work on the study of boundary conditions for the mean curvature flow problem and the study of a contact angle dynamics problem that admits facets in the flow. We hope that the COVID-19 restrictions will be lifted and allow me to travel to international conferences and visit my collaborators.
|
| Causes of Carryover |
Due to COVID-19 travel restrictions, my planned visits of my collaborators and attendance of international conferences have been cancelled both in 2020 and 2021. I hope to make these trips once the travel restrictions are lifted.
|
