| 研究課題/領域番号 |
26800068
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| 研究機関 | 金沢大学 |
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研究代表者 |
POZAR Norbert 金沢大学, 数物科学系, 助教 (00646523)
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| 研究期間 (年度) |
2014-04-01 – 2018-03-31
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| キーワード | homogenization / viscosity solutions / free boundary problems / Hele-Shaw problem / random media |
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| 研究実績の概要 |
With I. Kim (UCLA), we have submitted a paper on the convergence of the porous medium type equation to the Hele-Shaw type equation in the stiff pressure limit. This has an application to models of tumor growth. In these models, the tumor spreads as more cancer cells are born outside of the tumor in the so called pre-cancer zone. In the porous medium equation type model, the cancer cells can diffuse, which influences the spreading speed. On the other hand, in the purely mechanical model, a Hele-Shaw type equation, the cells only move to satisfy a constraint of a maximal cancer cell density. The boundary between the tumor and the surrounding pre-cancer zone is the free boundary in this problem. We have used viscosity solution techniques, in part developed in this project, to rigorously justify the relationship of these two models and we were able to identify the speed of tumor expansion in the purely mechanical model.
With Y. Giga (U. of Tokyo), we have submitted a paper where we develop a new notion of viscosity solutions for a model of a crystal growth in three dimensions. We establish a unique existence of viscosity solutions and a comparison principle. This is a major step that will allow us to apply the viscosity solution techniques considered in this project to such important problems. The paper has been accepted for publication.
I have participated in a few international conferences, presenting results related to this project on homogenization and free boundary problems.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
As explained above, with my collaborators we have made significant contributions to the theory of viscosity solutions for free boundary problems and surface evolution.
As anticipated last year, the homogenization of the G-equation with mean curvature seems to be currently too difficult to handle by our techniques and therefore with I. Kim we have shifted focus to the Hele-Shaw type problem and its homogenization. Throughout last year, and especially during my visit of UCLA, we have refined our geometric approach with viscosity techniques to this problem. We are currently working on the homogenization of similar free boundary problems with applications to motion of droplets on a surface and other related models. Our main goal is to treat nonmonotone free boundary motion.
The paper with W. Feldman (U. Chicago) on the homogenization of elliptic problems with mixed boundary condition has been delayed due to the length of the preprint and minor technical difficulties, however this does not delay the plan of the project.
During the visits to various conferences, I have been able to interact with other researchers in the field and keep up to date with the most recent developments and techniques.
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| 今後の研究の推進方策 |
The project will continue according to the original proposal. In particular, with I.Kim we will focus on extending the homogenization result to free boundary problems with nonmonotone free boundary motion (and more general problems if possible). We have already roughly checked the key steps in proving the homogenization result. In particular, the Lipschitz regularity of surfaces on large scales seems to extend to nonmonotone motions. We are hoping that we can finish most of the work during the first half of the coming fiscal year, with a high chance that we will be able to submit it for publication.
We will also further investigate the possibility of applying our techniques to the homogenization of the Hele-Shaw problem in random environments. In particular, we want to use the concentration inequality-type results of the probability theory.
Provided that there is sufficient time, I will consider the asymptotic homogenization of the Stefan problem for large times. This model has various applications to phase transitions and material science and is approachable by the techniques developed in this project.
I plan to attend at least one major international conference (AIMS Orlando 2016 Conference, Orlando, USA). I also plan to visit my collaborators at least once.
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| 次年度使用額が生じた理由 |
I originally planned to buy a new desktop computer for my research. However, it turned out that a better use of the funds for this project is to invite a visitor (Dohyun Kwon, KIAS/UCLA) to Kanazawa University and to attend a lecture series by X. Cabre (ICREA/UPC, Barcelona) at Tohoku University, March 28-30, 2016. The final total cost was lower than the anticipated cost of a new computer.
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| 次年度使用額の使用計画 |
I plan to use the additional money for an extra participation in a conference, or to invite a collaborator for a visit, depending on the progress of the project.
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