2014 Fiscal Year Research-status Report
Viscosity methods in homogenization of nonlinear PDEs
| Project/Area Number |
26800068
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| Research Institution | Kanazawa University |
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Principal Investigator |
POZAR Norbert 金沢大学, 数物科学系, 助教 (00646523)
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| Project Period (FY) |
2014-04-01 – 2018-03-31
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| Keywords | homogenization / viscosity solutions / free boundary problems / Hele-Shaw problem / random media |
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| Outline of Annual Research Achievements |
This is the first year of the project and therefore most of the results are not finished yet. However, the homogenization method for a special free boundary problem with periodic inhomogeneity, that is, the Hele-Shaw problem in a periodic medium that depends on both space and time, has been refined. The Hele-Shaw problem is used as a model of water flow through porous medium. The original homogenization result, which was obtained before the beginning of this project, has now been published.
Moreover, together with W. Feldman (UCLA), we have managed to obtain a homogenization result for a fully nonlinear elliptic problem posed on a bounded domain with a mixed periodic boundary data (Dirichlet and Neumann). This result is now being prepared for publication.
Finally, with I. Kim (UCLA), we started to work on the convergence of the porous medium equation to the Hele-Shaw problem in a model of tumor growth. As a first step, we have extended the notion of viscosity solutions to this type of Hele-Shaw problem, and proved well-posedness of solutions.
All of these results advance the theory of viscosity solutions and the viscosity methods in the homogenization theory.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
I obtained most of the partial results that I anticipated in the original proposal, as explained above. I visited my collaborators at UCLA. With W. Feldman, we are now preparing for publication a homogenization result for the elliptic problem with a mixed (Dirichlet and Neumann) boundary data. With I. Kim, we are continuing our attempts at homogenization of the G-equation with mean curvature. But due to the expected difficulty of this problem, we also started a parallel project on proving the convergence of the porous medium equation to the Hele-Shaw problem in a model of tumor growth. This result is now being prepared for publication.
However, I was forced to transfer some of the travel funds for this projects to the year 2015 because I was invited to two international meetings (2015 Banff Workshop "Developments in the theory of Homogenization", Canada, and ICIAM2015, Beijing, China). I also plan to participate "Oberwolfach Seminar: Stochastic Homogenization", Germany, September 2015.
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| Strategy for Future Research Activity |
The project will continue according to the original proposal. Part of the time will be devoted to finishing the papers mentioned above. To obtain new results, I will continue extending the method for the homogenization of the Hele-Shaw problem to be able to obtain convergence rates, and to investigate the possibility to treat random media.
Since the homogenization of the G-equation with mean curvature is a rather difficult problem, as expected, with I. Kim (UCLA) we will partially shift our focus to the model of tumor growth and study the asymptotic behavior of the porous medium equation when it converges to the Hele-Shaw problem. The viscosity approach, which takes advantage of the problem's maximum principle structure, is in many respects more powerful than the standard theory that relies on the integration by parts and energy methods. For example, the free boundary condition can be recovered more explicitly.
I plan at least one visit of my collaborators, as well as participation in international meetings to present the results of this project and exchange knowledge with other researchers in the field.
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| Causes of Carryover |
I originally anticipated that I would participate in at least one international meeting during the first year. However, in the end there was no suitable conference to attend. At the same time, I was invited to give lectures at two international conferences in 2015 (Banff workshop, Canada, and ICIAM2015, China). Furthermore, there is an intensive seminar "Oberwolfach Seminar: Stochastic Homogenization" in September 2015 that I would like to participate in. Finally, I will need to buy a new desktop computer during 2015. I therefore decided to transfer the unused funds towards the next fiscal year.
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| Expenditure Plan for Carryover Budget |
Participation in the following meetings: Developments in the Theory of Homogenization, Banff, Canada, July 26-31, 2015; ICIAM2015, Beijing, China, August 10-14, 2015; Oberwolfach Seminar: Stochastic Homogenization, Oberwolfach, Germany, September 9-16, 2015
I also plan at least one visit of my collaborators (UCLA, USA).
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Research Products
(6 results)
