Viscosity methods in homogenization of nonlinear PDEs

2017 Fiscal Year Final Research Report

• Project/Area Number: 26800068
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Kanazawa University
• Principal Investigator: Pozar Norbert 金沢大学, 数物科学系, 助教 (00646523)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: homogenization / crystal growth / viscosity solutions / Hele-Shaw problem / phase transitions / porous medium equation

Outline of Final Research Achievements

Many problems in applied sciences, for example the growth of tumors or crystals, are described by nonlinear differential equations with a moving interface. We worked on the analysis of such problems using the notion of viscosity solutions that rely on a order-preserving property of solutions (maximum principle) in these problems. We showed how a small-scale variations in the properties of ice influence the speed of melting using the homogenization approach. We also clarified the relation between sharp interface and diffusive interface models of tumor growth and population dynamics, including situations with a drift field. Finally, we introduced a new notion of viscosity solutions for a model of crystal growth (the crystalline mean curvature flow) in an arbitrary dimension and proved its existence, uniqueness and stability. This opens the possibility for further rigorous study of this model.

Free Research Field

Mathematical analysis, Applied analysis