Viscosity solution theory for quasilinear PDEs, free boundary problems and thier applications

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13436
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tottori University of Environmental Studies (2020-2022); Fukuoka Institute of Technology (2018-2019)
• Principal Investigator: Kosugi Takahiro 公立鳥取環境大学, 人間形成教育センター, 講師 (80816215)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: 粘性解 / 完全非線形方程式 / 準線形方程式 / 正則性

Outline of Final Research Achievements

Viscosity solution is a notion of weak solutions for second-order elliptic and parabolic equations and is one of the most studied fields, especially when considering (stochastic) optimal control and equations arising from differential games.
The obstacle problem is an important equation in applications that appears in the optimal stopping problem, and since it cannot be handled as is in numerical calculations, an approximate equation is often constructed via penalization. The rate of convergence of the solution to the viscosity solution of the original obstacle problem is investigated.
Sufficient conditions for the existence of global-in-time solutions of a weakly coupled system of fully nonlinear Fujita equations are also considered. The Fujita equation is one of the most studied in the field of parabolic equations, and we extend it to a fully nonlinear system.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

粘性解理論は準線形方程式に対する正則性など発散型方程式の弱解に比べて未知な部分が少なくないため，準線形方程式を粘性解で扱うことで粘性解理論の可能性を広げる価値がある．
数値計算を行う際は一旦離散化の必要があるが，収束の速さがわかることで離散化のサイズを決定できる可能性がある．
藤田型方程式は濃度により反応速度が変わる化学反応などを表すとされているが，その完全非線形化は分枝過程と関連づく可能性がある．