Construction of the viscosity solution theory connecting continuous problems and discrete problems

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17621
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Hokkaido University
• Principal Investigator: Hamamuki Nao 北海道大学, 理学研究院, 准教授 (70749754)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 粘性解 / 等高面法 / 動的境界値問題 / 決定論的ゲーム / 比較定理 / 平均曲率流方程式 / ハミルトン・ヤコビ方程式

Outline of Final Research Achievements

The goal of this research is to get a deeper understanding of continuous problems and discrete problems for nonlinear partial differential equations, especially surface evolution equations, by means of construction of the viscosity solution theory connecting both the problems. Typically, I studied the following two topics:
The first topic is improvement of level set equations. I modified classical level set equations so that it enables us to compute the level sets effectively in a discrete level.
The second topic is establishment and discretization of the viscosity solution theory for dynamic boundary value problems. I proved unique existence of viscosity solutions and gave deterministic discrete game-theoretic interpretations. Furthermore, applying the discrete game, I studied asymptotic behavior and geometric properties of solutions.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

身の回りの現象の時間変化を予測することは、自然科学全体に横たわる課題であり、社会生活にも深く関わる。本研究で得られた成果は、特に界面の形状の予測と理解を可能にするものである。
数学理論を実用する際は、離散問題を設定し、計算機で近似解を求めることになる。この実装のための適切な離散化と効率的な計算法を、本研究で提案できた。様々な初期値・境界値問題の数学的基礎を確立したことは、粘性解理論そのものの進展でもある。