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Asymptotic analysis of mean curvature flow equations via games
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| Title: | Asymptotic analysis of mean curvature flow equations via games |
| Other Titles: | ゲームを用いた平均曲率流方程式の漸近解析 |
| Authors: | 三栖, 邦康 Browse this author |
| Issue Date: | 25-Mar-2024 |
| Publisher: | Hokkaido University |
| Abstract: | In part I we consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, to which we extend that of Kohn and Serfaty [37]. The paper [37] gives a deterministic two-person zero-sum game whose value functions approximate the solution to the level set mean curvature flow equation without obstacle functions. We prove that moving curves governed by the mean curvature flow converge in time to the boundary of the convex hull of obstacles under some assumptions on the initial curves and obstacles. Convexity of the initial set, as well as smoothness of the initial curves and obstacles, are not needed. In these proofs, we utilize properties of the game trajectories given by very elementary game strategies and consider reachability of each player. Also, when the equation has a driving force term, we present several examples of the asymptotic behavior, including a problem dealt in [22]. In part II we study the initial value problem for a fully nonlinear degenerate parabolic equation with discontinuous source terms, to which a usual type of comparison principles do not apply. Examples include singular equations appearing in surface evolution problems such as the level set mean curvature flow equation with a driving force term and a discontinuous source term. By a suitable scaling, we establish weak comparison principles for a viscosity sub- and supersolution to the equation. We also present uniqueness and existence results of possibly discontinuous viscosity solutions. In part III we consider the asymptotic shape of solutions to the level set mean curvature flow equation with a negative driving force and a discontinuous source term. This is a model equation of crystal growth phenomenon called a two-dimensional nucleation. A typical source term in our mind is a characteristic function of a set Ω. It turns out that, if Ω satisfies some weak convexity condition, then the asymptotic shape of the solution is given by the unique solution of the corresponding stationary problem with the Dirichlet boundary condition. We also apply the game-theoretic interpretation established in [49]. By using the game, we construct a solution with non-trivial growth speed when Ω consists of two disks touching each other. We also give another non-uniqueness result by using the game, which is a counter-example to a weak comparison principle in [33] when the source term does not satisfy the assumption of the weak comparison principle. Each part I, II, and III of this doctoral thesis corresponds to the reference [49, 33, 31] respectively. Since all the parts are independent, there are some common definitions and similar arguments in them. Lastly we note the sign of the driving force term. Through the thesis, we consider both the competitive situation and the cooperative situation. Namely the mean curvature term and the driving force term are competitive or cooperative when considering a closed hypersurface ∂A of a bounded convex set A. We denote the driving force by ν ∈ R. The competitive situation corresponds to ν > 0 in part I and II and to ν < 0 in part III. |
| Conffering University: | 北海道大学 |
| Degree Report Number: | 甲第15736号 |
| Degree Level: | 博士 |
| Degree Discipline: | 理学 |
| Examination Committee Members: | (主査) 准教授 浜向 直, 特任教授 神保 秀一, 特任教授 栄 伸一郎 |
| Degree Affiliation: | 理学院（数学専攻） |
| Type: | theses (doctoral) |
| URI: | http://hdl.handle.net/2115/92228 |
| Appears in Collections: | 課程博士 (Doctorate by way of Advanced Course) > 理学院(Graduate School of Science)
学位論文 (Theses) > 博士 （理学）
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