Study of viscosity solutions for partial differential equations with subdifferential

次微分偏微分方程粘度解的研究

• 批准号: 13640200
• 负责人: TANAKA Naoto
• 金额: $ 1.92万
• 依托单位: Fukuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640200/
• 关键词: viscosity solution; subdifferential

The aim of present research is to extend the notion of viscosity solutions to multi-valued nonlinear partial differential equations (PDE's) with subdifferential and to investigate the existence, uniqueness and other properties of solutions. The theory of viscosity solutions has been applied to wide class of frilly nonlinear PDE's. Among them, the obstacle problem is an important class of applications of viscosity solutions. We have put emphasis of the research on following three points:1. We formulate various PDE's by subdifferential and apply the theory of viscosity solutions for it.2. We compare the formulation by viscosity solution with known ones and study an advantage for it.3. We prove solvability of various nonlinear PDE's and discuss the relation between the methods by viscosity solution.It is shown that the notion of viscosity solutions are extended for nonlinear second order PDE's with subdifferential whose typical example is obstacle problem, and unique, existence theorem and stability of the solution are proved. Moreover, we investigate convergence of Yoshida approximation for subdiffrential term.

本研究的目的是将粘性解的概念推广到具有次微分的多值非线性偏微分方程（PDE），并研究解的存在性、唯一性和其他性质。粘性解理论已被广泛地应用于一类褶边非线性偏微分方程。其中，障碍问题是粘性解的一类重要应用。本文的研究重点有以下三点：1.我们用次微分的方法构造了各种偏微分方程，并应用粘性解理论求解.我们将粘性溶液配方与已知配方进行了比较，并研究了其优势。3.利用粘性解证明了各种非线性偏微分方程的可解性，讨论了粘性解与各种方法之间的关系，将粘性解的概念推广到二阶次微分非线性偏微分方程，并以障碍问题为例，证明了解的唯一性、存在性定理和稳定性。此外，我们还研究了次微分项的Yoshida逼近的收敛性。

项目成果

石井克幸, 中村武: "An error estimate to Bence-Merriman-Osher algorithm for motion by mean curvature"神戸商船大学紀要、第2類、商船・理工学編. 51. 105-115 (2003)

Katsuyuki Ishii、Takeshi Nakamura：“通过平均曲率对 Bence-Merriman-Osher 运动算法进行误差估计”神户商船大学通报，第 2 类，商船科学与工程版 51. 105-115 (2003)。

草野 尚, J.Jaros, 吉田範夫: "Picone-type inequakities for nonlinear elliptic equations and their applications"J. Inequal. Appl.. 6. 387-404 (2001)

Takashi Kusano、J.Jaros、Norio Yoshida：“非线性椭圆方程的 Picone 型不等式及其应用”J. Inequal。6. 387-404 (2001)

田中尚人, 谷温之: "Surface waves for a compressible viscous fluid"Journal of Mathematical Fluid Mechanics. 5. 303-363 (2003)

Naoto Tanaka、Atsuyuki Tani：“可压缩粘性流体的表面波”数学流体力学杂志 5. 303-363 (2003)。

丸尾健二, 富田義人: "Remarks on viscosity solutions of the Dirichlet problem for quasilinear dgenerate elliptic equations"Funkcialaj Ekvacioj. 45. 89-101 (2002)

Kenji Maruo、Yoshito Tomita：“关于拟线性 d 生成椭圆方程的狄利克雷问题的粘度解的评论”Funkcialaj Ekvacioj 45. 89-101 (2002)

J.Jaros, 草野尚, 吉田範人: "Picone-type inequalities for nonlinear elliptic equations and their applications"Journal of Inequalities and Applications. 6. 387-404 (2001)

J.Jaros、Hisashi Kusano、Norito Yoshida：“非线性椭圆方程的 Picone 型不等式及其应用”《不等式与应用杂志》6. 387-404 (2001)。