open access

https://doi.org/10.14943/doctoral.k15729

https://hdl.handle.net/2115/91872

非有界領域上の楕円型方程式に対する種々の最大値原理

英語

2024-03-25

Hokkaido University

iii, 22p.

In this dissertation, we report some topics related to maximum principle. We deal with fully nonlinear second order elliptic partial differential equations on unbounded domains in the n-dimensional Euclidean space. Our argument is based on the viscosity solution theory. In Chapter 1, we establish two Phragmén–Lindelöf theorems for viscosity subsolutions to fully nonlinear elliptic equations with a dynamical boundary condition. The first result is for an elliptic equation on an epigraph in Rn. Because we assume a good structural condition, which includes wide classes of elliptic equations as well as uniformly elliptic equations, we can benefit from the strong maximum principle. The second result is for an equation that is strictly elliptic in one direction. Because the strong maximum principle does not need to hold for such equations, we adopt a strategy often used to prove the weak maximum principle. Considering such equations on a slab, we can approximate the viscosity subsolution by functions that strictly satisfy the viscosity inequality and obtain a contradiction. In Chapter 2, we establish the Hadamard three sphere theorem for viscosity supersolutions to fully nonlinear uniformly elliptic equations with a superlinear growth in the gradient. The classical Hadamard property asserts that the circumferential minimum of the supersolution of an elliptic equation has some convexity. We prove this assertion by constructing a radially symmetric solution on the annulus. Moreover, we derive Liouville type theorem by applying the Hadamard theorem. In addition, we apply the argument to singular or degenerate elliptic cases, the ellipticity of which depends on the gradient. Chapter 1 is essentially based on a paper [1], which is reproduced with permission from Springer Nature. Chapter 2 is based on [2], which is a joint work with Hamamuki. All sections, formulas, theorems, etc. are cited only in the chapter where they appear. References [1] K. Abiko. Phragmén–lindelöf theorems for a weakly elliptic equation with a nonlinear dynamical boundary condition. Partial Differential Equations and Applications, 4(3):24, 2023. [2] K. Abiko and N. Hamamuki. Hadamard and Liouville type theorems for fully nonlinear uniformly elliptic equations with a superlinear growth in the gradient, in preparation.

2024-03-25

甲第15729号

博士(理学)

(主査) 准教授 浜向 直, 特任教授 神保 秀一, 特任教授 栄 伸一郎

理学院（数学専攻）

博士論文

Version of Record