权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Research in viscosity solutions using the method of Functional Analysis.

使用泛函分析方法研究粘度解决方案。

基本信息

批准号：
10640169
负责人：
MARUO Kenji
金额：
$ 0.96万
依托单位：
Kobe University of Mercantile Marine
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640169/
关键词：
Viscosity Solution Degenerate Elliptic Equation Existence Theorem Uniqueness Theorem Semilinear Quasilinear Radial Solution 退化楕円型偏微分方程式退化楕円型半線型方程式 radial solution standard solution unbounded solution 退化型二階常微分方程式

项目摘要

We consider the Dirichelet problem for a semilinear degenerate elliptic equation (DP) :-g(|x|)Δu+f(|x|, u(x)) = 0, and Boundary Conditionwhere N【greater than or equal】2 and g(|x|), f(|x|, u) are continuous. We discuss the problem (DP) under the following assumption : 1)g is nonnegative. 2)f is strictly monotone for u. We first define a standard viscosity solution by the viscosity solution such that if g(|x|) = 0 then f(|x|, u(x)) = 0. Then we can prove that the any continuous standard viscosity solution is the radial solution and it is unique. We add an assumption : 3)∫ィイD1a-0ィエD1gィイD1-1ィエD1(s)ds = ∞ or ∫ィイD2a+0ィエD2gィイD1-1ィエD1(s)ds = ∞ for any a : g(a) = 0. Then We obtain that any continuous viscosity solution is the radial solution and it is unique. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not unique. Here, the domain is a bounded boall in n-dimension space.We next state the existence and uniqueness of the continuous unbounded viscosity solution in RィイD12ィエD1. We use the order of the infinite neighborhood of the solution as the boundary condition. We know that the existence or nonexistence of the solution are dependent on a kind of the order of the solution. Moreover, we get the results which the uniqueness or non-uniqueness are also dependent on a kind of the order of the solution. In case, we assume that g, f is sufficiently smooth.We now show the existence of a continuous viscosity solution to quasi-semilinear degenerate elliptic problem. Here, g(|x|, u), f(|x|, u) are continuous and f is strictly monotone for u. Moreover, we assume there exists an implicite function of f = 0 and the implicite function holds some smoothness. Then we can prove the existence of the continuous viscosity solution. But it is difficult to prove the uniqueness of the solution.

本文考虑一类半线性退化椭圆型方程（DP）的Dirichelet问题：-g（|X|）Δu+f（|X|，u（x））= 0和边界条件，其中N[大于或等于]2和g（|X|），f（|X|，u）是连续的。我们在以下假设下讨论问题（DP）：1）g是非负的。2）f对u是严格单调的。我们首先通过粘度溶液定义标准粘度溶液，使得如果g（|X|）= 0，则f（|X|，u（x））= 0。从而证明了任意连续的标准粘度解是径向解，并且是唯一的。我们增加一个假设：3）对于任意a：g（a）= 0，则等式D 1a-0等式D 1g等式D 1 -1等式D 1（s）ds = ∞或等式D 2a +0等式D 2g等式D 1 -1等式D 1（s）ds = ∞。得到了任意连续粘性解都是径向解，且是唯一的。如果不满足假设3），则存在连续粘性解不唯一的例子。这里，区域是n维空间中的一个有界球，我们接着讨论了R_∞ D_（12）_∞ D_（11）中连续无界粘性解的存在唯一性。我们使用解的无穷邻域的阶作为边界条件。我们知道解的存在或不存在依赖于解的某种阶。此外，我们还得到了解的唯一性或非唯一性还依赖于解的某种阶的结果。在情形下，我们假设g，f是充分光滑的，我们证明了拟半线性退化椭圆型方程连续粘性解的存在性。这里，g（|X|，u），f（|X|，u）是连续的，并且f对于u是严格单调的。此外，我们假设存在一个f = 0的隐函数，并且该隐函数具有一定的光滑性。从而证明了连续粘性解的存在性。但很难证明解的唯一性。