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.

我们考虑半线性简并椭圆方程 (DP) 的 Dirichelet 问题：-g(|x|)Δu+f(|x|, u(x)) = 0，且边界条件 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。我们首先定义 u 是严格单调的。由粘度解得到标准粘度解，如果 g(|x|) = 0 则 f(|x|, u(x)) = 0。那么我们可以证明任意连续的标准粘度解都是径向解，并且我们添加一个假设： 3)∫D1a-0D1gD1-1D1(s)ds = ∞ 或∫D2a+0D2gD1-1D1(s)ds = ∞ 对于任何 a : g(a) = 0。则我们得到任何连续粘度解都是径向解并且它是唯一的如果不满足假设 3) 则存在示例使得连续粘度解不唯一。这里，域是n维空间中的有界boall。接下来我们陈述存在性。 RiiD12D1 中连续无界粘度解的唯一性我们使用解的无限邻域阶数作为边界条件我们知道解的存在或不存在取决于解的一种阶数。，我们得到的结果的唯一性或非唯一性也取决于解的一种阶数。在这种情况下，我们假设g，f足够光滑。现在我们证明连续的存在性。拟半线性简并椭圆问题的粘性解这里，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 f 的隐函数。 = 0 且隐式函数具有一定的光滑性，则可以证明连续粘度解的存在性，但很难证明解的唯一性。