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

On symmetric and radial viscosity solutions for elliptic partial differential equation.

椭圆偏微分方程的对称和径向粘度解。

基本信息

批准号：
09640187
负责人：
MARUO Kenji
金额：
$ 1.47万
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：
1997
资助国家：
日本
起止时间：
1997 至 2000
项目状态：
已结题

项目摘要

We consider the Dirichelet problem for a semilinear degenerte elliptic equation (DP) : -g(|x|)Δu+f(|x|, u(x))=0, and Boundary Condition where N【greater than or equal】2 and g (|x|), f(|x|, u) are continuous and the domain is a bounded ball in N-dimensional space. We discuss the problem (DP) under the following assumptions : 1)g is nonnegative. 2) f is strictly monotone for u. We frist define a standard viscosity solution by the viscosity solution such that f (|x|, u(x))=0 if g(|x|)=0. Then we can prove that the any continuous standard viscosity solution is the radial solution and unique. We add an assumption : 3)∫^<a-0>g^<-1> (s) ds=∞ or ∫_<a+0> g^<-1> (s) ds=∞ for any a : g (a)=0. Then We obtain that any continuous viscosity solution is the radial solution and uniqne. If the assumption 3) is not satisfied there exist examples such that the continuous viscosity solutions are not uniqne.We next state the existence and uniqueness of the continuous unbounded viscosity solution in R^N. We u … More se the order of the infinite neiborhood of the solution as the boundary condition. We know that the existence or nonexistece 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 this case, we assume that g, f is sufficiently smooth.We now show the existence and uniquness of the continuous viscosity solution to quasi-semilinear degenrate 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 smootheness. Then we can prove the existence of the continuous viscosity solution. We next state the uniquenss of the continuous viscosity solution. Assume that g (|x|, u) and f (|x|, u) hold the some relations such that f (|x|, u)/g (|x|, u) is monotone for u. Then we have the uniquness theorem and get the result this viscosity solution is the radial solution. Less

本文考虑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，并且域是N维空间中的有界球。我们在下列假设下讨论问题（DP）：1）g是非负的。2)f对u是严格单调的我们首先定义一个标准粘度解，使得f（|X|，u（x））=0，如果g（|X|）=0。证明了任意连续的标准粘度解是径向解，且是唯一的。我们增加一个假设：3）对于任意a：g（a）=0，<$^<a-0>g^<-1>（s）ds=∞或<$_<a+0> g^<-1>（s）ds=∞。从而证明了任何连续粘性解都是径向解，且是唯一的。如果假设3）不满足，则存在连续粘性解不唯一的例子，接着我们讨论了R^N中连续无界粘性解的存在唯一性。我们使用 ...更多信息以解的无穷邻域的阶作为边界条件。我们知道解的存在性或不存在性依赖于解的某种阶。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。在这种情况下，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问题连续粘性解的存在唯一性。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。我们接着说明连续粘性解的唯一性。假设g（|X|，则f（|X|，u）保持一些关系，使得f（|X|，u）/g（|X|，u）对于u是单调的。然后利用唯一性定理得到了该粘性解是径向解的结论。少