Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence
Uniqueness of viscosity solutions of fully nonlinear second order parabolic equations with discontinuous time-dependence
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具有不连续时间依赖性的全非线性二阶抛物型方程粘度解的唯一性
DOI：
10.57262/die/1371586186
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发表时间：
1990
影响因子：
1.4
通讯作者：
D. Nunziante
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文献类型：
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作者：
D. Nunziante
其中 n 是 IRN 的开界子集，0 < T < oo 和函数 F : (t,x,u,p,X) E [O,T] X n X IR X IRN X MN--+ F(t,x,u,p,X) E IR 已给出。这里 MN 表示 N x N 对称矩阵的空间，Du = gradxu = (ux,, · · · , UxN )，D 2u 是 u 相对于 x = (x 1 , · · · ,xN)· 的 Hessian 矩阵 众所周知，粘性解的概念是由 Crandall 和 Lions [3] 引入 Hamilton-Jacobi 方程（另请参见 Crandall、Evans 和Liops [1] 提供了 [3] 中证明的一些等效公式和简化，然后由 Lions [14] 扩展到二阶情况；就此类解决方案的存在性和唯一性而言，我们还参考了[13]、[8]、[9]以及其中的参考文献。由于粘度解的定义涉及 F 的逐点值，因此 [1]、[3] 和 [14] 中考虑的函数 F 在所有参数中都应该是连续的。最近，H. Ishii在[7]中解决了(1.1)解的唯一性和存在性问题，其中F不依赖于D 2u，在(0, T)中可积，并且在其余参数中连续。为此，他通过观察以某种局部方式与 F 相关的一类连续函数的逐点行为，将粘度解的定义扩展到非连续 F 的情况。请注意，在 [15) 中提出了一些源自主动算子理论的等效定义。在这里，我们将粘度解的新定义扩展到二阶方程的情况（1.1）。我们做出以下假设
where n is an open-bounded subset of IRN, 0 < T < oo and a function F : (t,x,u,p,X) E [O,T] X n X IR X IRN X MN--+ F(t,x,u,p,X) E IRis given. Here MN denotes the space of N x N symmetric matrices, Du = gradxu = (ux,, · · · , UxN ), D 2u is the Hessian matrix of u with respect to x = (x 1 , · · · ,xN)· As it is well known, the notion of viscosity solution was introduced by Crandall and Lions [3] for the Hamilton-Jacobi equations (see also Crandall, Evans and Liops [1] for some equivalent formulations and simplifications of proofs in [3]) and th'en extended by Lions [14] to the second-order case; we also refer to [13], [8], [9] and the references therein as far as the existence and uniqueness of such solutions is concerned. Since the definition of viscosity solution involves pointwise values of F, the functions F considered in [1], [3] and [14] are supposed to be continuous in all arguments. Recently, H. Ishii in [7] has solved the problem of the uniqueness and the existence of solutions of (1.1) where F does not depend on D 2u, is integrable in (0, T) and continuous in the remaining arguments. To do so, he extended the definition of viscosity solution to the case of noncontinuous F, by looking at the pointwise behavior of a class of continuous functions related in some local way to F. Notice that in [15) are presented some equivalent definitions originating from the theory of acretive operators. Here we extend this new definition of viscosity solution to the case ( 1.1) of the second-order equations. We make the following assumptions
Ishii,H.：J.Geod.Soc.Japan，35，1989。
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