论文标题
关于均匀椭圆形差异算子的HOPF引理的最大原理和定量版本
On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators
论文作者
论文摘要
In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets.通常，操作员的系数不必是连续的，而只能是界限和可测量的。我们的某些结果可能被认为是HOPF引理的“定量”版本，因为它们在域的边界上的最大值在订阅点的最大点上，就其点的值提供了下限。我们还将通过与$ a $和域相关的主要本征函数在其最大点附近的订阅上显示下限。其他结果，其中包括薄弱和强大的最大原则，弱的Harnack不平等也被证明。
In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution %on a boundary of a domain in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with $A$ and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.
