论文标题
具有复杂系数的二阶差分运算符的功能耗散性标准
Criterion for the functional dissipativity of second order differential operators with complex coefficients
论文作者
论文摘要
在本文中，我们考虑了第二阶差异操作员$ e = \ nabla（a \ nabla）$的Dirichlet问题，其中$ a $是具有复杂值的$ l^\ infty $条目的矩阵。我们介绍了$ e $相对于给定函数$φ的耗散性的概念：r^+ \ to r^+ $。在假设$ im \，a $对称的假设下，我们证明条件$ | s \，φ'（s）| \，| \ langle im \，a（x）\，ξ，ξ\ rangle | \ leq 2 \，\ sqrt {φ（s）\，[s \，φ（s）]'} \，\ langle re \ \ langle re \，a（x） $ s> 0 $，$之一，在r^n $中）是必需的，对于$ e $的功能消散性是必要的。
In the present paper we consider the Dirichlet problem for the second order differential operator $E=\nabla(A \nabla)$,where $A$ is a matrix with complex valued $L^\infty$ entries. We introduce the concept of dissipativity of $E$ with respect to a given function $φ:R^+ \to R^+$. Under the assumption that the $Im\, A$ is symmetric, we prove that the condition $|s\, φ'(s)| \, | \langle Im\, A (x)\, ξ,ξ\rangle |\leq 2\, \sqrt{φ(s)\, [s\, φ(s)]'}\, \langle Re\, A(x) \, ξ,ξ\rangle $ (for almost every $x\inΩ\subset R^N$ and for any $s>0$, $ξ\in R^N$) is necessary and sufficient for the functional dissipativity of $E$.
