二阶复系数微分算子泛函耗散性的判别准则
Criterion for the functional dissipativity of second order differential operators with complex coefficients
摘要：
本文考虑二阶微分算子$E=\nabla(A \nabla)$的Dirichlet问题，其中$A$是具有复值$L^\infty$元素的矩阵.我们引入了$E$关于给定函数$φ:R^+ \to R^+$的耗散性的概念。在$Im\, A$是对称的假设下，证明了$E$的函数耗散性满足条件$|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 $（对几乎所有$x\inΩ\subset R^N$和任何$s>0$，$ξ\in R^N$）的充要条件.
英文摘要：
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 $E$.
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