二阶微分方程组的$L^{p}$ -耗散性的判定
Criteria for the $L^{p}$-dissipativity of systems of second order differential equations
摘要：
本文给出了一类具有$\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$形式的偏微分算子系统的Dirichlet问题的$L^{p}$耗散性的完整代数刻画，其中${\mathscr A}^{hk}(x)$是$m\times m$矩阵.首先，我们确定了一般复系数标量算子的耗散锐角。其次证明了二维弹性算子是$L^{p}$ -耗散的当且仅当$$ ({1\over 2}-{1\over p})^{2} \leq {2(ν-1)(2ν-1)\over (3-4ν)^{2}}, $$ $ν$是泊松比。最后，我们得到了算子$\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$的$L^{p}$ -耗散性的一个充分必要代数条件，其中${\mathscr A}^{h}(x)$是具有复$L^{1}_{\rm loc}$元素的$m\times m$矩阵，并描述了该算子的$L^{p}$ -耗散性的最大角.
英文摘要：
We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr A}^{hk}(x)$ are $m\times m$ matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is $L^{p}$-dissipative if and only if $$ ({1\over 2}-{1\over p})^{2} \leq {2(ν-1)(2ν-1)\over (3-4ν)^{2}}, $$ $ν$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$, where ${\mathscr A}^{h}(x)$ are $m\times m$ matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^{p}$-dissipativity for this operator.
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