二阶复系数微分算子的$L^{p}$ -耗散性的判定
Criterion for the $L^{p}$-dissipativity of second order differential operators with complex coefficients
摘要：
我们证明了代数条件$|p-2| |< {\mathscr Im}{\mathscr A}ξ,ξ>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}ξ,ξ>$（对于任何$ξ\in\mathbb{R}^{n}$）是微分算子$\nabla^{t}({\mathscr A}\nabla)$的Dirichlet问题的$L^{p}$耗散性的充分必要条件，其中${\mathscr A}$是一个元素为复测度且虚部对称的矩阵.这一结果是新的，即使是光滑系数，当它意味着相应的半群的$L^{p}$ -压缩性的一个准则。我们还考虑了算子$\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$，其中系数是光滑的，并且${\mathscr Im}{\mathscr A}$可能是不对称的。证明了该算子的$L^{p}$ -拟耗散性的充分必要条件。相同的条件是相应半群的$L^{p}$ -拟压缩性的充分必要条件。给出了常系数算子$\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$在$\mathbb{R}^{n}$中$L^{p}$ -耗散的一个充分必要条件.
英文摘要：
We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}ξ,ξ>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}ξ,ξ>$ (for any $ξ\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $\nabla^{t}({\mathscr A}\nabla)$, where ${\mathscr A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^{p}$-contractivity of the corresponding semigroup. We consider also the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^{p}$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^{p}$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^{p}$-dissipativity in $\mathbb{R}^{n}$ of the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$ with constant coefficients.
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