Strong positivity in $$C(\bar \Omega )$$ for elliptic systemsfor elliptic systems
Strong positivity in $$C(\bar \Omega )$$ for elliptic systemsfor elliptic systems
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DOI：
10.1007/bf02570833
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发表时间：
1992
影响因子：
0.8
通讯作者：
G. Sweers
中科院分区：
文献类型：
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作者：
G. Sweers
Protter 和 Weinberger 所著的关于极大值原理的书包含了本质上为正椭圆方程组的极大值原理。这些系统是wcakly耦合的，即：导数中没有耦合。最近这个问题已经被一些作者重新访问过，例如[21]和[28]。内格尔将半群理论用于算子矩阵，并发现了椭圆系统的正结果作为应用。 De Figueiredo 和 Mitidieri 使用最大原理进行一次计算。在本文中，我们将通过使用克雷因-鲁特曼定理的扩展来给出直接证明。底层空间将为 (C (O)) k。在我们的方法中，拥有具有连续系数的运算符就足够了。 wc usc 的三个条件可以描述为：(i)本质上为正耦合矩阵；(ii)完全耦合；(iii)存在正超解。我们将展示唯一的第一本征函数的存在。此外，我们还将研究三个基本条件的必要性。对于离本质正值不远的 somc 系统，将显示部分结果。（本质正值也称为合作性。）对于最后一个结果，我们需要对格林函数进行逐点估计。此类估计的最新结果列于附录中。将给出抛物线系统的含义。
The book on maximum principles by Protter and Weinberger contains a maximum principle for systems of essentially positive elliptic equations. These systems arc wcakly couplcd, that is: no coupling in the derivatives. Recently the problem has bccn rcvisitcd by several authors, eg [21] and [28]. Nagel uses semigroup theory for operator matrices and finds as an application a positivity result for thc clliptic system. De Figueiredo and Mitidieri use the maximum principle for onc cquation. In this note we will give a direct proof by using an extension of thc Krein-Rutman Theorem. The underlying space will be (C (O)) k. In our approach it is sufficient to have operators with continuous coefficients. The thrce conditions wc usc can be described by:(i) essentially positive coupling matrix;(ii) full coupling;(iii) existence of a positive supersolution. We will show thc cxistencc of a unique first eigenfunction. Furthermore we will investigate the necessity of the three basic conditions. A partial result will be shown for somc systems that are not far from essentially positive.(Essentially positive is also known as cooperative.) For the last result we need pointwise estimates for Green functions. Recent results for such estimates are listed in an appendix. Implications for the parabolic system will be given.