The Newton Polygon and Elliptic Problems with Parameter
The Newton Polygon and Elliptic Problems with Parameter
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带参数的牛顿多边形和椭圆问题
DOI：
10.1002/mana.19981920108
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发表时间：
1998
影响因子：
1
通讯作者：
L. Volevich
中科院分区：
文献类型：
--
作者：
R. Denk;R. Mennicken;L. Volevich
在研究标量椭圆算子的解的过程中，例如，在无边界流形上，有一个众所周知的带参数的椭圆性条件Agmon - Agranovich - Vishik，它保证了解的最小增长射线的存在。本文研究了Douglis - Nirenberg意义上的椭圆型系统的相同问题。我们寻找符号上的代数条件，以提供复平面上包含一条射线的解集的存在性。我们用牛顿多面体法来解决这个问题。该方法的思想是同时研究系统的所有准齐次部分，通过赋予相应牛顿多边形定义的谱参数不同的权值来获得系统的所有准齐次部分。通过这种方法，得到了保证系统符号存在的几个等价的充要条件和解的尖锐估计。其中一个等价条件可以表示为：符号的左上次元都满足椭圆性条件。A. KOZHEVNIKOV [K2]引入了Douglis‐Nirenberg意义上的椭圆型系统的这一子类。
In the study of the resolvent of a scalar elliptic operator, say, on a manifold without boundary there is a well‐known Agmon‐Agranovich‐Vishik condition of ellipticity with parameter which guarantees the existence of a ray of minimal growth of the resolvent. The paper is devoted to the investigation of the same problem in the case of systems which are elliptic in the sense of Douglis‐Nirenberg. We look for algebraic conditions on the symbol providing the existence of the resolvent set containing a ray on the complex plane. We approach the problem using the Newton polyhedron method. The idea of the method is to study simultaneously all the quasihomogeneous parts of the system obtained by assigning to the spectral parameter various weights, defined by the corresponding Newton polygon. On this way several equivalent necessary and sufficient conditions on the symbol of the system guaranteeing the existence and sharp estimates for the resolvent are found. One of the equivalent conditions can be formulated in the following form: all the upper left minors of the symbol satisfy ellipticity conditions. This subclass of systems elliptic in the sense of Douglis‐Nirenberg was introduced by A. KOZHEVNIKOV [K2].