MULTI-DIMENSIONAL MORSE INDEX THEOREMS AND A SYMPLECTIC VIEW OF ELLIPTIC BOUNDARY VALUE PROBLEMS
MULTI-DIMENSIONAL MORSE INDEX THEOREMS AND A SYMPLECTIC VIEW OF ELLIPTIC BOUNDARY VALUE PROBLEMS
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DOI：
10.1090/s0002-9947-2010-05129-3
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发表时间：
2011-03-01
影响因子：
1.3
通讯作者：
Jones, Christopher
中科院分区：
文献类型：
--
作者：
Deng, Jian;Jones, Christopher
在各种边界条件下证明了多维椭圆型边值问题的莫尔斯指数定理。这些定理适用于星形区域，并且基于测量在缩小边界上解集轨迹的“振荡”的新思想。通过在该边界上适当的Sobolev函数空间中建立马斯洛夫指数来测量振荡。狄利克雷边界条件和诺伊曼边界条件之间的根本区别是通过单调性暴露出来的，这种单调性只在前一种情况下成立。
Morse Index Theorems for elliptic boundary value problems in multi-dimensions are proved under various boundary conditions. The theorems work for star-shaped domains and are based on a new idea of measuring the "oscillation" of the trace of the set of solutions on a shrinking boundary. The oscillation is measured by formulating a Maslov index in an appropriate Sobolev space of functions on this boundary. A fundamental difference between the cases of Dirichlet and Neumann boundary conditions is exposed through a monotonicity that holds only in the former case.