Recent Progress on Some Problems in Several Complex Variables and Partial Differential Equations

2006 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Recent Progress ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Partial Differential

scholarly journals Decomposition theorems for the generalized metaharmonic equation in several independent variables

1972 ◽  
Vol 13 (1) ◽  
pp. 35-46
Author(s):  
David Colton
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Radiation Condition ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Entire Space ◽  
Partial Differential ◽  
Independent Variables ◽  
Image Position ◽  
Decomposition Theorems

In this paper solutions of the generalized metaharmonic equation in several independent variables where λ > 0 are uniquely decomposed into the sum of a solution regular in the entire space and one satisfying a generalized Sommerfeld radiation condition. Due to the singular nature of the partial differential equation under investigation it is shown that the radiation condition in general must hold uniformly in a domain lying in the space of several complex variables. This result indicates that function theoretic methods are not only the correct and natural avenue of approach in the study of singular ordinary differential equations, but are basic in the investigation of singular partial differential equations as well.

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