Wave equations in a complex domain
type: Item
instance of: scholarly article
zbMATH Open document ID: 1364.35062
author: Keisuke Uchikoshi
publication date: 2016-06-21T00:00:00Z
full work available at URL: jms230202.html
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35A20
Mathematics Subject Classification ID: 35A21
zbMATH DE Number: 6595969
zbMATH Keywords: complex Cauchy problem
zbMATH Keywords: light cone
MaRDI profile type: Publication
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title: Wave equations in a complex domain
published in: Journal of Mathematical Sciences. University of Tokyo
review text: In this paper the author considers the Cauchy problem for 2-dimensional wave operator in \(C^3\) as follows NEWLINE\[NEWLINE(\partial^2_0- \partial^2_1- \partial^2_2) u(x)+ P_1(x_0) u(x)+ P_0(x_0) u(x)= 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0, x_1,x_2)= u_0(x_1,x_2),\;\partial_0 u(0,x_1,x_2)= u_1(x_1,x_2)\tag{2}NEWLINE\]NEWLINE Then he proves the following theorem: Let \(Y= Y_0\cup Y_{1,+}\cup Y_{1,-}\cup Y_{2,+}\cup Y_{2,-}\), where NEWLINE\[NEWLINEY_0= \{x\in C^3; -x^2_0+ x^2_1+ x^2_2= 0\}NEWLINE\]NEWLINE is the light cone issuing from the origin, and NEWLINE\[NEWLINEY_{j,\pm}= \{x\in C^3; x_j\pm x_0= 0\},\;j= 1,2NEWLINE\]NEWLINE is a characteristic hypersurface issuing from \(\{(x_1,x_2)\in C^2; x_j\neq 0\}\) and propagating backwards or forwards. Assume that \(x^{n_0}_1 x^{n_0}_2 u_j(x_1,x_2)\), \(j= 0,1\) are holomorphic near the origin for some positive integer \(n_0\). Then the solution of (1) and (2) is holomorphic on the universal covering space \({\mathcal R}(\omega\setminus Y)\) of \(\omega\setminus Y\), where \(\omega\) is a small neighborhood of the origin.
reviewed by: Kunihiko Kajitani
description: scientific article; zbMATH DE number 6595969