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On the notion of elementary solution in the complex theory of differential equations. (English. Russian original) Zbl 0789.35005
Sov. Math., Dokl. 44, No. 2, 567-571 (1992); translation from Dokl. Akad. Nauk SSSR 320, No. 4, 825-829 (1991).
The principal object of consideration is the Cauchy problem \[ (-1)^ m {{\partial^ m u}\over {\partial t^ m}} -H \biggl( x,t,-{\partial\over {\partial x}},-{\partial\over {\partial t}}\biggr)u=0,\tag{1} \]
\[ u|_{t=0}= u_ 0, \qquad \partial u/\partial t|_{t=0}= u_ 1,\dots, \partial^{m-1} u/\partial t^{m-1}|_{t=0}= u_{m-1}, \] where \(H(x,t,-\partial/\partial x, -\partial/\partial t)\) is a differential operator of order \(m\) with coefficients that are analytic in \(\mathbb{C}_{x,t}^{n+1}\) and which does not contain the highest derivative with respect to \(t\).
Let \(X\subset\mathbb{C}_ x^ n\) be an analytic set of complex codimension 1 and let \({\mathcal H}\subset \mathbb{C}_{x,t}^{n+1}\) be the characteristic conoid of the set \(X\) in \(\mathbb{C}_{x,t}^{n+1}\) relative to the principal symbol of the operator in problem (1). The authors study questions of the solvability of (1) in spaces \(u\in {\mathcal A}_ q({\mathcal H})\) assuming that the Cauchy data \(u_ 0,u_ 1,\dots, u_{m-1}\) lie in the spaces \({\mathcal A}_ q(X),{\mathcal A}_{q-1}(X),\dots,{\mathcal A}_{q- m-1}(X)\) respectively, \(q>m-1\).
They introduce the concept of an elementary solution of the Cauchy problem, which plays the same role in the complex theory of partial differential equations as the fundamental solution does in the real case. This concept is based on the integral transformation of complex analytic functions which the authors introduced in [Math. USSR, Izv. 29, 407-427 (1987); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 50, 1054-1076 (1986; Zbl 0634.44002)].
MSC:
35A08 Fundamental solutions to PDEs
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
32G99 Deformations of analytic structures
35C20 Asymptotic expansions of solutions to PDEs
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
35A20 Analyticity in context of PDEs
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