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Problème de Cauchy ramifié à caractéristiques multiples holomorphes de multiplicité variable. (English) Zbl 0534.35040

Equations aux dérivées partielles hyperboliques et holomorphes, Sémin. Paris, Année 1981-82, Exp. No. 3, 38 p (1982).
[For the entire collection see Zbl 0497.00008.]
The A. considers the following non-characteristic Cauchy problem \[ (1)\quad a(x,D)\quad u(x)=0,\quad D^ h_ 0u(x)|_{x^ 0=0}=w_ h(x'),\quad 0\leq h<m \] where \(a=\sum_{| \alpha |<m}a_{\alpha}(x)D^{\alpha}\), \(D^{\alpha}=D_ 0^{\alpha_ 0}\times...\times D^{\alpha_ n}\), \(x=(x^ 0,x')\), \(x'=(x^ 1,...,x^ n)\). \(a_{\alpha}(x)\) are holomorphic functions and \(w_ h(x')\) have poles along \(x^ 0=x^ 1=0\). The main result of the paper is the following representation of the solution of the problem (1). \[ u(x)=\sum_{j\in J}\sum_{K\in M}\int_{S_ K}u^ K_ j(\phi_ j(\sigma,x')d\sigma^ K \] where J is a finite set, \(\phi_ j(\sigma,x')\), \(u^ K_ j(t,\sigma,x')\) are holomorphic functions of \(x'\in {\mathbb{C}}^ n,\quad \sigma =(\tau)^{\ell}\!_{\ell \in M^{\pm}}\in \Sigma =\oplus^{\infty}_{\ell =1}{\mathbb{C}}_{\tau^{\ell}}\) and \(S_ K\) is a singular simplex of dimension K in \(\Sigma\).
Reviewer: R.Salvi

MSC:

35J45 Systems of elliptic equations, general (MSC2000)
35C15 Integral representations of solutions to PDEs

Citations:

Zbl 0497.00008