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Propagation de l’analyticité des solutions d’équations non- linéaires de type principal. (French) Zbl 0566.35017
Let u be a \(C^{\infty}\) solution in \(\Omega\) for the nonlinear equation \(F(y,u(y),...,u^{(\alpha)}(y),...)=0,\) where F is an analytic function of the variables \((y,u^{(\alpha)}\), \(| \alpha | \leq m)\). Denote by P the linearization of F at the solution u and let \(\phi\) be some \(C^ 2\) function on \(\Omega\) with real values such that \(d\phi\) \(\neq 0\) on \(\Omega\). Further consider \(y^ 0\) with \(\phi (y^ 0)=0\), assume that P is of real principal type, that \(\{\phi =0\}\) is noncharacteristic for P and that all real characteristics for P are transverse to \(\{\phi =0\}\) at \(y^ 0\). The main result of the article is then the following: If u is analytic for \(\phi <0\), then it is also analytic in a full neighborhood of \(y^ 0\). The result extends a previous result of the same authors for nonlinear hyperbolic systems [Invent. Math. 75, 189-204 (1984; Zbl 0545.35063)].
Reviewer: O.Liess

MSC:
35F25 Initial value problems for nonlinear first-order PDEs
35A20 Analyticity in context of PDEs
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