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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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##### References:
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