Parametrix and propagation of singularities for the interior mixed hyperbolic problem. (English) Zbl 0375.35037

Eskin, Gregory

J. Anal. Math. 32, 17-62 (1977).

L’auteur considére la construction d’une parametrix et la propagation de singularités pour le problème intèrieur mixte hyperbolique: \[ A(x, D) u=0 \text { pour } x \in G, \quad u=g\left(x^{\prime}\right) \] \( \left.u\right|_{\Gamma}=0 \) pour \( x_{0}<0, x \in G=\left(-\infty, x_{0}\right) \times \Omega, x_{0} \) fixé où \( x=\left(x_{0}, x^{\prime}\right) \in \mathbb{R} \times \bar{\Omega} \), \( \Omega \) I’interieur d’un domaine borné dans \( \mathbb{R}^{n} \) avec la frontière \( \gamma \) plate, \( D=\left(i \partial / \partial x_{0}, i \partial / \partial x_{1}, \ldots, i \partial / \partial x_{n}\right) \) et \( g\left(x^{\prime}\right) \in D^{\prime}, g\left(x^{\prime}\right)=0 \) pour \( x_{0}<0 \) et \( A(x, D) \) un opérateur strictement hyperbolique d’ordre 2. On suppose de plus que si on pose \( \Gamma=\left(-\infty, x_{0}\right) \times \gamma \) alors \( \bar{\Gamma} \) est strictement convexe par rapport à toute bicaractéristique nulle de l’opérateur \( A(x, D) \) qui est tangente à \( \bar{\Gamma} \). L’auteur remarque que des résultats similaires mais obtenus par une autre voie sont celles de K. G. A \( n d e s s \circ n \) et R. Me I \( r \) o s e [Sémin.

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MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
35B99	Qualitative properties of solutions to partial differential equations

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