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Propagation de la régularité microlocale pour des problèmes de Dirichlet non linéaires d’ordre deux. (Propagation of microlocal regularity for nonlinear Dirichlet problems of the second order). (French) Zbl 0726.35026

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 20, 8 p. (1989).
Let \(\Omega\) be an open subset of \({\mathbb{R}}^ n\) with \(C^{\infty}\)- boundary function \(\phi\geq 0\) and let \(u\in H^ s_{loc}(\Omega)\) (satisfying \(u|_{\partial \Omega}\in C^{\infty}\) and \(s>5+n/2)\) be a real solution of the equation \(F(y,u,\nabla u,\nabla^ 2u)=0\) where F is a real \(C^{\infty}\)-function. Furthermore, let \[ L(y,D_ y)=\sum_{| \alpha | \leq 2}(\partial_{z_{\alpha}}F)(y,\partial^{\beta}u(y))\partial^{\alpha } \] and assume that \(\partial \Omega\) is noncharacteristic for L. Let \(\Gamma\) denote a generalized bicharacteristic of its principal symbol \(L_ 0\). Under suitable conditions, as main result of the paper, there is stated that all u, being microlocally of class \(H^{s'}\) \((s'<2s-n/2- 4-1/2)\) at some point of \(\Gamma\), are microlocally of class \(H^{s'- \delta}\) \((\delta >0\) arbitrary) for all points of \(\Gamma\). This is proved by utilizing results of J.-M. Bony, L. Hörmander, E. Leichtnam, and M. Sablé-Tougeron.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35G20 Nonlinear higher-order PDEs
35S99 Pseudodifferential operators and other generalizations of partial differential operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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