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Régularité microlocale pour des problèmes de Dirichlet non linéaires non caractéristiques d’ordre deux à bord peu régulier. (Microlocal regularity of nonlinear Dirichlet problems non characteristic of order two with non-smooth boundary). (French) Zbl 0636.35056

A non-linear second order partial differential equation \(F(x,\partial^{\alpha}u)=0\) with \(n\) independent variables and with \(F\in C^{\infty}\) is considered. It is assumed that the principal symbol \(p\) of a linearized operator is of real principal type and that \(d_{\xi}p\neq 0\) at every diffraction point of \(\partial \Omega\). Let \(u\) be an \(H^ s\) solution, \(\alpha_ 0\not\in WF^{ s''}(\partial \Omega)\cup WF^{s'}(u|_{\partial \Omega})\) where \(s<s'<s''\leq 2s- 2-n/2\) and \(s'\leq s''-1\) if \(\alpha_ 0\) is either hyperbolic or diffraction point and \(s'<s''-1\) if \(\alpha_ 0\) is a nondiffraction gliding point. A standard assertion about propagation of \(WF^{s'}(u)\) is valid; however at every diffraction point there is a loss) such that \(f_{\lambda}(Tx)=\lambda f_{\lambda}(x)\) a.e. \(m\). For a measure preserving transformation on a probability space \(e(T)\) is a countable subgroup of the circle \(S^ 1\). However, M. Osikawa [Publ. Res. Inst. Math. Sci., Kyoto Univ. 13, 167-172 (1977; Zbl 0369.28016)] gave examples which do not preserve any finite measure equivalent with \(m\) and for which \(e(T)\) is an uncountable subgroup of \(S^ 1\). The purpose of this paper is to survey some of the known results (some due to the authors) and to prove some new ones about non-singular transformations and their \(L_{\infty}\) eigenvalues. For example a proof that \(e(T)\) is a Borel set in \(S^ 1\) with a unique Polish topology is given. It is known [J.-F. Melá, C. R. Acad. Sci., Paris, Sér. I 296, 419-422 (1983; Zbl 0547.28011)] that \(e(T)\) is a saturated weak Dirichlet set and it is shown that \(e(T)\) cannot be generated by an uncountable Kronecker set. An analogue of the Halmos-von Neumann isomorphism theorem is given for non-singular transformations.
The authors study variations of Osikawa’s examples; special flows built over an adding machine, giving new properties (for example \(\hat e(T)={\mathbb{Z}})\). A class of ergodic group rotations with \(\sigma\)-finite, infinite invariant measures which are prime in the measure preserving sense is constructed. The interaction between \(L_{\infty}\) eigenvalue groups and \(L_ 2\) spectra using systems of imprimitivity is developed, leading to a result on the extension of cocycles. The paper finishes with examples illustrating the results and some open questions.
Reviewer: G.R.Goodson

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
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