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On the exact WKB analysis of microdifferential operators of WKB type

Takashi Aoki, Takahiro Kawai, Tatsuya Koike, Yoshitsugu Takei (2004)

Annales de l’institut Fourier

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

On the existence and the stability of solutions for higher-order semilinear Dirichlet problems

Marek Galewski, M. Płócienniczak (2007)

Czechoslovak Mathematical Journal

We investigate the existence and stability of solutions for higher-order two-point boundary value problems in case the differential operator is not necessarily positive definite, i.e. with superlinear nonlinearities. We write an abstract realization of the Dirichlet problem and provide abstract existence and stability results which are further applied to concrete problems.

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