Albeverio, S., Demuth, M., Schrohe, E., Schulze, B.-W. (Eds.)
2003, VII, 440 p.
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This book is the seventh volume of "Advances in Partial Differential Equations" , a series originating from the work of the research group "Partial Differential Equa tions and Complex Analysis" at the University of Potsdam. The present volume focuses on recent developments in nonlinear and hyper bolic equations. In the first contribution, P. Popivanov of Sofia studies the singu larities of solutions of several classes of nonlinear partial differential equations and systems. He begins with a survey of the known theory on propagation and inter action of singularities and then presents his own results which have applications to the Monge-Ampere equation, to quasi-linear systems arising in fluid mechanics as well as to integro-differential equations for mechanics of media with memory. There follows an article by F. Hirosawa (Tsukuba) and M. Reissig (Freiberg) on Lp - Lq decay estimates for Klein-Gordon equations with time-dependent coef ficients. They explain, in particular, the influence of the relation between the mass term and the wave propagation speed on the estimates. The third paper is by M. Dreher (Freiberg). He investigates quasi-linear weakly hyperbolic equations. His main topics are the local existence of solutions in Sobolev spaces and Coo, blow-up criteria, domains of dependence, and C= regularity. Spectral theory of semi bounded selfadjoint operators is the topic of the con tribution by A. Noll (Darmstadt). He proves upper and lower bounds for the bottom eigenvalue as well as an upper bound for the second eigenvalue in terms of capacitary estimates.
Nonlinear PDE. Singularities, Propagation, Applications.- From Wave to Klein—Gordon Type Decay Rates.- Local Solutions to Quasi-linear Weakly Hyperbolic Differential Equations.- An Approach to a Version of the S(M, g)-pseudo-differential Calculus on Manifolds.- Spectral Invariance and Submultiplicativity for the Algebras of S(M, g)-pseudo-differential Operators on Manifolds.- Domain Perturbations and Capacity in General Hilbert Spaces and Applications to Spectral Theory.- An Interpolation Family between Gabor and Wavelet Transformations. Application to Differential Calculus and Construction of Anisotropic Banach Spaces.- Formes de torsion analytique et fibrations singulières.- Regularisation of Secondary Characteristic Classes and Unusual Index Formulas for Operator-valued Symbols.