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Основание: книга 2

Union of Bulgarian Mathematicians

Mathematics and Its Applications, Volume 2

Geometrical Methods for
Solving of Fully Nonlinear
Partial Differential Equations
 

by P. POPIVANOV
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Sofia, Bulgaria
 

ISBN-10 954-8880-24-5
ISBN-13 978-954-8880-24-4
Sofia, 2006

This book deals with the method of characteristics and the envelope method for solving the Cauchy problem for first order fully nonlinear partial differential equations (PDE). Second order hyperbolic PDE of Monge-Ampere type and the Cauchy problem for them are considered too.

The solutions of these nonlinear PDE can be interpreted as smooth surfaces (integral surfaces) developing specific singularities. Local existence results based on geometrical ideas are well known (Cauchy, Lagrange-Charpit, Goursat, Darboux and many others). We give the corresponding proofs and moreover, in several cases we construct solutions with maximal domain of definition, multivalued solutions and find their smooth single valued branches.

An Appendix with some recent results on the generic singularities of ruled and developable surfaces is given at the end of the book. Possible applications are to the equations of Clairaut, eikonal and Monge-Ampere types.

The book contains different applications to geometry and mechanics, namely: geodesics, reconstruction of surfaces by their Gauss curvatures, canonical transformations, integration of the Hamilton-Jacobi systems, characteristics in the hodograph plane of two dimensional steady, isentropic, irrotational flow (the latter turning out to be epicycloids) and others. Anomalous singularities of the solutions of semilinear non-strictly hyperbolic systems with one or two space variables are studied too.

Audience:
The first four Chapters can be used by (graduate) students in Math., Physics, Engineerings as a manual on nonlinear PDE equipped with more than 50 exercises. The rest part of the book could be of interest to PhD students and researchers in the domain of Analysis and Geometry.
 

Prof. Petar Popivanov (born in 1946) is full member of the Bulgarian Academy of Sciences and works in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. His publications are mainly in the domain of Partial Differential Equations (PDE) and concern linear and nonlinear microlocal analysis (hypoellipticity, local solvability, propagation of singularities), solvability of linear PDE on compact manifolds, boundary value problems for elliptic PDE (tangential oblique derivative problem, classical and viscosity solutions in the nonlinear case). He has written about 110 papers and 2 monographs. P. Popivanpv has given lectures on  ODE and PDE in the Faculty of Mathematics and Informatics of the Sofia University "St. Kliment Ohridski" and in the South-West University  "Neofit Rilski".