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Partial differential equations of mixed elliptic-hyperbolic type arise in diverse areas of physics and geometry, including fluid and plasma dynamics, optics, cosmology, traffic engineering, projective geometry, geometric variational theory, and the theory of isometric embeddings. And yet even the linear theory of these equations is at a very early stage. This text examines various Dirichlet problems which can be formulated for equations of Keldysh type, one of the two main classes of linear elliptic-hyperbolic equations. Open boundary conditions (in which data are prescribed on only part of the boundary) and closed boundary conditions (in which data are prescribed on the entire boundary) are both considered. Emphasis is on the formulation of boundary conditions for which solutions can be shown to exist in an appropriate functions space. Specific applications to plasma physics, optics, and analysis on projective spaces are discussed. (From the preface)
List of contents
1 Introduction.- 2 Mathematical Preliminaries.- 3 The Equation of Cinquini-Cibrario.- 4 The Cold Plasma Model.- 5 Light near a Caustic.- 6 Projective Geometry.
Summary
Partial differential equations of mixed elliptic-hyperbolic type arise in diverse areas of physics and geometry, including fluid and plasma dynamics, optics, cosmology, traffic engineering, projective geometry, geometric variational theory, and the theory of isometric embeddings. And yet even the linear theory of these equations is at a very early stage. This text examines various Dirichlet problems which can be formulated for equations of Keldysh type, one of the two main classes of linear elliptic-hyperbolic equations. Open boundary conditions (in which data are prescribed on only part of the boundary) and closed boundary conditions (in which data are prescribed on the entire boundary) are both considered. Emphasis is on the formulation of boundary conditions for which solutions can be shown to exist in an appropriate function space. Specific applications to plasma physics, optics, and analysis on projective spaces are discussed. (From the preface)
Additional text
From the book reviews:
“The book is based on Otway’s lectures at Henan University in 2010. These notes are very well written and very useful for readers, particularly advanced researchers and Ph.D. students. Each chapter has its own list of references, its own dynamics, and its own interesting remarks.” (Henryk Leszczyński, Mathematical Reviews, May, 2014)
“The lecture notes were written by the author to supplement a series of ten lectures given at Henan University in the summer of 2010. They are intended for graduate students and researchers in pure or applied analysis with a background in functional analysis -- including Sobolev spaces -- and the basic theory of partial differential equations. The book contains 6 chapters (every one with its references), summary, addenda, directions for future research and index.” (Elena Gavrilova, Zentralblatt MATH, Vol. 1246, 2012)
Report
From the book reviews:
"The book is based on Otway's lectures at Henan University in 2010. These notes are very well written and very useful for readers, particularly advanced researchers and Ph.D. students. Each chapter has its own list of references, its own dynamics, and its own interesting remarks." (Henryk Leszczynski, Mathematical Reviews, May, 2014)
"The lecture notes were written by the author to supplement a series of ten lectures given at Henan University in the summer of 2010. They are intended for graduate students and researchers in pure or applied analysis with a background in functional analysis -- including Sobolev spaces -- and the basic theory of partial differential equations. The book contains 6 chapters (every one with its references), summary, addenda, directions for future research and index." (Elena Gavrilova, Zentralblatt MATH, Vol. 1246, 2012)
