|
Showing 1 - 2 of
2 matches in All Departments
This text is a concise introduction to the partial differential
equations which change from elliptic to hyperbolic type across a
smooth hypersurface of their domain. These are becoming
increasingly important in diverse sub-fields of both applied
mathematics and engineering, for example: * The heating of fusion
plasmas by electromagnetic waves * The behaviour of light near a
caustic * Extremal surfaces in the space of special relativity *
The formation of rapids; transonic and multiphase fluid flow * The
dynamics of certain models for elastic structures * The shape of
industrial surfaces such as windshields and airfoils * Pathologies
of traffic flow * Harmonic fields in extended projective space They
also arise in models for the early universe, for cosmic
acceleration, and for possible violation of causality in the
interiors of certain compact stars. Within the past 25 years, they
have become central to the isometric embedding of Riemannian
manifolds and the prescription of Gauss curvature for surfaces:
topics in pure mathematics which themselves have important
applications. Elliptic Hyperbolic Partial Differential Equations is
derived from a mini-course given at the ICMS Workshop on
Differential Geometry and Continuum Mechanics held in Edinburgh,
Scotland in June 2013. The focus on geometry in that meeting is
reflected in these notes, along with the focus on quasilinear
equations. In the spirit of the ICMS workshop, this course is
addressed both to applied mathematicians and to
mathematically-oriented engineers. The emphasis is on very recent
applications and methods, the majority of which have not previously
appeared in book form.
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)
|
|
Email address subscribed successfully.
A activation email has been sent to you.
Please click the link in that email to activate your subscription.