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Über dieses Buch

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)



Chapter 1. Introduction

In the introductory chapters to most plasma physics texts (e.g., [43,45]), an idealized model is presented in which the plasma ion and electron temperatures – rather than being millions of degrees – are set to absolute zero. This is done to reduce the mathematics to its simplest possible form, but even in this case a rigorous description of the plasma is problematic: the classical Dirichlet problem, which is the physically natural boundary value problem in this context, turns out to be ill-posed on typical domains.
Thomas H. Otway

Chapter 2. Mathematical Preliminaries

The purpose of this chapter is to emphasize some standard material in functional analysis and the theory of partial differential equations which will be particularly useful in subsequent chapters. In addition, a brief survey of applications is given in Sect. 2.7. Specialists in partial differential equations may prefer to skip Sects. 2.1– 2.6 of this chapter.
Thomas H. Otway

Chapter 3. The Equation of Cinquini-Cibrario

Although the study of mixed elliptic–hyperbolic equations goes back at least toRiemann’s computation of the Laplacian in toroidal coordinates (c.f. [47] or p. 461, (B) of [7]), the first systematic study of well-posedness for boundary value problemsappears to be the memoir by Tricomi [50]
Thomas H. Otway

Chapter 4. The Cold Plasma Model

Because a plasma is a fluid, its evolution must satisfy the equations of fluid dynamics. But because the fluid is composed of electrons and one or more species of ions, the charges on these particles act as sources of an electromagnetic field, which is governed by Maxwell’s equations. The presence of this intrinsic field leads to highly nonlinear behavior; and in fact, the dominance of long-range electromagnetic interactions over the short-range interatomic or intermolecular forces is often cited as the defining characteristic of the plasma state. In order to construct a mathematically rigorous model for the plasma which is also accessible to analysis, hypotheses must be imposed which control these nonlinearities. In Sect. 3.6 we assumed that the pressure on the plasma was zero and that magnetic forces dominated over other forces. Those hypotheses reduced the governing equations to the Beltrami equations (3.62), (3.65). In this section we impose a similar physical hypothesis: that the temperature of the plasma is zero.
Thomas H. Otway

Chapter 5. Light Near a Caustic

In the cold plasma model the sonic curve is a parabola. In the physical model presented in this chapter the sonic curve is a circle, and the elliptic region of the governing equation surrounds the hyperbolic region. Thus we can prescribe Dirichlet data on a suitable closed curve lying entirely in the elliptic region and obtain an elliptic–3hyperbolic boundary value problem. Eventually,we will construct such a problem and show that it possesses a weak solution. In the next chapter the sonic curve will also be a circle; but in that case the hyperbolic region of the governing equation will enclose the elliptic region, leading to a significant reduction in regularity for elliptic–hyperbolic Dirichlet problems.
Thomas H. Otway

Chapter 6. Projective Geometry

Projective geometry enters the study of partial differential equations of mixed elliptic–hyperbolic type by a very indirect route, beginning with the following geometric variational problem.
Thomas H. Otway


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