Partial Differential Equations III

1. Front Matter

   Pages i-xxi

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2. Function Space and Operator Theory for Nonlinear Analysis

   • Michael E. Taylor

   Pages 1-88

3. Nonlinear Elliptic Equations

   • Michael E. Taylor

   Pages 89-270

4. Nonlinear Parabolic Equations

   • Michael E. Taylor

   Pages 271-358

5. Nonlinear Hyperbolic Equations

   • Michael E. Taylor

   Pages 359-465

6. Euler and Navier-Stokes Equations for Incompressible Fluids

   • Michael E. Taylor

   Pages 466-523

7. Einstein’s Equations

   • Michael E. Taylor

   Pages 524-605

8. Back Matter

   Pages 607-611

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Partial differential equations is a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of math­ ematics, such as differential geometry, complex analysis, and harmonic analysis, as weIl as a ubiquitous factor in the description and elucidatiön of problems in mathematical physics. This work is intended to provide a course of study of some of the major aspects ofPDE.1t is addressed to readers with a background in the basic introductory grad­ uate mathematics courses in American universities: elementary real and complex analysis, differential geometry, and measure theory. Chapter 1 provides background material on the theory of ordinary differential equations (ODE). This includes both very basic material-on topics such as the existence and uniqueness of solutions to ODE and explicit solutions to equations with constant coefficients and relations to linear algebra-and more sophisticated resuIts-on ftows generated by vector fields, connections with differential geom­ etry, the calculus of differential forms, stationary action principles in mechanics, and their relation to Hamiltonian systems. We discuss equations of relativistic motion as weIl as equations of classical Newtonian mechanics. There are also applications to topological resuIts, such as degree theory, the Brouwer fixed-point theorem, and the Jordan-Brouwer separation theorem. In this chapter we also treat scalar first-order PDE, via Hamilton-Jacobi theory.

• Gaussian curvature
• Minimal surface
• Sobolev space
• calculus
• curvature
• differential equation
• differential geometry
• function space
• hyperbolic equation
• mathematical physics
• minimum
• operator theory
• partial differential equation

• Department of Mathematics, University of North Carolina, Chapel Hill, USA

  Michael E. Taylor

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