The Clairaut equation: A study in the geometry of partial differential equations.

摘要

The primary focus of this dissertation is the family of first order PDEs which we collectively call the Clairaut equation. Any Clairaut equation has a solution called the characteristic solution, which presents itself naturally as a subset of the zero set of the characteristic vector field for the equation. The culmination of our efforts is Theorem 6.1 where we use an abstract notion of perpendicularity to construct generalized solutions to the Clairaut equation starting from any submanifold of its characteristic solution.;The concept of contact makes it possible to formulate a first order PDE as an equation on the jet space. The latter is the natural setting for the characteristic vector field and the contact form. Chapter 2 develops the necessary ideas related to jets of functions, contact, and in addition, includes the concept of a submanifold of a vector space.;In Chapter 3, certain geometric aspects of first order PDEs are discussed. In particular, we show how the characteristic vector field of a PDE appears naturally in the 1-jets, and we produce the integral curves for the characteristic vector field of the Clairaut equation.;Chapter 4 develops the characteristic solution for Clairaut. The concept of a generalized solution is introduced and is reconciled with the notion of a "classical solution" by way of the contact condition. Also in Chapter 4 is a brief detour from the main theme of the work to look at the ;A formidable amount of underpinning is required to support a precise exposition of the main ideas of the dissertation. A mature understanding of the derivative is indispensible. Chapter 1 provides this foundation.;In Chapter 5, we give a thorough treatment of the second fundamental form of a submanifold of an inner product space. We discuss additional properties that this 2-tensor has under special circumstances.;Early in Chapter 6 we define what we mean by a Classical Embedded Clairaut Equation after which the main theorem of the dissertation is stated and proved.