Nonlinear First-Order Partial Differential Equations by the Characteristic Method applied to The Hamilton-Jacobi Equation

This master's thesis has two main goals. First, to give a rigorous presentation of the method of characteristics without losing the important intuitive aspects that accompany it. Second, to use the method of characteristics to present the Hamilton-Jacobi equation, solve it in a reasonable manner and then introduce some of its most important properties. These topics will be accompanied by enlightening examples. In the first part we develop the method of characteristics by transforming a first-order nonlinear PDE to a system of first-order ODE:s. We start by defining the notion of complete integrals as a sort of singleton solution to our PDE, and then combining these to a single family; an envelope encompassing the full solution. We then assume that the PDE itself can be written in a moving coordinate frame, and using this concept make an important assumption regarding the nature of the underlying curves at each moment. The full solution is finally achieved by weaving these curves to form the solution surface. To complete the theory, proper care needs to be taken of the boundary to make it compatible with our notion of curves. Lastly, all of the theory will be combined to make sure that the method actually produces well-defined local solutions. With the method of characteristics developed, we have a look at the Hamilton-Jacobi equation. We will give sufficient conditions on when this initial-value problem can be solved with the Hopf-Lax formula. Based on this formula, a notion of weak solution will be given with its uniqueness proof. The Hamilton-Jacobi initial-value problem will be approached with the tools of variational calculus and convex analysis. These tools will be used to intimately link the Hamiltonian and Lagrangian by the means of the Legendre transform. The Hopf-Lax formula will then be constructed with the aim of solving the Hamilton-Jacobi initial-value problem. The formula is shown to have a useful functional identity as well as being Lipschitz continuous. Finally the uniqueness of the solution will be achieved by assuming semiconcavity from the initial function, or uniform convexity from the Hamiltonian. The final chapter gives an insight as to how the developed theory can be further generalized and used. We will refer to some bibliography containing an abundance of further reading on semiconcave functions, optimal control theory and the Hamilton-Jacobi-Bellman equations